1 Introduction
The relativistic Vlasov–Maxwell system describes a collisionless plasma whose particles move under their self-consistent electromagnetic field. For momentum v\in\R^3, write
The particle density is f=f(t,x,v)\ge0; the electric and magnetic fields are E=E(t,x) and B=B(t,x). We use the one-species normalization
where
There is no background charge.
A smooth admissible datum is a triple (f_0,E_0,B_0) satisfying
Here C_b^\infty means that every spatial derivative is bounded. A classical solution on [0,T) is a C^1 triple solving (1.1) pointwise, with E,B\in C([0,T);L^2_x), such that the support of f in (x,v) is contained in a compact set on every compact time subinterval.
Theorem 1.1. Every smooth admissible datum (1.2) generates a unique global classical solution of (1.1). For every finite T, this solution satisfies
and f has compact phase-space support on [0,T]. The two divergence constraints hold for all time.
Theorem 1.1 resolves the large-data global classical regularity problem for this one-species model in three spatial and three momentum dimensions. The data have no size or symmetry restriction. The field assumptions allow a nonzero total charge and its Coulomb tail; they do not require square-integrability of all field derivatives. The momentum support bound is on each finite time interval and may grow with that interval.
1.1 Historical context
The distinction between weak existence and classical regularity has shaped the Cauchy problem for Vlasov–Maxwell. Wollman established local existence and uniqueness for classical solutions [29]. DiPerna and Lions developed global weak existence for large data [5]; a direct formulation for the relativistic one-species system is given by Rein [22]. These weak existence results do not furnish the regularity and uniqueness asserted in Theorem 1.1.
For classical solutions, Glassey and Strauss identified bounded momentum support as a continuation condition [11] and proved global existence for initially dilute plasmas [12]. Global classical existence was also established by Glassey and Schaeffer in dimensional reductions: one and one-half dimensions, two and one-half dimensions, and the planar problem [7, 6, 9, 10]. For the spherically symmetric electrostatic reduction, Glassey and Schaeffer proved global classical existence for the repulsive relativistic Vlasov–Poisson system [8]. Rein proved persistence of global existence under small perturbations of suitably decaying global reference solutions, including nearly symmetric configurations [21]. These results use restrictions on size, dimension, or the reference configuration.
The continuation theory also led to new methods. Klainerman and Staffilani gave a Fourier-based proof of the bounded-momentum criterion [13], while Bouchut, Golse and Pallard gave a physical-space proof using a division lemma for wave kernels [4]. Luk and Strain showed that it suffices to control the momentum support after projection onto a fixed two-dimensional plane [15, Theorem 1.2]. Moment and spatial-density conditions for continuation were developed by Pallard [18, 19], Sospedra–Alfonso and Illner [24], and Luk and Strain [16]. We use the Glassey–Strauss criterion as stated in Luk and Strain [15, Theorem 1.1]; Section 10.5 verifies its hypotheses for the normalization and initial field class considered here.
Small-data theory also developed beyond compact momentum support. Schaeffer allowed a rapidly decaying momentum tail, retaining compact spatial support and a suitable smallness condition [23]. Using a vector-field method with modified commutators, Bigorgne obtained global existence and sharp asymptotic estimates under weighted smallness assumptions [1]. His result permits nonzero charge and noncompact particle data. Wang proved a small-data result without compact support using vector-field and Fourier methods [25]. Wei and Yang allow arbitrarily large Maxwell fields with suitable spatial decay while requiring a sufficiently small weighted particle density [28]. Bigorgne subsequently proved global existence and modified scattering with a small distribution function and a large Maxwell field [3]. These theorems preserve a particle-smallness hypothesis. By prescribing asymptotic data and choosing a sufficiently late starting time, Bigorgne also constructs particular global solutions with large particle and field norms [2]. This construction does not prescribe arbitrary Cauchy data.
Wang’s July 2026 two-paper treatment establishes global existence for large cylindrically symmetric data [26, 27]. The first paper develops an iterative smoothing construction and the second supplies pointwise estimates. Their Theorem 1.1 assumes H^s fields, s\ge6, a weighted H^s particle density, and covariance of the data under rotations about a fixed axis. It allows localized noncompact particle data and has no smallness requirement. The theorem here instead concerns arbitrary compatible compactly supported particle data, without symmetry, and fields in the class (1.2). Thus the two data classes should not be identified.
The formulas used below have several classical antecedents. The retarded-field representation goes back to Glassey and Strauss [11]. Luk and Strain organize its most singular terms using electromagnetic components controlled by energy flux through light cones [15, Proposition 2.2 and Section 3]; Pallard also uses the orthogonality of the acceleration kernel to the cone’s spatial normal [19, Lemma 2.1 and Section 2.3]. The change from retarded phase points to initial particle labels appears in Lazarovici [14, Lemma 9.2]. The moving-characteristic cone change of variables is used by Pallard [18]; its proof is also given by Luk and Strain [15, Proposition 5.2]. We derive the required formulas in the normalization (1.1).
1.2 Proof strategy
The central estimate concerns the signed change of a particle’s momentum. Along a characteristic (X(t),V_X(t)), with X'=u(V_X) and V_X'=E(t,X)+u(V_X)\times B(t,X), it is the impulse
We take the norm after integration. Section 2 states the estimate precisely and shows that it places a lower bound on the time needed to double the largest particle momentum. These lower bounds have a divergent sum, which rules out unbounded momentum support in finite time and allows the continuation criterion to apply.
To estimate the impulse, we follow source particles to their intersections with the backward light cone of the particle receiving the force. We divide the source contribution into ranges of momentum, distance, and angles between the source velocity, receiver velocity, and light ray. Energy flux through the cone and bounds on the time that source particles spend in these ranges give the direct force estimates of Sections 4 and 5. At receiver energy comparable to w, the absolute-force bound has an extra factor \sqrt w compared with the signed estimate we need. The loss occurs when the source velocity is much closer to the light-ray direction than the receiver velocity is.
In this narrow angular region, we integrate the retarded force along a source trajectory before taking absolute values. The identity in Section 6 combines the part without source acceleration with the part containing it. The former cancels a term generated by differentiating the cone geometry. The force is thereby written as a total derivative plus less singular bulk terms, one of which contains the receiver’s acceleration. Expressing that acceleration through V_X' turns this into an error proportional to the receiver force. The proof must control its coefficient as well as the other integration-by-parts errors.
We use the signed identity twice, under an assumed bound on all signed momentum increments. First, after projection perpendicular to a particle’s momentum, it bounds the number of direction changes of a prescribed angular size. This step uses only the preliminary occupation and direct force estimates. When both source and receiver momenta are large, direction control gives a stronger occupation bound at intermediate relative angles: between changes of direction, relative motion makes a monotone crossing of the region contributing to a fixed range. Section 7 proves the direction bound before using it to improve occupation.
For the second use, Section 8 selects only those ranges whose direct force estimates exceed a summable threshold. The other ranges already satisfy the required bound. On the selected ranges, the coefficient multiplying the receiver force is at most a constant times w^{-1/2} after summation. This exactly compensates for the \sqrt w loss in the available absolute-force bound. The occupation improvement need only hold in the stated angular window: the selection inequalities force any possible obstruction to this coefficient bound into that window, where improved occupation rules it out. This yields an impulse estimate with a time weight that vanishes at the interval endpoints. Section 9 restores the short endpoint intervals using the assumed signed bound, obtains a strict improvement, and closes the bootstrap by continuity.
1.3 Organization
Section 2 states the signed estimate and deduces Theorem 1.1 from it and a continuation result. Sections 3–5 derive the retarded force, occupation estimates and direct bounds. Section 6 proves the signed cancellation; Section 7 bounds direction changes and improves occupation. Section 8 estimates the selected signed force, and Section 9 closes the bootstrap. Section 10 proves local existence, uniqueness, smooth persistence and the field modification needed to apply the bounded-momentum continuation theorem to (1.2).
2 The signed momentum estimate and its consequence
Our analytic goal is an estimate on momentum increments along characteristics. We state it first and explain why its logarithmic time scale excludes finite-time breakdown. Its proof occupies the central sections of the paper.
Fix a finite horizon T_0>0 and a compact interval [0,t_*]\subset[0,T_{\max}), with t_*\le T_0, of a classical solution. A characteristic with initial label z=(y_{\rm in},v_{\rm in}) solves
It is supported if z\in\supp f_0. A distinguished supported characteristic, along which we estimate the force, is written (X(t),V_X(t)) and called the receiver. We write
The signed increment is the vector impulse
Source characteristics continue to use the letters Y,V,u.
For nonempty \supp f_0, choose a dyadic number P such that
A dyadic number is a power of two. Receiver scales are w=2^k, k\ge0. In every nonempty range w/8\le q_X\le8w one has w\le P.
Proposition 2.1 (Signed momentum increments). For each smooth admissible datum and finite T_0, there are constants M,A\ge1 and P_0 with the following property. On every compact classical solution interval [0,t_*] as above, for every dyadic P\ge P_0 satisfying (2.1), every supported receiver, every w=2^k, k\ge0, and every interval [t_1,t_2]\subset[0,t_*] on which w/8\le q_X(t)\le8w throughout, one has
The constants are independent of t_* and P.
Proposition 2.2 (Local theory and continuation). Every smooth admissible datum has a unique maximal classical solution. It is smooth on each compact subinterval of its lifespan. If its maximal time T_{\max} is finite and
then the solution has a classical extension beyond T_{\max}.
The proof is given in the final section. It combines the Glassey–Strauss bounded-momentum criterion in the H^5 formulation quoted by Luk and Strain [15, Theorem 1.1 and Footnote 1, arXiv version] with a finite-horizon modification of the initial fields. The modification retains the initial Coulomb field and cuts off vector potentials for the divergence-free remainders. The removed part evolves as a vacuum Maxwell field that vanishes throughout the particle region for the chosen horizon. This supplies Sobolev field data for the continuation theorem while preserving the original particle dynamics.
Proof of Theorem 1.1 from Propositions 2.1 and 2.2. If f_0=0, the vacuum Maxwell solution is global and smooth, as shown in the final section. Suppose \supp f_0\ne\varnothing and, toward a contradiction, T_{\max}<\infty. Proposition 2.2 then forces unbounded momentum support. Set T_0=T_{\max} and
Continuity of the characteristic flow and compactness of the label set make Q continuous. For all sufficiently large integers n, let t_n<T_{\max} be the first time when Q(t_n)=2^n. Choose a label attaining this value. Along that label let s_n be its last time before t_n at energy 2^{n-1}. Such a time exists because the initial energy is smaller; moreover s_n\ge t_{n-1}. On [s_n,t_n] the energy lies in [2^{n-1},2^n].
Apply Proposition 2.1 on the prefix [0,t_n] with P=1024\cdot2^n and w=2^n. Since q is 1-Lipschitz, writing \Delta_n=t_n-s_n gives
At least one term on the right is at least 2^{n-2}. Consequently
for a fixed c>0. The sum of these lower bounds diverges, contradicting t_n<T_{\max}. Thus T_{\max}=\infty. Uniqueness, smoothness and propagation of the constraints follow from Proposition 2.2 and its proof.
To prove Proposition 2.1, we assume (2.2) for all such receivers, scales and intervals in [0,\tau], where \tau\le t_*, and derive a uniform strict improvement. Until that improvement is proved, (2.2) is a bootstrap assumption. In the estimates, C and the constants implicit in \lesssim or \asymp may depend on the datum, T_0, and fixed decomposition parameters, but not on t_*,P,M,A. Additional dependence is indicated explicitly. Lower thresholds on P may depend on M,A. The final choices are made in the order: a time-ramp fraction, then M, then A, then the threshold on P.
For nonnegative estimates we allow integration over any measurable subset of receiver times in an interval J\subset[0,\tau] of length at most I, provided those times satisfy w/8\le q_X\le8w and
The number I is an upper bound on length in these estimates. For the final signed estimate the interval has length exactly I and the receiver stays in the indicated range throughout. This distinction permits summation over disjoint time pieces without introducing a new constant for each piece.
Figure 1 records the conditional dependencies. We now prove Proposition 2.1. Energy and timelike geometry give preliminary bounds on absolute force. Those bounds lose a factor at large receiver momentum; the subsequent signed cancellation removes this loss.
Figure 1: Main dependencies in the conditional proof of the signed momentum estimate. The dashed box is a bootstrap assumption. The first use of signed cancellation controls particle direction from this assumption and the baseline estimates; improved occupation is obtained afterward. The second use yields a strict improvement, and a continuity argument establishes the initially assumed bound.
3 Energy and the retarded force
The estimates below use the conserved energy and two different changes of variables on a backward light cone. We first derive the force kernels without derivatives of the density. This representation belongs to the Glassey–Strauss approach [11]; the organization by cone-controlled field components also appears in [15, Proposition 2.2 and Section 3]. We give the source-label calculation explicitly, including its initial boundary term.
Throughout this section, fix a classical solution on a compact interval [0,t_*], with t_*\leq T_0<\infty, in the class specified in the main theorem. All limiting operations are justified on this fixed compact interval. Constants in the final bounds depend only on the initial data and T_0; constants used solely to remove a cutoff may also depend on the compact-interval momentum support. Retain the characteristic notation fixed in Section 2.
Lemma 3.1 (Transport and cone energy). The characteristic flow preserves phase volume and transports f_0. In particular 0\leq f\leq\|f_0\|_\infty. Define
Then
For every vertex (t,x) in this interval, put r=t-s, y=x-rn, n\in S^2, and
Then
Proof. The phase vector field has zero divergence: u is independent of x, E is independent of v, and \nabla_v\cdot(u\times B)=B\cdot(\nabla_v\times\nabla_v q)=0. Liouville’s formula gives volume preservation, and the Vlasov equation gives f(s,Y,V)=f_0(z). The flow exists backwards from every finite phase point on this compact time interval. Indeed, its spatial path stays in a bounded ball because |u|<1; the fields are bounded on the resulting compact spacetime set, so its momentum cannot escape in finite time.
Multiplication of Vlasov by q and integration in v, followed by Maxwell’s energy identity, give
There is no momentum boundary term because f has compact phase-space support on the interval. Integrate against spatial cutoffs equal to one on a ball of radius R, with gradient bounded by C/R. The error tends to zero: the fields are bounded in C([0,t_*];L^2) and the particle energy has compact support and is continuous in time. Passing to the limit proves conservation.
The outward spatial normal of the ball |y-x|<t-s is -n. Its energy derivative is therefore minus the boundary integral of \mathcal U-n\cdot\mathcal S. At the cone tip the ball energy tends to zero by local continuity. Hence the exact cone identity is
Its particle part is \int qd f\,dv, and its field part is
Since (a+b)^2\leq2(a^2+b^2), the last identity implies (3.2).
3.1 The retarded source labels
For fixed t,x, set r=|x-y|, s=t-r, and n=(x-y)/r when r>0. The zero-data wave potentials are
The following change to source labels also appears in [14, Section 9, Lemma 9.2]. We give the calculation for the present flow.
Lemma 3.2 (Retarded phase change). Away from the cone tip, the map sending (y,v) at time s=t-|x-y| to its initial label is one-to-one, with Jacobian
Its image consists, up to the null set of tip labels, of
For such a label, the retarded time solves
Consequently (3.3) equals
Proof. Let Z_s be the fixed-time inverse phase flow. Its determinant is one by Lemma 3.1, and \partial_sZ_s=-D Z_s\,(u,V'). Since \nabla_y s=n, the derivative of Z_{t-|x-y|}(y,v) is the fixed-time derivative multiplied by
The determinant of this rank-one perturbation is 1-n\cdot u=d.
Along a fixed label, the function on the left of (3.5) is strictly increasing. More explicitly, on any compact part of that characteristic, its increment between s_1<s_2 is at least (1-\sup|u|)(s_2-s_1)>0. It starts below t exactly for labels in \mathcal L_{t,x} and at s=t is at least t. Thus it has exactly one root. The root is a tip only if Y(t,z)=x. This set has phase Lebesgue measure zero, by the volume-preserving fixed-time flow and Fubini’s theorem. This proves injectivity and the image assertion. Transport of f and (3.4) now give (3.6).
3.2 The exact electric and magnetic kernels
Along a source characteristic write
where these fields, u, and q are evaluated at its retarded point.
Proposition 3.3 (Retarded field representation). Let E^{\rm fr},B^{\rm fr} be the homogeneous wave solutions with data
For t>0, define initial boundary terms by
Set these terms to zero at t=0. Then
where
All these integrals are well defined at the stated classical regularity. Moreover,
Proof. Wave representation. Kirchhoff’s formula gives \Box(\Phi,\mathcal A)=(\rho_f,j_f) with zero data, where \Box=\partial_t^2-\Delta. Maxwell’s equations give, distributionally,
Thus the fields are (-\nabla\Phi-\partial_t\mathcal A,\nabla\times\mathcal A) plus the stated homogeneous waves. Here it is important that the electric potential contribution has initial time derivative -j_f(0).
We justify this assertion with only C^1 sources. In the variables y=x-rn, the potentials integrate a C^1 source against r\,dr\,dn. Their first derivatives are continuous, and locally uniformly in x,
These formulas give the required zero initial values and initial difference quotients of the potential fields. For completeness, uniqueness of the resulting continuous distributional wave remainder can be reduced to classical uniqueness: convolve spatially with a compact smooth mollifier. The wave equation then supplies a continuous second time derivative, including at zero; the initial difference quotients identify its zero initial velocity. Local wave energy gives zero, and removal of the mollifier gives the assertion. No second derivative of f is used.
Interior kernels. To differentiate (3.6), hold the initial label fixed. Implicit differentiation of (3.5) gives
Write \ell=(rd)^{-1}. With s held fixed,
With x fixed,
Therefore the interior electric kernel is
which simplifies to (3.10). In the magnetic curl the corresponding expression is its cross product with n: indeed
Boundary terms and the cone tip. The label domain has indicator \mathbf1_{\{t>|x-y_0|\}}. Its time derivative is \delta(t-|x-y_0|) and its spatial gradient is -n_0\delta(t-|x-y_0|). The resulting electric factor is (n_0-u_0)/(td_0) and the magnetic one is its cross product with n_0. Integration over the initial sphere gives (3.8). The initial velocities are bounded on \operatorname{supp}f_0, so d_0 has a fixed positive lower bound. Sphere area and compact initial velocity support consequently give |E^{\rm bd}|+|B^{\rm bd}|\leq C_{f_0}t, uniformly in x.
The differentiations at the cone tip can be justified by inserting a smooth cutoff that vanishes for r<\gamma and equals one for r>2\gamma. On the supported labels of the fixed compact time interval, d is bounded below and u,b are bounded. Derivatives of this cutoff are at most C/\gamma. By (3.4), the omitted potential and cutoff errors are bounded by integrals of Cf/r over r<2\gamma, multiplied by at most C/\gamma for the latter. They are O(\gamma^2) and O(\gamma), respectively. The omitted kernel integrals are O(\gamma) because their worst radial singularity after this change of variables is r^{-2}. These bounds are locally uniform for t>0. The already established initial traces handle t=0. Thus no term supported at the cone tip remains.
Finally, Kirchhoff’s formula for a homogeneous wave W with data (W_0,W_1) bounds it by
The assumed bounded initial derivatives give (3.11). The homogeneous waves here need not together be a vacuum Maxwell solution; the representation requires only their displayed wave data.
3.3 Receiver geometry and pointwise force bounds
Let (X(t),V_X(t)) be a supported receiver characteristic and write
Since h+a\times(n\times h)=Qh, Proposition 3.3 and (3.4) give
where |K^{\rm data}|\leq C_{T_0,f_0,E_0,B_0} and
The density and source quantities in these integrals are evaluated at (t-r,y,v), not at the receiver.
Figure 2 shows the source and receiver geometry.
Figure 2: Retarded geometry along a receiver characteristic. The source event (s,Y(s)) lies on the backward light cone of (t,X(t)), with r=t-s=|X(t)-Y(s)|>0. The spatial unit vector n points from Y(s) to X(t). Panel (b) shows the source velocity u(V(s))=Y'(s) and the receiver velocity a(t)=X'(t) at these respective events. The curves and vector directions are schematic.
Lemma 3.4 (Pointwise force bounds). Let \theta,\phi>0 satisfy \theta^2\asymp d and \phi^2\asymp D, with fixed comparison constants. The transport kernel splits into two terms, according to the normal and tangential parts of n-u, bounded by
With G as in Lemma 3.1, the source acceleration satisfies
Consequently
If V_X\ne0 and P_X is orthogonal projection onto V_X^\perp, then |P_XS_{\rm ker}|\leq C\phi(S_g+S_b).
Proof. The exact identities
give |n-u|\leq C\theta and |n-a|\leq C\phi. Also Qn=n, whereas for h\perp n,
The normal component of n-u is dn and its tangential component has norm at most C\theta. This proves (3.14).
Put u=(1-d)n+u_\perp and H=E+n\times B. The force at the source is
Its tangential and normal components satisfy
Indeed the tangential part of u_\perp\times B involves only n\cdot B, while its normal part is bounded by C\theta|B|. For L_u=\Id-u\otimes u we have
Combining these component bounds with b=q^{-1}L_uF_{\rm src} proves the first two inequalities in (3.15). The last follows from q'=u\cdot E=u\cdot H-u\cdot(n\times B).
The numerator N=(n-u)(n\cdot b)-db is tangent to n, since n\cdot N=d(n\cdot b)-d(n\cdot b)=0. This cone tangency is also used in [19, Lemma 2.1]. The preceding estimates give
The tangential bound for Q now proves (3.16). Finally P_Xa=0, so |P_Xn|=|P_X(n-a)|\leq C\phi. For h\perp n,
Applying this to N gives the additional projected factor.
Lemma 3.5 (Source time along a receiver). Along a retarded source label satisfying t=s+|X(t)-Y(s,z)| away from collisions,
For f_0(z)dz-almost every label there is no same-time collision Y(t,z)=X(t) anywhere in the compact time interval. Such a label has one continuously differentiable retarded branch after its initial-cone entry, when that entry occurs, and r is bounded below on each closed part of this branch in the compact interval.
Proof. Differentiate t-s-|X(t)-Y(s,z)|=0 to obtain D\,dt=d\,ds. Both factors are positive. Combining this with (3.4) gives the measure identity in (3.17).
To exclude same-time collisions, let R_v bound the supported momenta on [0,t_*]. Use a time grid of mesh at most \zeta with O(1+t_*/\zeta) points. If a collision occurs, at a nearest grid time the two positions are at distance at most 2\zeta, since both spatial speeds are at most one. At each grid time the mass of labels satisfying this distance bound is at most C\|f_0\|_\infty R_v^3\zeta^3, by phase-volume preservation. The union has mass O(\zeta^2) and tends to zero.
For the remaining labels, strict timelikeness gives uniqueness and continuous differentiability of the retarded root. The entry time is unique because t-|X(t)-Y(0,z)| is strictly increasing. At entry r=t>0, since an entry at zero would be an excluded initial collision. On any closed part of the branch, continuity and the absence of collisions give a positive minimum of r. This assertion is labelwise; no uniform lower bound over all labels is used.
4 Occupation of retarded bins
We estimate how much particle mass a receiver can meet in a prescribed momentum, angle, and distance range. Two changes of variables give crude bounds. The signed increment assumption then limits how long a source can remain close to the receiver. A time-averaged energy budget will make the result summable over the small angular sector.
The residence-time and relative-velocity viewpoint is inspired by Pfaffelmoser’s Vlasov–Poisson argument [20]. This is methodological inspiration, not a theorem input for the Vlasov–Maxwell system: the retarded geometry and source-acceleration terms require the estimates proved here.
Throughout this section assume (2.2) up to a time \tau\leq t_*. Fix a supported receiver X, a dyad w, and a closed interval J\subset[0,\tau] of length at most I, where
We integrate over any measurable subset of \{t\in J:w/8\leq q_X(t)\leq8w\}. The receiver need not remain in this momentum range at other times of J. All constants below may depend on the fixed horizon, initial energy, \|f_0\|_\infty, and the fixed partition parameters, but not on M,A,P, \tau,J, or the measurable subset. In particular, they are uniform over supported receivers. We use the already established bounds
4.1 The momentum and angular ranges
Let H_0=\max(1,T_0). At a source point on the receiver’s backward cone write
and retain the notation
Choose smooth nonnegative dyadic partitions for r/H_0,\sqrt{d/2},\sqrt{D/2},\sqrt{e/2}, with representatives h,\theta,\phi,\sigma\in\{1,2^{-1},2^{-2},\ldots\}. Each factor is supported where its argument lies between one half and twice its representative. The partitions sum to one, including near the upper endpoint 1, have bounded overlap, and have uniformly bounded derivatives with respect to the logarithm of each variable. For example, normalized dilates of a fixed bump supported in (1/2,2) and positive on [3/4,3/2] give such partitions. Use the analogous partition of q\geq1 with representatives p_0=1,2,4,\ldots, but write p=64p_0. On its support,
The last inequality follows from the assumed bound on all supported momenta. We use closed support ranges for upper estimates; this preserves bounded overlap.
These are angular scales with the inverse-momentum deficits included, not literal geometric angles. The exact identities
imply, on a nonempty bin,
All angles have at most C\log(2+P) nonempty dyadic ranges. For the relations between them, apply the triangle and reverse triangle inequalities to
Their norms are \sqrt{2d},\sqrt{2D}, and the norm of their sum is \sqrt{2e}. Consequently \sigma\leq C(\theta+\phi), and \sigma is comparable to the larger of \theta,\phi when those two angles are sufficiently separated.
Fix a sufficiently small numerical \kappa>0. The small sector consists of representatives \theta<\kappa\phi; here \sigma\asymp\phi, and we omit the \sigma partition entirely. All small-sector estimates below remain valid on the enlarged range \theta\leq16\kappa\phi. The remaining sector has \theta\geq\kappa\phi and retains the \sigma partition. In particular it includes \theta\gg\phi. For the two sectors define
The parameter \omega will be the velocity tolerance in the occupation argument.
Let N(t,s,n) be the integral of f(s,X(t)-rn,v) over the momentum restrictions of one bin, set to zero when the remaining restrictions fail. Then
Indeed the momentum shell has volume at most Cp^3. If \theta is small, (4.2) restricts its velocity direction to a spherical cap of radius C\theta about n. If \sigma is small, both velocity magnitudes are close to one and their directions differ by at most C\sigma, giving a cap about the receiver direction. Use whichever cap is smaller. When the controlling parameter is bounded below, the shell bound alone proves the same estimate with a larger constant. The same argument shows that the allowed n have spherical area at most C\phi^2.
4.2 Two changes of variables
Define the occupation measure of the bin by
where t ranges over the specified receiver times and 0\leq s<t. The constants below absorb H_0 in the comparison r\asymp h. The factor D makes this cone integral equal to particle mass integrated over source-time hits, as the following changes of variables show.
Lemma 4.1 (Crude occupation bounds). For every bin and every allowed set of receiver times,
Proof. The characteristic change of variables below is used in Pallard’s work [18, Lemma 2.1]; see also Luk–Strain [15, Proof of Proposition 5.2]. Fix s. The map
is injective. Indeed, for fixed y, the function t-s-|X(t)-y| is strictly increasing: wherever differentiable its derivative is at least 1-|a(t)|>0. Its absolute Jacobian is r^2D. This follows by taking two tangent vectors to the sphere and the time derivative a-n, whose normal component is -D. Hence (4.6) is the integral of f(s,y,v)\,ds\,dy\,dv over the corresponding hits. The possible source times lie in an interval of length at most C(I+h). On the momentum bin q\geq cp, so the particle energy bound proves (4.7).
Passing at each fixed s to the initial phase labels z, whose measure is preserved by the characteristic flow, gives equivalently
By Lemma 3.5, labels having a same-time collision have zero f_0(z)\,dz measure. Every remaining label has a single increasing C^1 retarded branch, and
The branch begins when it enters the initial cone, if such an entry occurs. Since its hit receiver times form a subset of J, their total length is at most I. On the bin, D/d\leq C\phi^2/\theta^2. Thus the hit duration of each label is at most CI\phi^2/\theta^2. Integrating in (4.9) and using the conserved finite particle mass proves (4.8).
The three changes of variables. The preceding formulas use different fixed variables. For later reference, their Jacobians are collected here:
| Fixed | Change of variables | Jacobian |
| (t,x) | Retarded (y,v) to the initial label z | dz=d\,dy\,dv |
| s | (t,n)\mapsto X(t)-(t-s)n | dy=r^2D\,dt\,dn |
| Label z | t=s+|X(t)-Y(s,z)| | dt/ds=d/D |
In particular the combined measure identity is ds\,f_0(z)dz=D\,dt\,f\,dy\,dv.
4.3 Stability on a short cell
Fix large constants C_0,C_1, chosen below, and set
Call a bin actual if \omega>C_0/m, and deficit otherwise. In an actual bin (4.2) implies |u(s)-a(t)|\geq c\omega at each hit: the two inverse-momentum terms are smaller than a fixed small multiple of \omega^2 once C_0 is large. In the small sector use additionally \theta\leq16\kappa\phi. In a deficit bin,
We record explicitly the stability supplied by the bootstrap. Given a fixed constant C_* and a hit, all source times within C_*\lambda of its source time, and all receiver times within C_*\lambda of its receiver time, clipped to [0,\tau], satisfy
Here \varepsilon_0>0 can be prescribed by choosing C_1 sufficiently large in terms of C_* and the fixed comparisons, independently of M,A. To prove this, choose a dyad w' nearest q_{\rm hit} and stop at the first exit from [q_{\rm hit}/2,2q_{\rm hit}]. Before exit the bootstrap applies with w'\asymp q_{\rm hit}\geq cm. Its two terms on an interval of length at most C_*\lambda are bounded by
For sufficiently large C_1, this excludes the first exit, because q is 1-Lipschitz and \omega\leq1. The same stopping argument runs backwards from the hit. The segment between the two momentum vectors has q\geq cq_{\rm hit}; integrating \|D_vu\|\leq1/q on that segment then proves the velocity estimate. This proves (4.13). We now fix C_* large enough to cover the cell and retarded-time separations used below, and then fix C_1 accordingly.
Lemma 4.2 (Occupation on stable cells). In every actual bin,
Proof. Suppose first h\leq\lambda, and partition [0,\tau] into intervals of length \lambda, with the last interval clipped if necessary. For upper bounds we may enlarge the receiver times to all eligible times in J and use closed bin ranges. The labels with some hit in a closed cell form a measurable set: it is the projection of a compact subset of the product of the compact initial support, the cell, J, and S^2; the fixed bin has r\geq ch>0.
For any label with a hit (s_0,t_0) in the cell, use that hit as a reference. At every same-time point s of the cell, both u(s,z) and a(s) are within a small multiple of \omega of their respective reference velocities: the time separations are at most \lambda+C_{T_0}h\leq C_*\lambda. The actual velocity difference therefore has projection at least c\omega in one fixed direction throughout the cell. This direction may depend on the label and its reference hit. At any hit in the cell,
using |a(t)-n|\leq C\phi and receiver stability between s and t. The projected same-time separation is strictly monotone with derivative of magnitude at least c\omega. Its values at hits lie in an interval of length Ch(\phi+\omega), so their total s-duration is at most Ch(1+\phi/\omega)\leq Chk_1. This holds even if the hit set is disconnected.
All labels that hit the cell have q\asymp p at any one common time in the cell, by stability. Conservation of phase volume and the energy bound give their total label mass at most C/p. The source-time range of all hits has length at most C(I+h), so at most C\max(1,I/\lambda,h/\lambda) cells can occur. Formula (4.9) proves (4.14). If h>\lambda, instead use (4.7): its right-hand side is bounded by that of (4.14), since k_1\geq1 and h\max(I,h)/\lambda\geq\max(I,h).
4.4 One energy budget for all distance and source-angle bins
Choose a fixed enlargement
with its constant large enough for every source cell considered below when h\leq\lambda\leq I. In particular |J^*|\leq CI. For fixed s,y, let t_{\rm ret}(s,y)\in(s,\tau] be the unique solution of t-s=|X(t)-y|, if it exists; omit its tip y=X(s). This definition deliberately imposes no restriction that t_{\rm ret} belong to J or to the eligible receiver subset.
The first indicator below is used to average the mass of hit labels over a stable cell, using the velocity separation that persists there. The second records the receiver angle at a retarded hit and is used when the spatial change of variables estimates hits directly. For each p,\phi define these indicators with fixed sufficiently enlarged comparison constants:
The second indicator is zero when its retarded time does not exist. The strictly monotone implicit equation makes its non-tip solution continuous when the root lies in (s,\tau) and continuously extendible to a root at \tau. The existence condition is a closed inequality at \tau, after excluding the tip; thus the indicators are measurable. Set
These numbers depend on the receiver, J,I,p,\phi, but not on \theta,h or on the eligible measurable subset. Their comparison constants, and hence the bound
are independent of M,A. Indeed at each (s,y,v) only a bounded number of p and \phi dyads can occur in either family. For the second family there is just one retarded time. Summing the indicators and using the energy bound and |J^*|\leq CI proves (4.16).
Proposition 4.3 (Baseline occupation with a joint energy budget). Suppose h\leq I, excluding only deficit bins with h\leq\lambda. Then
Consequently, with R=1+\sqrt{I/\lambda}\sqrt{\mathcal E},
for all h, except deficit bins with h\leq\min(I,\lambda).
Proof. In the remaining sector, (4.14) proves the claim for actual bins; if h>\lambda, (4.7) gives \mathsf M\leq CI/p\leq Ch(I/\lambda)/p. This also treats the allowed deficit bins.
Consider the small sector. When h\leq\lambda\leq I and the bin is actual, denote the set of hit labels in a cell C by Z_C. At every time s\in C all these labels have both q\asymp p and |u-a(s)|\asymp\phi, by the same stability argument as in Lemma 4.2. For a full cell, averaging its mass over its whole length therefore gives
All such cells lie in J^* and their interiors are disjoint. The at most two cells clipped by the endpoints of [0,\tau] have mass at most C/p by the common-time argument, without averaging. Each label has hit duration at most Ch in its cell. Summing (4.9) proves \mathsf M\leq Chp^{-1}(1+(I/\lambda)\mathcal E_{p\phi}). If \lambda>I, the source-time range has length at most CI and meets only a bounded number of cells. Their crude mass and duration bounds already give \mathsf M\leq Ch/p; no averaging outside J^* is used.
Finally suppose \lambda<h\leq I, whether actual or deficit. In the space map of Lemma 4.1, each hit belongs to H^{(2)}_{p\phi} and its source time belongs to J^*. Hence
Since h/\lambda>1, this is bounded by the right-hand side of (4.17). The estimate 1+(I/\lambda)\mathcal E\leq R^2 proves (4.18) when h\leq I. For h>I, (4.7) gives \mathsf M\leq Ch/p, and k_1R^2\geq1 completes the proof. The budget (4.15) can be defined for every bin, regardless of whether h\leq I; its use in this last case requires only nonnegativity.
5 Direct bounds for the retarded force
We first estimate the retarded force by absolute values. These estimates control every term except the narrow-angle part of S_g at the scale needed for the signed bootstrap. They also give an absolute bound for the full force, with an additional factor \sqrt w.
Retain the decomposition and occupation estimates of the preceding section. Let J\subset[0,\tau] have length at most I, where
and let \mathcal T\subset J be any measurable set of times at which w/8\le q_X\le8w. Every integral below is restricted to t\in\mathcal T. Estimates for a nonnegative integrand remain valid if it is multiplied by any measurable weight between zero and one.
Proposition 5.1 (Integrated force bounds). Assume the bootstrap (2.2). The bounded data term, T_r, T_\perp, S_b, and the remaining-sector part of S_g have combined integral at most
The integral of the full force satisfies
The constant C is independent of P,w,I,M,A.
We prove the proposition in several steps. All dyadic sums below have fixed overlap constants. The normalization of the radius by \max(1,T_0) is absorbed into C, so that r\asymp h.
5.1 Field budgets and a single bin
We require field budgets shared by all the momentum and source-scale bins. Write \chi_\phi(t,s,n) for the indicator of a fixed enlarged support of the receiver-angle bin. The overlap of these supports is bounded independently of all parameters. Define
The cone-energy estimate (3.2) gives \sum_\phi\mathcal G_\phi\le C. In particular this budget does not depend on p,\theta,\sigma,h.
For the magnetic field, let J^* be the common enlargement of J from the preceding section, chosen to contain all source times when h\le I; thus |J^*|\le CI. At a point (s,y) with a retarded receiver time t_{\rm ret}(s,y)\in(s,\tau], put
Set the indicator in the following definition to zero when the retarded time does not exist:
The retarded time is unique and is a measurable function on its domain. Bounded overlap and conserved spatial energy (3.1) imply \sum_\phi\mathcal F_\phi\le C. Again the budget is independent of p,\theta,\sigma,h. The Jacobian r^2D of the map (t,n)\mapsto X(t)-(t-s)n, together with D\asymp\phi^2, gives
For h>I the source times lie in an interval of length O(I+h), so conserved spatial energy proves the second case directly. The budgets in (5.4)–(5.5) may always be enlarged by fixed factors when supports are enlarged.
For a fixed bin, let N=\int_{\mathrm{bin}}f\,dv and denote its hit measure by \mathsf M. By (4.5), the angular cap bound, and the definition of \mathsf M,
Here and below N vanishes outside the indicated time and spatial restrictions. In the small sector \nu=\theta and the \sigma partition is omitted.
The force majorants on this bin are, up to fixed constants,
Using dy=r^2ds\,dn yields the following bounds for their integrals. Both columns apply on every bin: the near bound uses the pointwise density estimate, while the far bound also uses the occupation measure. Their names indicate which estimate is useful at smaller or larger radii.
The first two rows follow from (5.7). For S_g, apply Cauchy–Schwarz to rGN: the first squared factor is bounded by I\mathcal G_\phi and the second by either bound for \int N^2 in (5.7). The prefactor is \phi/(p\theta^2). For S_b, use r\phi|B| as the first factor, apply (5.6), and use the prefactor 1/p. This proves every entry of the table.
5.2 Summing the radius
Recall the occupation factor
It is independent of h, and also of \theta in the small sector. The refined occupation estimate (4.17) gives
when h\le I, except in deficit bins with h\le\lambda. For h>I the same inequality follows from (4.7), since R,k_1\ge1.
For positive a,b,\alpha,\beta, the elementary dyadic estimate
follows by splitting at h=(b/a)^{1/(\alpha+\beta)} and summing geometric series on both sides. It remains an upper bound when the allowed radii form only a subset of the dyadic scales. Substituting (5.9) into (5.8) and applying (5.10) gives
The first line balances h with h^{-1}; the other two balance h^{1/2} with h^{-1/2}. The exceptional deficit radii are excluded from these balances and are treated below.
For S_b with h>I, use instead the time-change bound (4.8), namely \mathsf M\le CI\phi^2/\theta^2. The corresponding minimum in (5.8) is
Its radius sum is at most
If one of \theta,\phi is much larger than the other, \sigma has bounded multiplicity or is absent, and the angular factor is at most \theta^{2/3}\phi^{1/3}. If \theta\asymp\phi, it is at most C\theta(\sigma/\theta)^{4/3}, with \sigma\lesssim\theta. The angular sums are geometric in both cases. Summing p proves that all large-radius S_b terms cost at most CP\sqrt I.
In an exceptional deficit bin, h\le\min(I,\lambda) and \lambda\le CP^{-2} by (4.11). Summing the near bounds over these radii gives, per remaining bin,
Indeed, in the remaining sector \phi\lesssim\theta and \nu\lesssim m^{-1}, whereas in the small sector \nu=\theta, \phi\lesssim m^{-1}, and \theta^{-1}\le p. The individual field budgets are bounded by their total budgets. There are at most CL^4 remaining bins. Since
the ratio of their total cost to I S(w) is at most
Thus these terms are absorbed uniformly into CI S(w).
The radius sum for T_r is also immediate from its near bound. When \theta\asymp\phi, summing (\nu/\theta)^2 over \sigma is a geometric sum; otherwise \sigma has bounded multiplicity or is omitted. Summing h, the O(L) values of \theta, then \phi^2 and p, gives
5.3 Angular and momentum sums
The radius sums are complete, including the deficit exception and the magnetic terms with h>I. It remains to sum the angular and momentum factors in (5.11). We keep the small-sector S_g contribution separate because it yields the absolute-force bound with its additional factor \sqrt w. The following elementary momentum sums will be used repeatedly:
To verify them, split at p=w. For p\le w one has m=p; for p>w one has m=w. Each resulting positive-power sum is geometric, while a constant summand costs at most CL.
Write \omega=\sigma in the remaining sector and \omega=\phi in the small sector. The definition of \lambda gives the exact parameter dependence
For the second line we used \mathcal E\le C; the first line retains the joint small-sector budget when it is needed.
The small-sector transport term. Here \nu=\theta and k_1\asymp1. Summing the first line of (5.11) over \theta\ge1/p gives
The term with R replaced by one sums to CP\sqrt I. The M term in (5.16) costs
Indeed, Cauchy–Schwarz and the joint budget reduce the sum to the square root of \sum_{p,\phi}(p/m)^2, which is at most C[L^2+L(P/w)^2] before taking the square root. For the \sqrt{AL} term the weight has an additional factor \sqrt\phi; its squared sum is at most C[L+(P/w)^2]. That contribution is therefore at most
Since 1\le w\le P and L\ge1, both parameter terms are bounded by C(M+\sqrt A)I S(w).
The remaining-sector transport term. Split into \theta>16\phi, where \sigma\asymp\theta, and \theta\asymp\phi, where \sigma\lesssim\theta. Fixed comparison constants may depend on \kappa. The angular factors in the plain term of the first line of (5.11) are respectively
Their sums are bounded uniformly for each p: sum \phi\lesssim \theta in the first expression, and \sigma\lesssim\theta in the second, then use \theta\ge1/p. Thus the plain contribution is at most CL\sqrt I\le CP\sqrt I.
After extracting CIM P(p/m), the M terms have angular factors
The second \sigma sum is at most C\sqrt{m\theta}\le C\sqrt{p\theta} because \sigma\gtrsim1/m; hence both remaining angular sums are bounded. After extracting CI\sqrt{AL}\,P\sqrt p/m, the corresponding factors are
The number of allowed \sigma values in the second case is at most C[1+\log_+(m\theta)]\le C\sqrt{p\theta} on a nonempty bin, so these angular sums are bounded as well. Here \log_+s=\max(0,\log s). By (5.15), the two parameter contributions are at most
Both are bounded by C(M+\sqrt A)I S(w).
The remaining-sector good-field term. Apart from CI^{3/4}pR^{1/2}\sqrt{\mathcal G_\phi}, the angular factors for S_g in the same two cases are
At fixed \phi their \theta,\sigma sums are at most C\sqrt\phi. Consequently the plain term is at most CI^{3/4}P\le CP\sqrt I, by Cauchy–Schwarz with \sum_\phi\mathcal G_\phi\le C and \sum_\phi\phi\le C. Dividing the displayed factors by \sqrt\sigma leaves, up to constants,
whose sums at fixed \phi are bounded. Division by \sigma^{1/4} gives no larger bound, since \sigma\le1. There are at most CL receiver-angle bins, so \sum_\phi\sqrt{\mathcal G_\phi}\le C\sqrt L. The parameter terms in (5.16) therefore cost at most
The last inequality follows from w\le P, L\ge1, and M,A\ge1.
The magnetic remainder for h\le I. For S_b the base angular factors in the two remaining-sector subcases and in the small sector are respectively
Their sums at fixed \phi are bounded. This remains true after division by \sqrt\sigma in the remaining-sector subcases, giving \theta and \theta(\sigma/\theta)^{3/4}, or by \sqrt\phi in the small sector, where the \theta sum is at most C\phi. The quarter-power divisions give no larger bound. Using \sum_\phi\sqrt{\mathcal F_\phi}\le C\sqrt L thus gives the parameter estimate (5.18) again. The plain term is at most CI^{3/4}P\sqrt L. Uniformly in I,
as follows from x\le1+x^2 with x=I^{1/4}\sqrt L. Since PL\le S(w), this completes the bound for S_b.
The small-sector good-field term. This term is needed only for the absolute estimate (5.3). Its base angular factor is \phi/\sqrt\theta, whose \theta sum is at most C\sqrt p\,\phi. The plain contribution is therefore bounded by CI^{3/4}P^{3/2}. The time restriction (5.1) gives
For the two parameter terms the \phi factors after this sum are \phi^{1/2} and \phi^{3/4}. Their sums against \sqrt{\mathcal G_\phi} are bounded by Cauchy–Schwarz and geometric summation. The last momentum sum in (5.15) gives a total of at most
The exceptional bins were already included in (5.13).
Combining these bounds with (5.12), (5.13), and (5.14) proves (5.2). The data force contributes at most CI, which is bounded by CP\sqrt I since I\le1. Finally the force representation is bounded by these nonnegative majorants; adding the small-sector S_g estimate and using w\ge1 proves (5.3). This proves Proposition 5.1.
6 Signed cancellation along the source trajectories
The direct force estimates lose an angular factor when the source velocity is much closer to the cone direction than the receiver velocity is. In this region we integrate along the source trajectory before taking absolute values. The transport part of the force must be included: it cancels a term produced by differentiating the cone geometry.
Keep a supported receiver X(t), with velocity a=X', energy q_X and momentum derivative K=V_X'. A source label has position Y(s), velocity u=Y' and energy q. Along its retarded branch, set
A prime denotes differentiation in s along this branch, and a_t denotes the receiver derivative in its own time. By Lemma 3.5, the retarded measure identities are
In particular d,D>0. Differentiating the retarded relation gives
Define the matrices and vectors
Thus Hg=k_0/d.
6.1 The exact identity
After the change to ds\,f_0(z)dz, the two interior force terms in (3.13) together have kernel
Here \partial_s^{(u)} differentiates through u alone. The second term is the transport kernel. As in the force representation, we suppress the common constant 1/(4\pi) throughout this section.
Lemma 6.1 (Signed force identity). At every non-tip point of a retarded branch,
Consequently the two nondifferentiated terms, in dt\,dy\,dv measure, are
Proof. The product rule first rewrites the source-acceleration term as
For a variation A of a, with u,n fixed,
Since a'=a_t d/D, its contribution is the last term of (6.5). The terms from r,n, including the transport term, are
Multiplying by r^2d^2 gives
Directly from (6.2) and |u|^2=1-q^{-2},
Finally, |a|^2=1-q_X^{-2} gives
These identities prove the middle term in (6.5), with its stated sign. Multiplication by the factor fD in (6.1) gives (6.6).
The two residual densities \mathcal R and \mathcal A contain neither the source acceleration b nor inverse powers of the source deficit d. The geometric residual \mathcal R will be estimated directly. The receiver term \mathcal A contains a_t, so its coefficient must be small enough to use an absolute bound on the receiver force. The primitive k_0/(rd) retains the source singularity: after integration by parts it creates boundary and cutoff-derivative terms. We now keep track of those terms.
6.2 Smooth cutoffs and the small angular sector
Use the dyadic representatives p,\theta,\phi,h for, respectively,
The fixed factor in the convention for p is retained. Thus q\asymp p, q_X\asymp w, d\asymp\theta^2, D\asymp\phi^2 and r\asymp h, with constants depending on T_0 where necessary. The small-sector representatives satisfy \theta<\kappa\phi. All estimates below remain valid on the wider overlap range \theta\leq16\kappa\phi, after choosing the fixed \kappa>0 sufficiently small.
Use the smooth partitions chosen in Section 4, whose factors have uniformly bounded derivatives in the logarithm of each variable. For an index j=(p,\theta,\phi,h), let \beta_j be the product of its four factors. These products satisfy
The sum is locally finite for r>0, with uniformly bounded overlap. Derivative bounds are taken on the closed support indicators; no estimate of a derivative by the bump itself is assumed.
Lemma 6.2 (Derivatives of a selected sum). Let \mathcal S be any fixed set of representative indices, all in the small sector, and put \chi=\sum_{j\in\mathcal S}\beta_j. Selection may depend on the interval and fixed parameters, but not on the variables being differentiated. For differentiation through source variables (q,u) alone, or through geometric variables (r,n) alone,
This derivative vanishes unless selected and unselected supports meet. On their intersection, representatives in each coordinate differ by at most a factor of four. In particular all relevant supports remain in the wider small-sector range above.
Proof. Differentiate the locally finite identity (6.7) before restricting to a retarded trajectory. This proves (6.8) separately for each indicated partial derivative. A smooth nonnegative function and its first derivatives vanish at every point where the function vanishes. Thus if either family has no positive summand, the corresponding derivative sum is zero. Meeting supports constrain each representative to within a factor of two of the same variable, proving the factor-four comparison. The ratio \theta/\phi therefore changes by at most a factor of sixteen.
The cancellation applies to the cutoff derivative only after its common physical multiplier has been factored out. It does not compare derivatives of two different kernels. An additional \sigma-partition may subsequently be inserted into a nonnegative remaining-sector bound at the structural edge; it is not differentiated in (6.8).
6.3 Bounds for each derivative
The Lorentz acceleration satisfies
The source estimates needed here, with G=|E+n\times B|+|n\cdot B| at the source point, are
These are (3.15) with q\asymp p. For nonzero V_X, let P_X denote orthogonal projection perpendicular to V_X. In the small sector,
Indeed e-D=a\cdot(n-u); splitting a=n+(a-n) proves the first bound, and then k_0=(u-n)+n(1-e/D) proves the next two. Also P_Xa=0. Since q_X^{-2}\lesssim D,
For the last two bounds, |a_t|\lesssim|K|/w and |n\cdot a_t|\lesssim\phi|K|/w. Insert the decomposition of k_0 in (6.10) to obtain |a_t\cdot k_0|\lesssim\theta|K|/w.
After integration by parts, a cutoff derivative in source time has common multiplier fDk_0/(rd) in dt\,dy\,dv measure. Its size is at most Cf\phi/(r\theta), with an additional factor \phi after projection. The separate logarithmic rates and resulting bounds are as follows.
Source derivatives. From (6.9),
Thus their error kernels are bounded by
They coincide with the direct source-force majorants. By Lemma 6.2 they may be assigned to overlapping unselected support indicators.
Geometric derivatives. Equation (6.2) gives |r'|\lesssim1 and |n'|\lesssim\theta/r. Since n'\perp n,
Together with the logarithmic rate of r, these give
These errors can also be assigned to unselected neighbors by Lemma 6.2.
Receiver derivatives. Only D in the cutoff depends directly on a. Its source-time rate is
Multiplication by fDk_0/(rd) gives
This is estimated on selected supports directly. Every bound (6.12)–(6.14) gains a factor \phi after projection by P_X.
Time weights and a varying projection. If \psi(t) is a Lipschitz weight, its derivative contributes
For |V_X|\asymp w, put \Pi=wP_X/|V_X|. Its norm is bounded and \|\Pi_t\|\lesssim |K|/w, by differentiation of the unit direction V_X/|V_X| and of |V_X|^{-1}. Consequently
This has the size of the projected receiver term. The estimate uses |k_0|\lesssim\theta/\phi, so it retains the longitudinal part of k_0.
6.4 Integration by parts and the initial cone
Write \mathcal K_t=fD\mathcal K for the interior force density in dt\,dy\,dv measure; the subscript identifies the measure and does not denote a derivative.
Proposition 6.3 (Weighted signed formula). Let J=[t_1,t_2] lie in a compact classical time interval. Let 0\leq\psi\leq1 be Lipschitz on J, vanish at its endpoints, and be extended by zero outside J. Let \chi be a selected sum as in Lemma 6.2. One may integrate (6.5) labelwise against \psi\chi and a bounded C^1 matrix \Pi(t), including \Pi=\Id. More precisely,
where the spatial integral has r=|X(t)-y|<t, all source quantities are evaluated at s=t-r, and \mathcal B_0 is the initial-cone boundary term. If \|\Pi\|\lesssim1, that boundary term is bounded in norm by
The constant in (6.18) depends only on the datum and fixed time horizon. For \Pi=\Id, and for \Pi=wP_X/|V_X| when |V_X|\asymp w, the derivative kernels satisfy the bounds above, uniformly in the fixed selected index set. For a general \Pi, its derivative term retains the explicit bound Cf\theta\|\Pi_t\|/(r\phi).
Proof. The null set of labels with a same-time collision with the receiver can be discarded by Lemma 3.5. For any remaining label, continuity on the compact time interval gives \min_t|Y(t)-X(t)|>0. At a retarded point,
so r is bounded below on that label’s branch. The branch is C^1 and increasing. Its entry at source time s=0, when present, has receiver time t_0 determined by t_0=|X(t_0)-Y(0)|. All other integration endpoints occur where \psi=0. Products with Lipschitz \psi are absolutely continuous along the branch. Integrating the total derivative in (6.5) is therefore legitimate for each such label. The factors \psi_s=\psi_t d/D and \Pi_s=\Pi_t d/D, followed by (6.1), give exactly ().
The sign at s=0 is positive. Its integrand relative to initial labels is
For fixed initial velocity, parameterize the initial position by y_{\rm in}=X(t_0)-t_0n. This map is one-to-one on the initial cone and has absolute Jacobian t_0^2D. The initial momentum support bounds d(0) below, and
is bounded in norm. Since \chi\leq1, boundedness and compact velocity support of f_0, and t_0\leq T_0, give (6.18). The same proof allows disjoint time weights to be summed, using their sum at the initial boundary.
Finally these operations also hold after integrating the labels and summing the dyadic products. On a fixed compact classical interval, d,D have positive lower bounds on the relevant supported trajectories, and source and receiver accelerations are bounded. After conversion to dt\,dy\,dv, all differentiated kernels have an integrable majorant C_{\rm comp} f(r^{-2}+r^{-1}), using bounded partition overlap. Indeed dy=r^2dr\,dn and the momentum support is bounded. The constant here may depend on compact classical bounds and on the fixed weight; it is used only to justify the identity, not in the uniform estimates. For a completely truncated argument, insert a radial cutoff vanishing for r<\gamma and equal to one for r>2\gamma. Its new derivative term is bounded by C_{\rm comp}f/(\gamma r) on r\asymp\gamma, whose integral is O_{\rm comp}(\gamma). All remaining terms converge by domination. Local finiteness for r>0 and the same majorants justify the infinite sum of distance bins and its differentiated version. Thus no cone-tip boundary remains.
6.5 Coefficients used in the later selections
We record a coefficient bound used in both selections, so that their receiver-derivative costs can be read without repeating the geometry. At a fixed receiver time, the coefficient on |K| in the receiver term of (6.11) or in (6.14), integrated over one small bin, obeys
For the first bound use the density estimate \int_{\rm bin}f\,dv\lesssim p^3\theta^2, the n-cap area O(\phi^2), and dy=r^2ds\,dn. For the second, divide the kernel f\theta/(wr\phi^2) by the particle cone-flux density qdf and use the cone budget. Projection adds \phi to the first bound; the \Pi_t term has the same projected size.
The direction estimate in Section 7 will use the first, projected coefficient with its own selection of indices; that argument also accounts for the varying projector. The final signed-increment estimate in Section 8 will use the two-sided minimum in (6.19). Both applications keep their selected index sets fixed while differentiating the cutoffs.
7 Direction changes and improved occupation
We now improve the occupation estimate when both particles have high energy and their relative angle is intermediate. The additional information is a bound on the number of direction changes of a prescribed angular size. Its proof uses the signed bootstrap and the baseline estimates already established; it does not use the improved occupation estimate proved at the end of this section.
We retain the notation of the preceding sections. In particular, L=\log(2+P), S(w)=P^2L/w, and m=\min(p,w). All constants below may depend on the datum, the fixed time horizon, and the fixed decomposition parameters. A constant written C(M,A) may also depend on the two bootstrap parameters. The lower threshold for P may depend on M,A. Fix a bootstrap endpoint \tau: throughout this section the bootstrap is assumed only on [0,\tau], and all time intervals are clipped to this range.
7.1 A bound on changes of direction
Lemma 7.1 (Direction changes). Assume the signed increment bootstrap (2.2) on [0,\tau]. Let w be a dyad with w\ge P^{3/5}, and let J\subset[0,\tau] be a closed interval of length at most (w/P)^2 on which a supported characteristic satisfies w/8\le q_X\le8w. Suppose
Starting at the left endpoint of J, stop whenever the direction \xi=V_X/|V_X| first has chordal distance \delta from its direction at the preceding stopping time. Include the last, possibly incomplete, interval. For sufficiently large P, the number of intervals is at most
Proof. Put I=(w/P)^2. Throughout J, |V_X|\asymp w, so normalization of nonzero vectors gives
Define
The bootstrap and (7.2) bound the direction change on an interval of length c\ell by C(\sqrt c+c)\delta. Choosing c>0 sufficiently small proves that each complete first-exit interval has length at least c\ell. In particular their number, denoted by N_c, is finite.
On each complete interval choose an endpoint-zero Lipschitz function 0\le\psi\le1 with monotone ramps of length c'\ell and value one between the ramps. Choose c' sufficiently small that the ramps are disjoint. On a ramp, integrate the removed weight against \xi_t after subtracting the direction at its outer endpoint. The derivative of the removed weight has L^1 norm one, so the removed signed integral is bounded by C(\sqrt{c'}+c')\delta. Here the bootstrap bounds endpoint-relative increments; no bound on absolute direction variation is used. After reducing the fixed constant c' once more, each complete interval satisfies
Let P_X=\Id-\xi\otimes\xi and \Pi=wP_X/|V_X|, so w\xi_t=\Pi K. Apply Proposition 6.3 with this multiplier and with the fixed selection
The same representative index set is used on every interval. The weights have disjoint interiors and sum to at most one. Thus, after taking norms, all bulk errors can be bounded over their union with the single upper length I. Only derivatives of the weights will produce a factor N_c.
First consider the receiver and time-weight derivatives. Their bounds explain the radial threshold in (7.4). The signed primitive contains k_0=u-ne/D, with
The projected receiver-acceleration and receiver-cutoff kernels have size at most
The moving projector has precisely the same size: writing \rho=|V_X| gives
The representative w is fixed here; no dyad indicator is differentiated.
At fixed receiver time, integrating these kernels over a bin uses the radial factor Ch^2, the velocity density bound Cp^3\theta^2, and an angular cap of area C\phi^2. Their coefficient on |K| is therefore
The sums in h,\phi are geometric. The remaining p,\theta count is O(L^2), so the total coefficient is CL^2/w. The baseline absolute-force bound on our interval is
Thus all receiver and projector derivative terms contribute at most C(M,A)w^{1/2}L^3. They do not contribute a factor N_c.
For a weight derivative, the kernel is f\Pi k_0/r, and its fixed-time coefficient is instead
Its sum is CL^2. Equation (7.3) consequently gives a total weight-derivative cost CL^2N_c. Initial-cone boundary terms are bounded by C\int\sum\psi\,dt\le CI and create no further cell count.
We now estimate the remaining terms. On the complementary small-sector bins, p\theta\sqrt h>1 will make the projected good-field far bound summable. Outside the small-scale deficit exception, the baseline occupation estimates give
For h>I this follows instead from the elementary energy bound \mathsf M\le Ch/p. We use the unrefined R in (7.7); in particular no enhanced occupation estimate enters this proof. The definition of \lambda in the small sector yields
where wp\le mP was used in the last inequality.
The baseline direct force bound contributes at most C(M,A)wL. For the small-sector good-field term, perpendicular projection supplies an additional factor \phi. Its far bound outside (7.4) is therefore
When the second condition in (7.4) fails, the denominator is greater than one and h>P^{-2}. Hence only O(L) radial dyads occur. There are also O(L) momentum dyads and O(L) dyads for each angle. Summing gives at most C(M,A)wL^5.
The small deficit exception is entirely within (7.4) after increasing the fixed constant C_1 in the definition of \lambda. Indeed, when h\le\lambda and \phi\le C_0/m,
Source cutoff derivatives are assigned to overlapping unselected bins by Lemma 6.2. At the structural edge \theta\asymp\kappa\phi, use the direct bounds for the remaining sector. Fixed-factor neighboring indices change only constants in these estimates.
By (6.13), the projected spatial cutoff kernel is bounded by Cf\phi^2/(r^2\theta). Its baseline near/far balance, followed by the geometric sum in h, is
per remaining indices. By (7.8), the sum over p,\theta, \phi is at most C(M,A)wL^4. The projected central residual is Cf\phi/r^2, which is smaller in this sector. For the exceptional deficit scales, use their near bounds directly. Summing h\le\lambda\le C/P^2 and then \theta\le C\phi gives
These terms fit the same bulk bound after summing the remaining dyads. This separate near estimate avoids using (7.7) on its exceptional set.
Summing (7.3) now yields the conservative bound
Since w\delta\ge w^{2/5}\ge P^{6/25}, the last term is absorbed for sufficiently large P. Division by w\delta and addition of the last incomplete interval prove (7.1).
7.2 Occupation at intermediate angles
The baseline cell argument keeps each velocity within a small multiple of \phi of its value at a reference hit, so its cell length \lambda shrinks with that tolerance. We now use cells whose length is chosen to keep only the energies comparable to p and w. Within each such cell, Lemma 7.1 partitions the source and receiver motions into intervals of nearly fixed direction. Their common refinement permits the same monotone-crossing argument as before; the number of direction intervals controls the total hit duration.
Lemma 7.2 (Direction-based occupation at intermediate angles). Assume the signed increment bootstrap on [0,\tau]. Let a small-sector bin satisfy
Let the eligible receiver times lie in a closed interval J\subset[0,\tau] of length at most I\le(w/P)^2, with receiver energy in [w/8,8w] at those times. Set I_m^0=(m/P)^2 and \beta=29/25. Then, for P\ge P_*(M,A), the hit measure satisfies
The constant C is independent of M,A.
Proof. The window (7.9) implies
Thus the bins are actual for large P, and speed-magnitude deficits are small compared to \phi.
Choose a sufficiently large fixed constant C_2 and put
For h>\lambda_0, the elementary energy estimate already suffices. Indeed, for sufficiently large P, (7.11) implies \phi^{-\beta}\ge Q_0, and hence h\phi^{-\beta}\ge I_m^0. Consequently
which proves (7.10) in this case.
Suppose now that h\le\lambda_0. For upper bounds we may enlarge to all eligible receiver times in J and use closed bin support ranges. Partition source time into intervals of length \lambda_0, clipped to [0,\tau]. All hit source times lie in an interval of length at most C(I+h), so at most C(1+(I+h)/\lambda_0) original intervals can contribute. Fix one such interval F. Extend it to the right by C_T h, where C_T bounds r/h, and clip this extension to [0,\tau].
If F has a hit, the receiver energy is comparable to w throughout the extended interval. For every label hitting F, its source energy is comparable to p throughout F. To see this, apply the bootstrap up to the first exit from a fixed enlarged range around a reference hit energy. On a time interval of length C_T\lambda_0, the two bootstrap terms are at most Cm/\sqrt{C_2} and Cm/C_2. Increasing C_2 excludes the first exit. The argument works in both time directions and is unaffected by clipping.
For each use of Lemma 7.1, choose a dyad j' nearest the relevant reference-hit energy. It is comparable to w for the receiver and to p for a source. The entire interval stays within [j'/8,8j']. Increasing C_2 also ensures that its length is at most (j'/P)^2. For large P, j'\ge P^{3/5}. Choose a small fixed c_2>0 and use \delta=c_2\phi. Uniformly for j'\asymp j,
Therefore all the hypotheses of Lemma 7.1 hold.
Choose one common receiver direction partition on the extended interval. Its number of cells is at most C(M,A)L^6/\phi. For each hit label, independently choose its source direction partition on F, with the same bound on the number of cells. We shall prove that the hit duration for each such label is at most
Call a hit bad if a boundary b of the receiver direction partition lies in [s,t], and call all other hits good. Since 0<t-s\le C_T h, the source time of a bad hit belongs to [b-C_T h,b]. The total duration of bad hits is thus bounded by C(M,A)hL^6/\phi. This bound uses only the time difference t-s; it is independent of the retarded derivative dt/ds.
At a remaining hit, s,t lie in one receiver direction cell. Throughout [s,t] its direction varies by at most 2c_2\phi, and its speed differs from one by O(w^{-2}). The bin condition |a(t)-n|\le C\phi and (7.11) therefore give
On F, refine the source and receiver direction partitions together. The number of resulting intervals is bounded by the sum of the two numbers of cells plus one. In one such interval containing a good hit, choose that hit as reference. Actual small-angle geometry gives |u(s)-a(t)|\ge c\phi. Replacing a(t) by a(s) changes this vector by at most Cc_2\phi+Cm^{-2}. At any other time in the combined interval, the change of relative velocity has the same bound, since each direction varies by at most 2c_2\phi and both speed deficits are O(m^{-2}). Choose c_2 small and then P large. There is a fixed unit vector e_F for this label and combined interval such that
throughout it. Thus e_F\cdot(Y-X) is strictly increasing with derivative at least c\phi. At good hits it lies in an interval of length at most Ch\phi, by (7.13). Their total duration in this combined interval is consequently at most Ch. This remains true when the good-hit set has several components. Summing the combined intervals and adding the bad-hit bound proves (7.12).
The set of labels hitting F has mass at most C/p: at any one time of F, all of these labels have energy at least cp, by the preceding stability argument, and total particle energy is bounded. The closed support convention makes the set of hit labels a compact projection before the null collision labels are discarded, as in the baseline occupation argument. The hit set is measurable, and the preceding reasoning bounds each of its source-time sections. The label-dependent reference hits and direction partitions serve only to prove this numerical bound; no measurability of those choices is required.
Multiplying the mass bound by (7.12) and then counting the original intervals yields
Clipped intervals and final direction remainders satisfy all the same estimates. Finally, since \beta-1=4/25, (7.11) gives
This positive power absorbs the factor C(M,A)L^7 for P\ge P_*(M,A). Equation (7.14) becomes (7.10), with a constant independent of the bootstrap parameters.
The gain in Lemma 7.2 comes from controlling how often relative motion can change its direction. Within each combined direction interval, a hit is confined to a short monotone crossing. Only after this estimate has been proved will it be used in the final selection of bins for the signed momentum bound.
8 Selecting the signed part of the force
Fix a supported receiver and an interval J=[t_1,t_2] of length I>0 on which w/8\le q_X\le8w. Throughout this section we assume (2.2) and
Write L=\log(2+P) and S(w)=P^2L/w. The estimates proved above give
as a bound, up to a fixed factor, for the direct transport and magnetic-remainder terms and the remaining-sector good-field term. They also give
The extra factor \sqrt w in this absolute estimate will be canceled by the small coefficient of the receiver terms after signed integration by parts.
Only the narrow-angle contribution controlled by S_g remains outside the baseline bound. We estimate bins directly when a near or far bound for S_g is already below a summable threshold. On the other bins we apply the signed identity to the original force kernels, including the transport term. The main task is to show that these selected bins have a sufficiently small coefficient on the receiver acceleration.
We use the small-sector representatives p,\theta,\phi,h, with \theta<\kappa\phi, \theta\ge1/p, p\le P, and \phi,h\le1. All constants below may depend on the fixed decomposition and on
but are independent of M,A,P,I,w. A sufficiently large lower threshold for P may depend on M,A.
8.1 The selection and its summable weight
Put m=\min(p,w) and recall the small-sector stability length
Put I_m^0=(m/P)^2. The three occupation bounds available for selection come from energy, stable source-time cells, and the direction estimate, respectively. Define \mathsf M_* to be the minimum of the applicable entries in the following table. The displayed entries contain no implicit constants.
The occupation estimates give \mathsf M\le C\mathsf M_* in every small bin. The distinction between this inequality and the exact entries of (8.6) is useful: the selection below is an exact inequality between representative quantities.
We assign a summable weight to each bin:
The same weight has two roles. The good-field contribution of an unselected bin will be at most CP\sqrt I\,W by direct estimation. For a selected bin, we will prove that the coefficient of |K| in the receiver terms is at most Cw^{-1/2}W. Summing the latter coefficients will cancel the factor \sqrt w in (8.3). The remaining task will then be to control the central residual and the cutoff and time-weight derivatives in the signed formula.
A small bin is selected if
This is a fixed set of indices during J. In particular, taking a partial derivative of a cutoff does not differentiate the selection.
Lemma 8.1. The sum of W over all small-sector indices is bounded by a constant depending only on \epsilon and the fixed decomposition. The unselected small-sector good-field terms have total integral at most CP\sqrt I.
Proof. The \theta sum satisfies
The remaining sums in p\le P, \phi\le1, and h\le1 are geometric. The small sector has no additional \sigma partition. For an unselected bin, the near and far good-field estimates, together with \mathsf M\le C\mathsf M_*, bound its integral by CP\sqrt I\,W. Summing proves the second assertion.
8.2 The receiver coefficient on selected bins
We first verify the coefficient bound promised by the selection. The source-time identity produces both the receiver-acceleration term \mathcal A and a derivative of the receiver-angle cutoff. By (6.11) and (6.14), both have majorant Cf\theta|K|/(wr\phi^2). At a fixed receiver time, the density bound \int_{\rm bin}f\,dv\le Cp^3\theta^2 and the angular cap give a coefficient at most Cw^{-1}h^2(p\theta)^3 on |K|. Alternatively, the particle cone-flux bound gives C/(wp\theta h\phi^2). Thus the coefficient, as recorded in (6.19), is at most a fixed multiple of
Lemma 8.2 (Receiver coefficient on selected bins). For fixed M,A, there is a threshold P_*(M,A) such that every selected index with P\ge P_*(M,A) satisfies the exact representative inequality
Proof. We argue by contradiction: a selected bin violating (8.10) will first be forced into the actual regime, then into the improved-occupation window, where its exponent inequalities are incompatible.
Step 1: an unsafe selected bin has actual velocity separation. Write the representatives as exact powers of P:
All terminating decimals below denote exact rational numbers. Here \alpha,z,c,b,y,H are nonnegative and z+c=1. The weight is W=P^{-\Delta}. Near selection and (8.1) give
Far selection with the first pure bound in (8.6) gives
Indeed, when H\le2c one has h\ge I, whereas otherwise \max(I,h)\le P^{-2c}. In either case b\le H-\alpha+\Delta. Putting d_*=\alpha+b+y+H, we obtain
It follows that
Only the pure energy bound has been used so far.
Suppose (8.10) fails. Both terms of the minimum in (8.9) must exceed w^{-1/2}W, so
If H\le2c, the first inequalities of (8.13) and (8.15) force H+z+6\alpha<8\Delta<.008z, which is impossible. Hence H>2c, and the other far bound gives
Combining this with (8.12),
In particular,
and
Thus m\phi>P^{.3}, and the bin is actual for sufficiently large P. This conclusion precedes any absorption of constants into powers of P.
Step 2: stable-cell occupation forces that bin into the improved window. Put \rho=\max(\alpha,c), so P/m=P^\rho. Equation (8.5) and \phi\le1 give
once P\ge P_*(M,A). The last threshold is uniform over the indices because z>.66. Since I,h\le P^{-2c} and \rho\ge c, the actual pure bound gives
Far selection therefore implies
Comparing (8.18) with the first inequality in (8.15), then using (8.12), yields
Together with (8.16), this gives
The two inequalities in (8.15) also imply
We now check both ends of the improved-occupation window for both momenta. The last inequality gives y>.171>.16\max(z,1-\alpha). For the opposite end put Y=.377z-1.75\alpha-.75c; by (8.17), y<Y, and
Therefore
Together with (8.20), these are exactly the conditions for including the improved entry of (8.6).
Step 3: improved occupation contradicts an unsafe coefficient. Since I_m^0=P^{-2\rho}, that entry gives
Far selection and (8.15) consequently give
On the other hand, multiplying (8.12) by \beta/2=.58 and using (8.14) yields
Thus H<(378/935)z<.41z, contradicting (8.16). This proves (8.10).
8.3 The spatial transition between selected and unselected bins
The selected receiver terms now have the coefficient needed to use (8.3). We next estimate the source, spatial, and time-weight derivatives, together with the central residual \mathcal R, in the signed formula. The spatial transition requires particular care: its angular sum must retain the joint energy budget without introducing another logarithmic factor.
Equation (6.13) gives the spatial-cutoff majorant
By Lemma 6.2, the sum of the partial source or spatial derivatives of selected cutoff products is the negative of the corresponding sum over unselected products. It is supported where selected and unselected products overlap. Indeed, the full smooth partition sums to one, and a nonnegative smooth cutoff has zero derivative where it vanishes. Thus we estimate these errors on overlapping unselected supports. Neighboring representatives have ratios bounded by four.
For fixed p,\phi, let
Outside the deficit exception \phi\le C_0/m and h\le\min(I,\lambda), the near and far bounds for (8.21) are
For h>I the pure energy bound \mathsf M\le Ch/p gives the second bound because R\ge1. Hence R is independent of both \theta and h in every use of (8.23).
Lemma 8.3. The total spatial-cutoff error is at most C\mathcal B(I,w).
Proof. First consider an unselected structural neighbor, with \theta\ge\kappa\phi, overlapping a selected support. Its ratio \theta/\phi lies within fixed factors of \kappa, so for each \phi there are only a bounded number of such \theta. Balancing (8.23) in h gives C\sqrt I\,Rp\phi, which fits the shared-budget calculation following (5.17): the sum of C\sqrt I\,Rp\phi over p,\phi is at most C\mathcal B(I,w).
For another overlapping unselected bin, fix p,\phi,\theta and define h_0>0 by
The ratio of the near quantity to its threshold is (h/h_0)^{1/2-\epsilon}. Overlap with a selected neighbor therefore implies h\ge c h_0, for a fixed c>0. If the unselected bin fails the near inequality, it also has h\le h_0. Only a bounded number of radial dyads occur in this case.
Put
The near-failing transition bins cost at most C\min(n_0,f_*). If instead the bin fails the far inequality, then
Substitution into the far spatial bound gives, by (8.24), the additional estimate
Together with (8.23), this bounds the sum over h\ge ch_0 by
To see this, split the dyadic sum of \min(n_0t^{2\epsilon},f_*/t) at t=(f_*/n_0)^{1/(1+2\epsilon)}, where t=h/h_0. Both resulting sums are geometric. The same bound includes the near-failing transition bins. No restriction on the size of h_0 is needed; enlarging the actual radial range to t\ge c only increases this upper bound.
Equation (8.24) gives
Since D_0 is independent of \theta, writing x=n_0/D_0 turns (8.26) into
Both \theta tails are geometric. Their sum is at most CD_0, with no additional logarithm. Summing C\sqrt I\,Rp\phi over p,\phi as above gives C\mathcal B(I,w).
Finally, in the omitted deficit exception one has \lambda\le CP^{-2} and \phi\le C/m. The near spatial bound, summed over h and \theta, is at most CIp^3\phi^4/P^2. The remaining momentum weights are bounded by 1/(P^2p) when p\le w and by P/w^4 when p>w. These fit CI S(w), including the fixed logarithmic counts from the decomposition. This treats the exception without using the unavailable far occupation bound there.
Source-cutoff derivatives have the direct good-field and magnetic-remainder majorants in (6.12). On overlapping unselected small bins, Lemma 8.1 applies to the good-field part; the magnetic-remainder part and structural neighbors are bounded by (8.2). The central term in (6.11) has majorant Cf/r^2. Its near/far balance is C\sqrt I\,Rp\theta, whose \theta sum is C\sqrt I\,Rp\phi. Its deficit exception is bounded by the same near estimate used at the end of Lemma 8.3.
8.4 Time weights
Choose 0<\eta<1/2 and an endpoint-zero Lipschitz function \psi on J, equal to one away from monotone ramps of length \eta I, with 0\le\psi\le1 and |\psi_t|\le C/(\eta I). Equation (6.15) gives the time-weight majorant C_\eta f\theta/(Ir\phi).
Lemma 8.4. The total time-weight derivative is at most C_\eta\mathcal B(I,w).
Proof. If h\le I, the ratio of this majorant to f/r^2 is bounded by C_\eta h\theta/(I\phi)\le C_\eta. The preceding f/r^2 estimate therefore applies.
If h>I, the near integral, divided by P\sqrt I, is at most
The first inequality is the near-selection inequality. For the second, use far selection and the pure bound \mathsf M_*\le h/p to obtain P\theta\sqrt h<W^{-1}. Equivalently,
Summing the increasing factor (P\theta)^{4\epsilon} in the last line of () leaves
Both exponents are positive. The sums in h,\phi,p are geometric, so these terms cost at most C_\eta P\sqrt I.
8.5 The weighted signed estimate
Proposition 8.5. For the weight \psi above and P\ge P_*(M,A),
The constant C_\eta is independent of M,A.
Proof. Let \chi=\sum_{j\in\mathcal S}\beta_j be the sum of the selected cutoffs, with the index set \mathcal S fixed throughout J. Split the interior force density as
Although the selection criterion uses the good-field majorant S_g, the selected term contains the transport kernel as required by Lemma 6.1. Apply Proposition 6.3 to that term with \Pi=\Id, and use the direct bounds on its complement. Lemmas 8.1, 8.3, and 8.4 bound the unselected good-field terms and all spatial and time-weight derivatives. The source derivatives, the central signed term, the unselected transport and magnetic-remainder terms, and structural neighbors were bounded above by C_\eta\mathcal B(I,w). By (6.18), the initial-cone boundary has a bounded-data cost CI, which fits CP\sqrt I by (8.1); the endpoint weights remove the other time boundaries.
For the receiver terms, Lemmas 8.1 and 8.2 give, at every receiver time,
Multiplying this coefficient by (8.3) bounds their total integral by C\mathcal B(I,w). All sums used fixed geometric exponents; their constants are independent of M,A. The only enlargement of the large-P threshold occurred in Lemma 8.2. This proves (8.28).
9 Closing the signed-increment bootstrap
We now prove Proposition 2.1. The preceding estimates were conditional on the signed bound (2.2). Their constants permit a strict improvement, uniformly over the chosen compact solution interval. A continuity argument then removes that assumption.
Fix an interval [t_1,t_2] of length I on which w/8\le q_X\le8w. Write the full bootstrap bound as
If \mathcal B_w(I)\ge32w, then
Otherwise, once M\ge32, we have I\le(w/P)^2, as required in the estimates. The case I=0 is immediate.
Let 0<\eta<1/2 and choose a Lipschitz function \psi on [t_1,t_2] that vanishes at the endpoints, equals one off the two edge intervals of length \eta I, and has linear monotone ramps on those intervals. The weighted impulse estimate (8.28) gives
We restore each edge using the signed bootstrap, without estimating the absolute force there. On the left edge put \chi=1-\psi and F(t)=V_X(t)-V_X(t_1). Since F(t_1)=0 and \chi vanishes at the other edge endpoint,
The monotonicity of the ramp gives \int|\chi'|=1; every such F(t) has norm at most \mathcal B_w(\eta I) by (2.2). The right edge has the same bound, anchoring at t_2. Hence
Choose \eta so that 2\sqrt\eta\le1/8 and 2\eta\le1/8. Fix M\ge\max(32,8C_\eta) and then
The right side of (9.2) is at most \mathcal B_w(I)/4, and in particular at most \mathcal B_w(I)/2. Finally choose P_0 large enough for all preceding estimates with these fixed M,A. These choices depend only on the datum and T_0, and not on t_*. Thus every bootstrap interval satisfies the uniform improvement
For completeness, fix P\ge P_0 satisfying (2.1) on [0,t_*]. Let \mathcal T be the set of \tau\in[0,t_*] for which (2.2) holds for all eligible receivers, scales and intervals contained in [0,\tau]. It contains zero and is an initial interval. It is closed: truncate any nontrivial eligible interval at an increasing sequence of endpoints from below and use continuity of V_X. The range condition is preserved by truncation.
It is also relatively open to the right. On this fixed compact solution interval there is a finite bound C_* for |K| on all supported characteristics. Consequently some \ell_0>0 ensures (2.2) for every interval of length at most \ell_0, since C_*I\le MP\sqrt I there. If \tau\in\mathcal T, a longer eligible interval ending in [\tau,\tau+\varepsilon] has a prefix ending at \tau. The strict estimate (9.3) on that prefix is at most half the full interval’s bound. The suffix has impulse at most C_*\varepsilon, which is smaller than the remaining margin when \varepsilon is sufficiently small, uniformly because the full bound is at least MP\sqrt{\ell_0}. An interval starting after \tau is short if \varepsilon<\ell_0. This proves relative openness. Hence \mathcal T=[0,t_*] and Proposition 2.1 follows.
The compact-time force bound C_* was used only to propagate the bootstrap for a fixed P. It does not enter M,A or P_0. This distinction preserves the uniformity needed in the momentum-doubling argument of Section 2.
10 Local theory and continuation for finite-energy fields
We prove Proposition 2.2. The distinction between bounded smooth derivatives and square-integrable derivatives matters here: the initial fields in the theorem need not belong to H^5. We first give the local theory for Sobolev fields, then replace the fields outside a sufficiently large ball while preserving the divergence constraints. For any prescribed finite horizon, a vacuum field transfers solutions between the original and modified data on every interval of existence within that horizon.
Throughout this section, F=(E,B), u(v)=v/(1+|v|^2)^{1/2}, and
The data are 0\le f_0\in C_c^\infty(\R^6) and E_0,B_0\in C_b^\infty(\R^3)\cap L^2(\R^3), with every spatial derivative bounded and with \nabla\cdot E_0=\rho_{f_0}, \nabla\cdot B_0=0. Choose R_0,R_v so that
As in Proposition 2.2, a classical solution is a C^1 pointwise solution whose fields are continuous into L^2_x and whose density has compact phase-space support on every compact time interval.
10.1 Characteristics and the Maxwell evolution
For any C^1 fields on a closed finite time interval, the characteristic system is
It has a unique solution through every phase point throughout that interval. Indeed, |\dot X|\le1 confines X to a compact spatial ball; the fields are bounded on the resulting compact spacetime cylinder, so |\dot V| is bounded there. The usual local ODE solution therefore cannot escape in finite time. Transport along these characteristics gives
On every closed finite slab the same argument gives a common momentum bound, provided the C^1 solution exists on that closed slab. It does not give a bound up to an excluded endpoint of existence. The phase-space vector field in (10.1) has zero divergence: u=\nabla_v(1+|v|^2)^{1/2} and hence \nabla_v\cdot(u\times B)=0. Consequently positivity and the L^\infty norm of f are preserved. Integration of the transport equation gives \partial_t\rho_f+\nabla\cdot j_f=0, which, together with Maxwell’s equations, propagates both divergence constraints.
We shall use the homogeneous first-order Maxwell evolution on F=(E,B). If C_\xi w=\xi\times w, its Fourier generator is
Thus U(t), defined by the multiplier \exp(tA(\xi)), is a strongly continuous unitary group on L^2 and on every H^k. No divergence constraint is needed for this assertion. The formula
therefore yields
We also need finite propagation without any global integrability assumption. For a C^1 Maxwell field with current j, set e=(|E|^2+|B|^2)/2. Direct differentiation gives
Integrating over the shrinking balls B(x,r+t-s), 0\le s\le t, shows that a field with zero initial data and zero current in this backward cone vanishes on the final ball B(x,r): the boundary term is -\int_{\partial B}(e+(E\times B)\cdot\nu)\le0. This local argument uses neither global L^2 bounds nor divergence constraints.
Lemma 10.1 (Exterior comparison). Suppose that a C^1 solution exists on [0,S], that f_0 is compactly supported as above, and that F(0)\in H^k(\R^3) for every integer k. Then the fields belong to C([0,S];L^2) and have bounded first spatial derivatives on the entire slab. More precisely,
Proof. The characteristic argument gives compact phase-space support on the whole slab, so j_f is C^1 and vanishes outside |x|\le R_0+t. The homogeneous field F^{\rm h}(t)=U(t)F(0) is smooth and has bounded spatial derivatives on [0,S]\times\R^3, by its uniform Sobolev bounds. The difference W=F-F^{\rm h} has zero initial data and current j_f. If |x|>R_0+t, choose r>0 smaller than |x|-R_0-t. Every point (s,y) of the backward cone from B(x,r) satisfies |y|>R_0+s, so its current is zero. The local energy argument gives W(t)=0 on B(x,r). Thus W is supported in the fixed ball |x|\le R_0+S on this slab. Its C^1 regularity on that compact cylinder gives W\in C([0,S];L^2) and uniform bounds on its first derivatives. Adding F^{\rm h} proves the claims. Notice that F^{\rm h} can have nonzero electric divergence; only the first-order Maxwell equations, not a component free-wave equation, were used for this comparison.
10.2 Classical uniqueness and local Sobolev solutions
Uniqueness holds in the full classical class stated above. To see this, let (f_i,F_i), i=1,2, have the same data, and work on an arbitrary compact common time interval. Choose R containing both momentum supports, and put g=f_1-f_2, \delta F=F_1-F_2. With the transport operator of the first solution,
The phase-space divergence is zero, g is compactly supported, and
Since \nabla_v f_2 is bounded on this slab, integration of the transport and Maxwell energies gives, with a slab-dependent finite constant C,
For completeness, the Maxwell integration is justified with cutoffs \chi(x/N), where \chi=1 on the unit ball and |\nabla\chi(x/N)|\le C/N. Its boundary error is at most C N^{-1}\int_0^S\|\delta F(s)\|_2^2\,ds, which tends to zero because \delta F\in C_tL^2_x. No square-integrable field derivatives are required. Gronwall’s inequality in (10.5) proves uniqueness.
We next record the estimate needed both to construct smooth solutions and to preserve their regularity. For an integer k\ge6, write
For a smooth solution with momentum support in |v|\le R,
The H^k norm of F is the Hilbert norm of the six-component vector (E,B); this is the norm preserved by U(t). The coefficient in (10.6) involves no derivatives above first order. For smooth solutions in these Sobolev spaces, bounds on the momentum support and this coefficient give a bound on Y_k by Gronwall’s inequality. The local construction below will use the Y_6 bound to obtain a common restart time.
To retain this dependence in the commutators, we use the integer Gagliardo–Nirenberg inequality on the whole space and include its proof; see also [17, Lecture II] for its classical context. For d\ge1, an integer s\ge1, a scalar or finite-dimensional vector-valued g\in H^s(\mathbb R^d)\cap L^\infty, and an integer 0<j<s,
Here D^j g denotes the array of all ordered jth derivatives, with its Euclidean pointwise norm; this convention is equivalent to the usual multiindex convention up to constants depending on d,s. For g\in C_c^\infty put p_j=2s/j and A_j=\|D^jg\|_{p_j}, with A_0=\|g\|_\infty. Integrating one derivative by parts in \int|D^jg|^{p_j} gives
The last step is Hölder’s inequality, since 1/p_{j-1}+1/p_{j+1}+(p_j-2)/p_j=1, with p_0=\infty. Thus A_j^2\le C_{d,s}A_{j-1}A_{j+1}. For a nonzero compactly supported g all A_j are positive. Choose C\ge1 valid for these inequalities and define
Then b_0=b_s=0 and 2b_j-b_{j-1}-b_{j+1}\le\log C. The discrete maximum principle, applied after subtracting \tfrac12(\log C)j(s-j), gives b_j\le\tfrac12(\log C)j(s-j), which proves (10.7); the zero function is immediate. To pass to H^s\cap L^\infty, first multiply by smooth compactly supported cutoffs bounded by one and then mollify with nonnegative unit-mass kernels. A diagonal sequence converges in H^s, has L^\infty norm at most \|g\|_\infty, and has the relevant derivatives converging almost everywhere along a subsequence. Fatou’s lemma proves (10.7) in the stated class.
We give the commutator argument explicitly. If s\ge1 is an integer, |\alpha|+|\beta|=s, and the functions are in H^s\cap L^\infty, then the Gagliardo–Nirenberg interpolation inequality and Hölder’s inequality give
Indeed, put j=|\alpha|. For 0<j<s apply (10.7) to g and the analogous bound with derivative order s-j to h, then use weighted arithmetic–geometric mean. The cases j=0,s are direct. The same estimate for lower positive total derivative orders follows by using that order in place of s; total order zero is immediate.
Differentiate the Vlasov equation in (x,v) to order at most k and pair with the corresponding derivative of f in L^2. The leading transport term vanishes by phase-space divergence. Derivatives falling on u are bounded and cost only \|f\|_{H^k}. For the force terms, choose a smooth momentum cutoff \zeta_R equal to one on a neighborhood of |v|\le R, and set a_R=\zeta_R(v)(E+u\times B). It is used only in the commutators; the leading transport operator is unchanged. We have
Each commutator product is a derivative of \nabla a_R times a derivative of \nabla f, with total derivative order at most k-1. Equation (10.8) therefore bounds their sum by
Finally, \|j_f\|_{H^k_x}\le C_R\|f\|_{H^k_{x,v}} by Cauchy–Schwarz in v, and (10.3) completes (10.6). The estimates for unsquared norms follow from the squared energy estimates by regularizing a vanishing norm and taking the regularization parameter to zero.
Here are sufficient details of the local construction to make the restart time in this argument explicit. Suppose first that F_0 belongs to every H^k. Starting with the time-independent data, solve successively the linear transport equation for f^{n+1} with force F^n, and the linear Maxwell equations for F^{n+1} with current j_{f^n}; every iterate has the same initial data. Characteristics and (10.3) produce smooth iterates. Because H^6(\R^6) controls first derivatives in L^\infty, the preceding estimates give, while all momentum supports lie in a fixed ball,
Choose M>2(Y_6(0)+1), and then choose \tau>0 small in terms of M and R_v. Induction gives Y_6^n\le M on [0,\tau]; shrinking \tau further makes the momentum increase at most one, since |\dot V|\le |E^n|+|B^n|\le C M. This closes the support condition with R=R_v+1.
The L^2 difference estimate for consecutive iterates has the form
The term involving f^n uses its uniform first-derivative bound, exactly as in (10.5); the Maxwell estimate is linear. Taking C_{R,M}\tau<1/2 makes the iterates Cauchy in C([0,\tau];L^2). For every higher k, the uniform C^1 bounds already obtained give
The exponential Y_k(0)e^{2C_{k,R,M}t} is a common upper bound, by induction and the scalar comparison inequality. Interpolating the L^2 convergence with the uniform H^{k+1} bounds gives convergence in C([0,\tau];H^k) for every k. The limit solves the equations; it preserves positivity, compact phase-space support, and the constraints. Sobolev embedding and the equations give smoothness in both space and time. Crucially, the common lifespan \tau depends only on Y_6(0) and the initial momentum radius, not on higher norms.
10.3 Changing the fields outside the particle region
We first construct the vacuum evolution needed for data whose bounded derivatives need not lie in L^2.
Lemma 10.2 (Smooth finite-energy vacuum fields). Let D_E,D_B\in C_b^\infty(\R^3)\cap L^2(\R^3) be divergence-free. There is a global smooth vacuum Maxwell solution on [0,\infty) with these data, continuous into L^2, whose spatial derivatives of every order are bounded on each finite slab. If the data vanish on B(0,L), the solution vanishes wherever |x|+t<L.
Proof. For M_tg(x)=(4\pi)^{-1}\int_{S^2}g(x+t\omega)\,d\omega, define
Kirchhoff’s formula gives smooth free-wave solutions with all the asserted finite-slab derivative bounds. Their Maxwell residuals E^{\rm v}_t-\nabla\times B^{\rm v} and B^{\rm v}_t+\nabla\times E^{\rm v} solve the wave equation with zero initial values. Their initial time derivatives are respectively \nabla(\nabla\cdot D_E) and \nabla(\nabla\cdot D_B), hence zero. Local wave uniqueness makes both residuals vanish. The divergences vanish as well. The spherical-mean formula proves the stated finite propagation.
To justify strong L^2 continuity without assuming square-integrable derivatives, convolve the data with a smooth compactly supported approximate identity \eta_\epsilon. The resulting divergence-free data belong to every H^k, converge in L^2, and converge uniformly with every fixed spatial derivative: the next bounded derivative makes each derivative uniformly continuous. Their Kirchhoff solutions converge with all fixed derivatives on finite slabs to the displayed solution. The same solutions are U(t)(\eta_\epsilon*D_E, \eta_\epsilon*D_B), whose L^2 convergence is uniform in t by unitarity. The two limits agree locally, so the displayed solution is U(t)(D_E,D_B). It is therefore continuous into L^2 and preserves its L^2 norm.
Fix a finite horizon H>0. Set \rho_0=\rho_{f_0} and take its Coulomb field
This field belongs to every H^s, s\ge0. Indeed, its Fourier transform is -i\xi|\xi|^{-2}\widehat\rho_0(\xi). Near zero its size is O(|\xi|^{-1}), whose square is integrable in three dimensions, and at infinity \widehat\rho_0 decays faster than any power. Sobolev embedding gives bounded smooth derivatives. This argument allows arbitrary total charge.
For a smooth divergence-free vector field G on \R^3, define
Then \nabla\times\mathcal A_G=G. To verify this, put V(x)=\int_0^1tG(tx)\,dt. We have \nabla\cdot V=0 and
Choose \chi\in C_c^\infty(\R^3) equal to one on B(0,L), where L=R_0+2H+1, and set
These fields belong to every H^k, satisfy the original divergence constraints, and agree with E_0,B_0 on B(0,L). Their difference from the original fields is divergence-free, smooth with bounded derivatives, and in L^2. Let D(t) be the vacuum solution of Lemma 10.2 with initial data D(0)=(E_0-\overline E_0,B_0-\overline B_0). Then
because |x|+t\le R_0+2H<L. Consequently
is an equivalence, up to time H, between classical solutions with the original and modified initial fields. Maxwell’s equations and both constraints are preserved by subtracting a divergence-free vacuum solution. The transport equation is unchanged on the particle region by (10.12); outside that region f and its momentum derivatives vanish. The equivalence preserves C_tL^2 and compact phase-space support.
The Sobolev construction above, applied to (10.11) with, for example, H=1, and followed by addition of D, now proves local classical existence for the full initial field class. The uniqueness already proved permits these local solutions to be joined into a unique maximal classical solution.
10.4 Smoothness through a classical interval
Suppose first that the initial fields belong to every H^k and that a classical solution exists on [0,S]. Lemma 10.1 gives a uniform global W^{1,\infty}_x bound for its fields on this slab. Compact phase-space support and C^1 regularity give uniform bounds for the momentum radius and \nabla_{x,v}f. Wherever the solution is smooth, (10.6) therefore bounds every Y_k, in particular Y_6, by a constant depending only on this slab and the initial norm.
To continue smoothness, suppose a first loss could occur before or at S. Choose times increasing to that first time from below. The Y_6 bounds and momentum bounds just obtained give the same positive local lifespan for the Sobolev construction at all these times. Restarting sufficiently close to the alleged loss extends a smooth solution beyond it. Classical uniqueness identifies this solution with the given one on their common interval, a contradiction. This argument requires no prior existence of an H^6 limit at the endpoint. The higher estimates on the same lifespan give persistence of every Sobolev order, and the equations give all time derivatives.
For the original field class, fix any compact classical interval, choose a larger finite horizon in (10.11), and subtract the associated vacuum field. The modified solution has all-Sobolev initial fields, so the preceding argument applies. Adding back the smooth vacuum proves smoothness on the chosen slab for the original solution.
The solutions just constructed satisfy the hypotheses of Lemma 3.1, so their total energy is conserved. Here C_tL^2 was established independently; no square-integrable field derivatives are needed to apply that energy identity.
10.5 The bounded-momentum criterion
The continuation input is the Glassey–Strauss criterion in [15, Theorem 1.1 and Footnote 1, arXiv version]. In its normalization, nonnegative compactly supported H^5 particle data and H^5 field data satisfying the constraints have the following property: a unique classical solution on [0,T), T<\infty, extends uniquely as a C^1 solution past T if there is a bounded continuous function \kappa:[0,T)\to(0,\infty) such that f(t,x,v)=0 whenever |v|\ge\kappa(t). The H^5 formulation in that statement does not require a separate momentum bound for approximation iterates. Its density and current are 4\pi\int f_{\rm LS}\,dv and 4\pi\int u f_{\rm LS}\,dv, so it applies to the present normalization with
Now let a classical solution for the original data exist on [0,T), where T<\infty, and suppose its momentum support is uniformly bounded there by P_*<\infty. Perform (10.11) with H=T+1. The modified solution (f,F-D) has all-Sobolev initial fields; its density is smooth and compactly supported initially, and the constraints and uniqueness have already been checked. After (10.14), all hypotheses of the cited theorem hold with the constant continuous choice \kappa=P_*+1. It gives a C^1 extension of the modified solution to [0,T+\epsilon); shrink \epsilon so that \epsilon<1.
It remains to check that this extension is in the classical class used here. On any closed subinterval of [0,T+\epsilon), its characteristics give compact phase-space support, as in (10.2). Lemma 10.1 supplies C_tL^2 for its fields, including continuity through T. Thus it is a classical solution in the stated class. Adding back D, which vanishes throughout its particle region up to T+\epsilon<H, gives a classical extension for the original data. Section 10.4 then proves its smoothness on every compact slab; that conclusion was not imported from the cited theorem.
Finally, if f_0=0, the constraints make both initial fields divergence-free. Lemma 10.2 directly gives the global solution with f=0, and classical uniqueness applies to it. This proves the local existence, uniqueness, persistence, and bounded-momentum continuation assertions of Proposition 2.2.
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