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 (Wollman 1984). DiPerna and Lions developed global weak existence for large data (DiPerna and Lions 1989); a direct formulation for the relativistic one-species system is given by Rein (Rein 2004). These weak existence results do not furnish the regularity and uniqueness asserted in Theorem 1.

For classical solutions, Glassey and Strauss identified bounded momentum support as a continuation condition (Glassey and Strauss 1986) and proved global existence for initially dilute plasmas (Glassey and Strauss 1987). 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 (Glassey and Schaeffer 1990; Glassey and Schaeffer 1997; Glassey and Schaeffer 1998a, 1998b). For the spherically symmetric electrostatic reduction, Glassey and Schaeffer proved global classical existence for the repulsive relativistic Vlasov–Poisson system (Glassey and Schaeffer 1985). Rein proved persistence of global existence under small perturbations of suitably decaying global reference solutions, including nearly symmetric configurations (Rein 1990). 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 (Klainerman and Staffilani 2002), while Bouchut, Golse and Pallard gave a physical-space proof using a division lemma for wave kernels (Bouchut et al. 2003). Luk and Strain showed that it suffices to control the momentum support after projection onto a fixed two-dimensional plane (Luk and Strain 2014, Theorem 1.2). Moment and spatial-density conditions for continuation were developed by Pallard (Pallard 2005, 2015), Sospedra–Alfonso and Illner (Sospedra-Alfonso and Illner 2010), and Luk and Strain (Luk and Strain 2016). We use the Glassey–Strauss criterion as stated in Luk and Strain (Luk and Strain 2014, 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 (Schaeffer 2004). Using a vector-field method with modified commutators, Bigorgne obtained global existence and sharp asymptotic estimates under weighted smallness assumptions (Bigorgne 2020). His result permits nonzero charge and noncompact particle data. Wang proved a small-data result without compact support using vector-field and Fourier methods (Wang 2022). Wei and Yang allow arbitrarily large Maxwell fields with suitable spatial decay while requiring a sufficiently small weighted particle density (Wei and Yang 2021). Bigorgne subsequently proved global existence and modified scattering with a small distribution function and a large Maxwell field (Bigorgne 2025). 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 (Bigorgne 2023). This construction does not prescribe arbitrary Cauchy data.

Wang’s July 2026 two-paper treatment establishes global existence for large cylindrically symmetric data (Wang 2026a, 2026b). 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 (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 (Glassey and Strauss 1986). Luk and Strain organize its most singular terms using electromagnetic components controlled by energy flux through light cones (Luk and Strain 2014, Proposition 2.2 and Section 3); Pallard also uses the orthogonality of the acceleration kernel to the cone’s spatial normal (Pallard 2015, Lemma 2.1 and Section 2.3). The change from retarded phase points to initial particle labels appears in Lazarovici (Lazarovici 2016, Lemma 9.2). The moving-characteristic cone change of variables is used by Pallard (Pallard 2005); its proof is also given by Luk and Strain (Luk and Strain 2014, Proposition 5.2). We derive the required formulas in the normalization (1).

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 \[V_X(t_2)-V_X(t_1) =\int_{t_1}^{t_2} \bigl(E(t,X(t))+u(V_X(t))\times B(t,X(t))\bigr)\,dt.\] 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.

Organization

Section 2 states the signed estimate and deduces Theorem 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 (2).

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 \[ (\Phi,\mathcal A)(t,x)=\frac1{4\pi} \int_{|x-y|<t}\int_{\mathbb R^3} \frac{f(t-|x-y|,y,v)}{|x-y|}(1,u(v))\,dv\,dy. \tag{8}\]

The following change to source labels also appears in (Lazarovici 2016, sec. 9, Lemma 9.2). We give the calculation for the present flow.

The exact electric and magnetic kernels

Along a source characteristic write \[ b=u'=\frac{\mathrm{Id}-u\otimes u}{q}(E+u\times B), \tag{12}\] where these fields, \(u\), and \(q\) are evaluated at its retarded point.

Receiver geometry and pointwise force bounds

Let \((X(t),V_X(t))\) be a supported receiver characteristic and write \[\begin{gathered} a=u(V_X),\qquad q_X=q(V_X),\qquad K=V_X',\\ D=1-n\cdot a,\qquad e=1-a\cdot u,\qquad Q=D\mathrm{Id}+n\otimes a. \end{gathered}\] Since \(h+a\times(n\times h)=Qh\), Proposition 6 and (9) give \[ K=K^{\rm data}+\frac1{4\pi}\int_{|X(t)-y|<t}\int_{\mathbb R^3} (T_{\rm ker}+S_{\rm ker})\,dv\,dy, \tag{17}\] where \(|K^{\rm data}|\leq C_{T_0,f_0,E_0,B_0}\) and \[ T_{\rm ker}=\frac{fQ(n-u)}{r^2q^2d^2},\qquad S_{\rm ker}=\frac{fQ\bigl((n-u)(n\cdot b)-db\bigr)}{rd^2}. \tag{18}\] 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.

The momentum and angular ranges

Let \(H_0=\max(1,T_0)\). At a source point on the receiver’s backward cone write \[r=t-s>0,\qquad y=X(t)-rn,\qquad n\in S^2,\] and retain the notation \[u=v/q(v),\quad a=a(t),\quad d=1-n\cdot u,\quad D=1-n\cdot a,\quad e=1-a\cdot u.\] 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, \[p/128\leq q\leq p/32,\qquad 64\leq p\leq P.\] 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 \[ \begin{split} 2d&=|n-u|^2+q^{-2},\\ 2D&=|n-a|^2+q_X^{-2},\\ 2e&=|a-u|^2+q^{-2}+q_X^{-2} \end{split} \tag{24}\] imply, on a nonempty bin, \[ \theta\geq1/p,\qquad \phi\geq c/w,\qquad \sigma\geq c/m,\qquad m=\min(p,w). \tag{25}\] 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 \[(u-n,q^{-1},0),\qquad(n-a,0,q_X^{-1})\in\mathbb R^5.\] 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 \[ \begin{array}{c|c|c|c} &\nu&k_1&\omega\\ \hline \text{small}&\theta&1&\phi\\ \text{remaining}&\min(\theta,\sigma)&1+\phi/\sigma&\sigma . \end{array} \tag{26}\] 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 \[ N\leq C p^3\nu^2. \tag{27}\] Indeed the momentum shell has volume at most \(Cp^3\). If \(\theta\) is small, (24) 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 \(C\phi^2\).

Two changes of variables

Define the occupation measure of the bin by \[ \mathsf M=\int dt\,ds\,dn\ r^2 D N(t,s,n), \tag{28}\] 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.

Stability on a short cell

Fix large constants \(C_0,C_1\), chosen below, and set \[ \lambda=\left[ C_1\frac Pm\left(\frac M\omega+ \sqrt{\frac{AL}{\omega}}\right) \right]^{-2}. \tag{33}\] Call a bin actual if \(\omega>C_0/m\), and deficit otherwise. In an actual bin (24) 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, \[ \lambda\leq\frac{C_0^2}{C_1^2P^2}. \tag{34}\]

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 \[ |\Delta V|\leq\varepsilon_0 m\omega, \qquad |\Delta u|\leq\varepsilon_0\omega, \qquad q\asymp q_{\rm hit}. \tag{35}\] 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 \[MP\sqrt{C_*\lambda}\leq \frac{\sqrt{C_*}}{C_1}m\omega, \qquad C A C_*\lambda\frac{P^2L}{m} \leq\frac{CC_*}{C_1^2}m\omega.\] 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 (35). We now fix \(C_*\) large enough to cover the cell and retarded-time separations used below, and then fix \(C_1\) accordingly.

One energy budget for all distance and source-angle bins

Choose a fixed enlargement \[J^*=\{s\in[0,\tau]:\operatorname{dist}(s,J)\leq C_{T_0}I\},\] 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: \[\begin{split} H^{(1)}_{p\phi}(s,y,v) &=\mathbf1\{q(v)\asymp p, |u(v)-a(s)|\asymp\phi\},\\ H^{(2)}_{p\phi}(s,y,v) &=\mathbf1\left\{q(v)\asymp p, t_{\rm ret}\text{ exists}, \sqrt{\frac{1-n_{\rm ret}\cdot a(t_{\rm ret})}{2}}\asymp\phi \right\},\\ n_{\rm ret}&=\frac{X(t_{\rm ret})-y}{t_{\rm ret}-s}. \end{split}\] 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 \[ \mathcal E_{p\phi}=\frac1I\int_{J^*}\int_{\mathbb R^3\times\mathbb R^3} qf\bigl(H^{(1)}_{p\phi}+H^{(2)}_{p\phi}\bigr)\,dy\,dv\,ds. \tag{37}\] 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 \[ \sum_{p,\phi}\mathcal E_{p\phi}\leq C, \tag{38}\] 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 (38).

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 \[ \mathcal G_\phi =\frac1I\int_{\mathcal T}dt\int_0^t ds\int_{\mathbb S^2}dn\, r^2\chi_\phi(t,s,n)G(s,X(t)-rn,n)^2. \tag{44}\] The cone-energy estimate (7) 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 \[n_{\rm ret}(s,y) =\frac{X(t_{\rm ret}(s,y))-y}{t_{\rm ret}(s,y)-s}.\] Set the indicator in the following definition to zero when the retarded time does not exist: \[ \mathcal F_\phi =\frac1I\int_{J^*}ds\int_{\mathbb R^3}dy\,|B(s,y)|^2 \chi_\phi\bigl(t_{\rm ret}(s,y),s,n_{\rm ret}(s,y)\bigr). \tag{45}\] The retarded time is unique and is a measurable function on its domain. Bounded overlap and conserved spatial energy (6) 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 \[ \int_{\mathrm{bin}} r^2\phi^2|B|^2\,dt\,ds\,dn \le C(I+h)\mathcal F, \qquad \mathcal F= \begin{cases} \mathcal F_\phi,&h\le I,\\ 1,&h>I. \end{cases} \tag{46}\] 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 (44)–(45) 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 (27), the angular cap bound, and the definition of \(\mathsf M\), \[ \begin{split} N&\le Cp^3\nu^2,\qquad \int_{\mathrm{bin}}dt\,ds\,dn\le CIh\phi^2,\\ \int N\,dt\,ds\,dn&\le \frac{C\mathsf M}{h^2\phi^2},\\ \int N^2\,dt\,ds\,dn &\le C\min\left(Ih\phi^2p^6\nu^4, \frac{p^3\nu^2\mathsf M}{h^2\phi^2}\right). \end{split} \tag{47}\] 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, \[T_r=\frac{f}{r^2p^2\theta^2},\qquad T_\perp=\frac{f\phi}{r^2p^2\theta^3},\qquad S_g=\frac{f\phi G}{rp\theta^2},\qquad S_b=\frac{f\phi|B|}{rp}.\] 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. \[ \renewcommand{\arraystretch}{1.5} \begin{array}{c|c|c} &\text{near bound}&\text{far bound}\\ \hline T_r&Ihp(\nu/\theta)^2\phi^2&-\\ T_\perp&Ihp\theta^{-3}\nu^2\phi^3 &\mathsf M/(h^2\phi p^2\theta^3)\\ S_g&I\sqrt h\,p^2(\nu/\theta)^2\phi^2\sqrt{\mathcal G_\phi} &\sqrt I\sqrt{p\mathsf M}\,\nu(\theta^2h)^{-1} \sqrt{\mathcal G_\phi}\\ S_b&\sqrt{I(I+h)}\sqrt h\,p^2\nu^2\phi\sqrt{\mathcal F} &\sqrt{I+h}\sqrt{p\mathsf M}\,\nu(\phi h)^{-1} \sqrt{\mathcal F}. \end{array} \tag{48}\] The first two rows follow from (47). 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 (47). The prefactor is \(\phi/(p\theta^2)\). For \(S_b\), use \(r\phi|B|\) as the first factor, apply (46), and use the prefactor \(1/p\). This proves every entry of the table.

Summing the radius

Recall the occupation factor \[R=1+\sqrt{I/\lambda}\sqrt{\mathcal E}, \qquad \mathcal E= \begin{cases} 1,&\theta\ge\kappa\phi,\\ \mathcal E_{p\phi},&\theta<\kappa\phi, \end{cases} \qquad \sum_{p,\phi}\mathcal E_{p\phi}\le C.\] It is independent of \(h\), and also of \(\theta\) in the small sector. The refined occupation estimate (39) gives \[ \mathsf M\le C h k_1R^2/p \tag{49}\] when \(h\le I\), except in deficit bins with \(h\le\lambda\). For \(h>I\) the same inequality follows from (29), since \(R,k_1\ge1\).

For positive \(a,b,\alpha,\beta\), the elementary dyadic estimate \[ \sum_{h\in2^{\mathbb Z}} \min(ah^\alpha,bh^{-\beta}) \le C_{\alpha,\beta} a^{\beta/(\alpha+\beta)}b^{\alpha/(\alpha+\beta)} \tag{50}\] 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 (49) into (48) and applying (50) gives \[ \begin{aligned} T_\perp:&\quad C\sqrt I\,R p^{-1}\theta^{-3}\nu\phi\sqrt{k_1},\\ S_g:&\quad C I^{3/4}R^{1/2}p\phi\nu^{3/2}\theta^{-2} k_1^{1/4}\sqrt{\mathcal G_\phi},\\ S_b\ (h\le I):&\quad C I^{3/4}R^{1/2}p\nu^{3/2}k_1^{1/4} \sqrt{\mathcal F_\phi}. \end{aligned} \tag{51}\] 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 (30), namely \(\mathsf M\le CI\phi^2/\theta^2\). The corresponding minimum in (48) is \[C\sqrt I\min\left(hp^2\nu^2\phi, h^{-1/2}\sqrt p\,\nu/\theta\right).\] Its radius sum is at most \[ C\sqrt I\,p\nu^{4/3}\theta^{-2/3}\phi^{1/3}. \tag{52}\] 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 (33). Summing the near bounds over these radii gives, per remaining bin, \[ \begin{aligned} S_g+S_b&\le CI\frac{p^2}{Pm^2},\\ T_\perp&\le \begin{cases} CIp/P^2,&\theta\ge\kappa\phi,\\ CIp^2/P^2,&\theta<\kappa\phi. \end{cases} \end{aligned} \tag{53}\] 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 \[\frac{p^2}{Pm^2}\le \frac1P+\frac{P}{w^2},\qquad \frac{p^2}{P^2}\le1,\] the ratio of their total cost to \(I S(w)\) is at most \[CL^3\left(\frac{w}{P^3}+\frac1{Pw}+\frac{w}{P^2}\right) \le \frac{CL^3}{P}.\] 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 \[ \int T_r\le CIPL\le CI S(w). \tag{54}\]

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 (51). 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: \[ \begin{aligned} \sum_p p&\le CP,& \sum_p\frac p m&\le C(L+P/w),& \sum_p\frac{\sqrt p}{m}&\le C(1+\sqrt P/w),\\ \sum_p p\sqrt{P/m}&\le CP\sqrt{P/w},&& \sum_p p^{3/2}\sqrt{P/m}&\le CP^2/\sqrt w. \end{aligned} \tag{55}\] 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 \[ \begin{split} R-1 &\le C\sqrt I\frac Pm \left(\frac M\omega+ \frac{\sqrt{AL}}{\sqrt\omega}\right) \sqrt{\mathcal E},\\ R^{1/2} &\le C\left[1+I^{1/4}\sqrt{P/m} \left(\frac{\sqrt M}{\sqrt\omega}+ \frac{(AL)^{1/4}}{\omega^{1/4}}\right) \right]. \end{split} \tag{56}\] For the second line we used \(\mathcal E\le C\); the first line retains the joint small-sector budget when it is needed.

The exact identity

After the change to \(ds\,f_0(z)dz\), the two interior force terms in (18) together have kernel \[ \mathcal K=-\frac Hr\,\partial_s^{(u)}g -\frac{Hg}{r^2q^2d}. \tag{63}\] 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.

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.

Smooth cutoffs and the small angular sector

Use the dyadic representatives \(p,\theta,\phi,h\) for, respectively, \[q,\qquad \sqrt{d/2},\qquad \sqrt{D/2},\qquad r/\max(1,T_0).\] 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 \[ \beta_j\geq0,\qquad \sum_j\beta_j=1. \tag{66}\] 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.

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 (67).

Bounds for each derivative

The Lorentz acceleration satisfies \[a_t=(\mathrm{Id}-a\otimes a)K/q_X.\] The source estimates needed here, with \(G=|E+n\times B|+|n\cdot B|\) at the source point, are \[ |u'|\lesssim p^{-1}(G+\theta^2|B|),\quad |n\cdot u'|\lesssim p^{-1}\theta(G+\theta^2|B|),\quad |q'|\lesssim G+\theta|B|. \tag{68}\] These are (20) with \(q\asymp p\). For nonzero \(V_X\), let \(P_X\) denote orthogonal projection perpendicular to \(V_X\). In the small sector, \[ |e-D|\lesssim\phi\theta,\quad |k_0|\lesssim\theta/\phi,\quad |P_Xk_0|\lesssim\theta,\quad |P_Xn|\lesssim\phi. \tag{69}\] 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 \(q_X^{-2}\lesssim D\), \[ |\mathcal R|\lesssim f/r^2,\quad |P_X\mathcal R|\lesssim f\phi/r^2,\quad |\mathcal A|\lesssim\frac{f\theta|K|}{wr\phi^2},\quad |P_X\mathcal A|\lesssim\frac{f\theta|K|}{wr\phi}. \tag{70}\] 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 (69) 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.

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.

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 (70) or in (73), integrated over one small bin, obeys \[ \frac Cw\min\left(h^2(p\theta)^3, \frac1{p\theta h\phi^2}\right). \tag{78}\] 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 (78). Both applications keep their selected index sets fixed while differentiating the cutoffs.

A bound on changes of direction

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 16 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.

The gain in Lemma 17 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.

The selection and its summable weight

Put \(m=\min(p,w)\) and recall the small-sector stability length \[ \lambda^{-1/2} =C_1\frac Pm\left(\frac M\phi+\sqrt{\frac{AL}{\phi}}\right). \tag{97}\] 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. \[ \begin{array}{c|l} \text{entry}&\text{condition for inclusion}\\ \hline \max(I,h)/p&\text{always}\\[2pt] (h/p)\max(1,I/\lambda,h/\lambda)&\phi>C_0/m\\[2pt] (h/p)\phi^{-\beta}\max(1,I/I_m^0,h/I_m^0) &p,w\ge P^{.7},\quad j^{-.49}\le\phi\le j^{-.16}\ (j=p,w), \end{array} \tag{98}\] The occupation estimates give \(\mathsf M\le C\mathsf M_*\) in every small bin. The distinction between this inequality and the exact entries of (98) is useful: the selection below is an exact inequality between representative quantities.

We assign a summable weight to each bin: \[ W=(P\theta)^{-\epsilon}\phi^\epsilon h^\epsilon. \tag{99}\] 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 (95). 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 \[ \min\left\{I\sqrt h\,p^2\phi^2, \frac{\sqrt I\sqrt{p\mathsf M_*}}{\theta h}\right\} >P\sqrt I\,W. \tag{100}\] This is a fixed set of indices during \(J\). In particular, taking a partial derivative of a cutoff does not differentiate the selection.

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 (70) and (73), 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 (78), is at most a fixed multiple of \[ U=w^{-1}\min\left\{h^2(p\theta)^3, (p\theta h\phi^2)^{-1}\right\}. \tag{101}\]

The spatial transition between selected and unselected bins

The selected receiver terms now have the coefficient needed to use (95). 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 (72) gives the spatial-cutoff majorant \[ C\frac{f}{r^2}\frac\phi\theta. \tag{113}\] By Lemma 14, 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 \[ R=1+\sqrt{I/\lambda}\sqrt{\mathcal E_{p\phi}}, \qquad \sum_{p,\phi}\mathcal E_{p\phi}\le C. \tag{114}\] Outside the deficit exception \(\phi\le C_0/m\) and \(h\le\min(I,\lambda)\), the near and far bounds for (113) are \[ C\min\left\{Ihp^3\theta\phi^3, \frac{R^2}{ph\theta\phi}\right\}. \tag{115}\] 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 (115).

Source-cutoff derivatives have the direct good-field and magnetic-remainder majorants in (71). On overlapping unselected small bins, Lemma 18 applies to the good-field part; the magnetic-remainder part and structural neighbors are bounded by (94). The central term in (70) 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 20.

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 (74) gives the time-weight majorant \(C_\eta f\theta/(Ir\phi)\).

The weighted signed estimate

Characteristics and the Maxwell evolution

For any \(C^1\) fields on a closed finite time interval, the characteristic system is \[ \dot X=u(V),\qquad \dot V=E(t,X)+u(V)\times B(t,X). \tag{124}\] 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 \[ f(t,X(t),V(t))=f_0(X(0),V(0)),\qquad \operatorname{supp}_x f(t)\subset\{|x|\le R_0+t\}. \tag{125}\] 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 (124) 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 \[A(\xi)=\begin{pmatrix}0&iC_\xi\\-iC_\xi&0\end{pmatrix}, \qquad A(\xi)^*=-A(\xi).\] 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 \[ F(t)=U(t)F(0)+\int_0^t U(t-s)(-j_f(s),0)\,ds \tag{126}\] therefore yields \[\|F(t)\|_{H^k}\le\|F(0)\|_{H^k} +\int_0^t\|j_f(s)\|_{H^k}\,ds.\]

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 \[ \partial_t e+\nabla\cdot(E\times B)=-E\cdot j, \qquad |E\times B|\le e. \tag{127}\] 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.

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, \[(\partial_t+u\cdot\nabla_x+(E_1+u\times B_1)\cdot\nabla_v)g =-(\delta E+u\times\delta B)\cdot\nabla_v f_2.\] The phase-space divergence is zero, \(g\) is compactly supported, and \[\|j_{f_1}-j_{f_2}\|_{L^2_x}\le C_R\|g\|_{L^2_{x,v}}.\] Since \(\nabla_v f_2\) is bounded on this slab, integration of the transport and Maxwell energies gives, with a slab-dependent finite constant \(C\), \[ \|g(t)\|_2^2+\|\delta F(t)\|_2^2 \le C\int_0^t\bigl(\|g(s)\|_2^2+\|\delta F(s)\|_2^2\bigr)\,ds. \tag{128}\] 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 (128) proves uniqueness.

We next record the estimate needed both to construct smooth solutions and to preserve their regularity. For an integer \(k\ge6\), write \[Y_k=\|f\|_{H^k(\mathbb R^6)}+\|F\|_{H^k(\mathbb R^3)}.\] For a smooth solution with momentum support in \(|v|\le R\), \[ Y_k(t)\le Y_k(0)+C_{k,R}\int_0^t \bigl(1+\|F(s)\|_{W^{1,\infty}_x} +\|\nabla_{x,v}f(s)\|_\infty\bigr)Y_k(s)\,ds. \tag{129}\] 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 (129) 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 (Nirenberg 1959, 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 \(0<j<s\), \[ \|D^j g\|_{L^{2s/j}} \le C_{d,s}\|g\|_\infty^{1-j/s} \|D^s g\|_2^{j/s}. \tag{130}\] Here \(D^j g\) denotes the array of all ordered \(j\)th 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 \[A_j^{p_j} \le C_{d,s}\int |D^{j-1}g|\,|D^{j+1}g|\, |D^jg|^{p_j-2} \le C_{d,s} A_{j-1}A_{j+1} A_j^{p_j-2}.\] 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 \[b_j=\log A_j-(1-j/s)\log A_0-(j/s)\log A_s.\] 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 (130); 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 (130) 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 \[ \|D^\alpha g\,D^\beta h\|_2 \le C_s\bigl(\|g\|_\infty\|h\|_{H^s} +\|h\|_\infty\|g\|_{H^s}\bigr). \tag{131}\] Indeed, put \(j=|\alpha|\). For \(0<j<s\) apply (130) 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 \[\|a_R\|_{H^k(\mathbb R^6)}\le C_{k,R}\|F\|_{H^k(\mathbb R^3)},\qquad \|\nabla a_R\|_\infty\le C_R\|F\|_{W^{1,\infty}_x}.\] 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 (131) therefore bounds their sum by \[C_{k,R}\bigl(\|F\|_{W^{1,\infty}_x}\|f\|_{H^k} +\|F\|_{H^k}\|\nabla f\|_\infty\bigr).\] Finally, \(\|j_f\|_{H^k_x}\le C_R\|f\|_{H^k_{x,v}}\) by Cauchy–Schwarz in \(v\), and (126) completes (129). 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 (126) produce smooth iterates. Because \(H^6(\mathbb R^6)\) controls first derivatives in \(L^\infty\), the preceding estimates give, while all momentum supports lie in a fixed ball, \[Y_6^{n+1}(t)\le Y_6(0)+C_R\int_0^t \bigl((1+Y_6^n)Y_6^{n+1}+Y_6^n\bigr)\,ds.\] 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 \[D_{n+1}(t)\le C_{R,M}\int_0^t D_n(s)\,ds, \qquad D_n=\|f^n-f^{n-1}\|_2+\|F^n-F^{n-1}\|_2.\] The term involving \(f^n\) uses its uniform first-derivative bound, exactly as in (128); 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 \[Y_k^{n+1}(t)\le Y_k(0)+C_{k,R,M}\int_0^t (Y_k^{n+1}(s)+Y_k^n(s))\,ds.\] 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.

Changing the fields outside the particle region

We first construct the vacuum evolution needed for data whose bounded derivatives need not lie in \(L^2\).

Fix a finite horizon \(H>0\). Set \(\rho_0=\rho_{f_0}\) and take its Coulomb field \[ E_{\rm C}=-\nabla(-\Delta)^{-1}\rho_0, \qquad \nabla\cdot E_{\rm C}=\rho_0. \tag{132}\] 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 \(\mathbb R^3\), define \[ \mathcal A_G(x)=\left(\int_0^1 tG(tx)\,dt\right)\times x. \tag{133}\] 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 \[\nabla\times(V\times x)=2V+(x\cdot\nabla)V =\int_0^1\frac{d}{dt}\bigl(t^2G(tx)\bigr)\,dt=G(x).\] Choose \(\chi\in C_c^\infty(\mathbb R^3)\) equal to one on \(B(0,L)\), where \(L=R_0+2H+1\), and set \[ \begin{split} \overline E_0&=E_{\rm C}+ \nabla\times\bigl(\chi\mathcal A_{E_0-E_{\rm C}}\bigr),\\ \overline B_0&=\nabla\times\bigl(\chi\mathcal A_{B_0}\bigr). \end{split} \tag{134}\] 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 24 with initial data \(D(0)=(E_0-\overline E_0,B_0-\overline B_0)\). Then \[ D(t,x)=0\quad\text{if}\quad 0\le t\le H,\quad |x|\le R_0+t, \tag{135}\] because \(|x|+t\le R_0+2H<L\). Consequently \[ (f,F)\longleftrightarrow(f,F-D) \tag{136}\] 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 (135); 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 (134) 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.

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 23 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, (129) 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 (134), 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 4, 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.

The bounded-momentum criterion

The continuation input is the Glassey–Strauss criterion in (Luk and Strain 2014, 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 \[ f_{\rm LS}=f/(4\pi),\qquad E_{\rm LS}=E, \qquad B_{\rm LS}=B. \tag{137}\]

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 (134) 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 (137), 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 (125). Lemma 23 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 24 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 3.