THE RELATIVISTIC VLASOV-MAXWELL-BOLTZMANN SYSTEM FOR SHORT RANGE INTERACTION

. We are concerned with the Cauchy problem of the relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. For pertur- bative initial data with suitable regularity and integrability, we prove the large time stability of solutions to the relativistic Vlasov-Maxwell-Boltzmann sys- tem, and also obtain as a byproduct the convergence rates of solutions. Our proof is based on a new time-velocity weighted energy method and some op- timal temporal decay estimates on the solution itself. The results also extend the case of “hard ball” model considered by Guo and Strain [Comm. Math. Phys. 310: 49–673 (2012)] to the short range interactions.

1.1. The Cauchy problem. We consider the following relativistic Vlasov-Maxwell-Boltzmann system (1.1) The self-consistent electromagnetic field satisfies the Maxwell system Here F ± = F ± (x, p, t) are the number density functions for ions (+) and electrons (−) at the phase-space position (x, p) = (x 1 , x 2 , x 3 , p 1 , p 2 , p 3 ) ∈ R 3 × R 3 , at time t ∈ R + , and E(t, x), B(t, x) are the electric and magnetic fields, respectively. p 0 = 1 + |p| 2 is the energy of a particle, here and in the sequel, we denotep = p p0 . The initial data of the coupled system above are given by F ± (x, v, 0) = F 0,± (x, p), (E, B)(x, 0) = (E 0 (x), B 0 (x)), (1.3) satisfying the compatibility conditions The relativistic Boltzmann collision operator Q(·, ·) in (1.1) takes the form of Here the "transition rate" W = W (p, q|p , q ) is defined as The delta functions express the conservation of momentum and energy: p + q = p + q, p 0 + q 0 = p 0 + q 0 . (1.6) The quantity s in (1.5) is the square of the energy in the "center of momentum", p + q = 0, and is given as s = s(p, q) = −(p 0 + q 0 , p + q) (p 0 + q 0 , p + q) = 2(p 0 q 0 − p · q + 1) ≥ 4 where denotes the Lorentz inner product, which is given by (p 0 , p) (q 0 , q) = −p 0 q 0 + p · q.
And the relativistic momentum g in (1.5) is denoted The angle θ in (1.5) is then given by (1.7) The relativistic differential cross section σ(g, θ) depends only on the relative momentum g and the deviation angle θ. Here, we use the following short rang interaction form cf. [13,31] σ(g, θ) = C θ s , (1.8) where C θ is a constant.
In the next sub-sections, we will present the reductions of the collision operator (1.4).
The post collision momentum in the expression (1.9) can be written: (1.10) The energies are then (1.11) The angle θ in the reduced expression (1.9) is defined by where v ∈ R 3 has a complicated expression as given in [37] but its precise form will be inessential. Now we turn to the expression given by Glassey-Strauss in [17].
1.3. Glassey-Strauss reduction. Glassey-Strauss have derived an alternative form for relativistic collision operator without using the center-of-momentum. We will skip their argument and write down the result as follows. (1.12) where the kernel is In this expression, the post collisional momentum are given as follows . and the energies can be expressed as These formula clearly satisfy the collisional conservations (1.6). The angle (1.7) in (1.12) can then be reduced to Moreover, assuming the collisions are elastic as in (1.6), we have the invariance ω · (p 0 q − q 0 p) = ω · (p 0 q − q 0 p ) Therefore, for fixed ω ∈ S 2 , B(p, q, ω) = B(p , q , ω). The Jacobian for the transformation (p, q) → (p , q ) in these variables [16] is Then, it is easy to see that where (p , q ) on the right hand side are given by (1.10) and G(p, q, p , q ) : R 3 × R 3 × R 3 × R 3 → R is a given function. Moreover the Jacobian (1.13) effectively works for (p , q ) in (1.10) as dωv(p, q)σ(g, θ)G(p , q , p, q).
(1.15) 1.4. Reformulation, weight functions and norms. We now turn to the presentation of our main result. The global relativistic Maxwellian (the Jöttner solution) is given by We set the perturbation in a standard way Use [·, ·] to denote the column vector in Then the Cauchy problem (1.1), (1.2), (1.3) can be reformulated as satisfying the compatibility condition Here, ζ = diag(1, −1), ζ 1 = [1, −1], and the linearized collision term Lf and the nonlinear collision term Γ(f, f ) are respectively defined by with As in [19], the null space of the linearized operator L is given by Let P be the orthogonal projection from L 2 one can decompose f uniquely as f = Pf + {I − P}f with Pf = [P + f, P − f ] and Here, the coefficient functions [a ± , b, c](t, x) are determined by f in the way (3.2).
In what follows, we introduce the weight functions and norms used throughout the paper. First of all, we define The relativistic Vlasov-Maxwell-Boltzmann system is one of the central equations in relativistic collision kinetic theory. Standard references which discuss relativistic kinetic theory include [1,2,9,12,14,15,16,17,18,22,23,24,26,35,37,32,38,39,45]. We mention some works related to the topic in the paper. When the relativistic effects are not considered, Guo [20] and Strain [30] obtained the global classical solutions of the Vlasov-Maxwell-Boltzmann system with angular cutoff on torus and in the whole space, respectively. Then, the rate of convergence to Maxwellians with any polynomial speed in large time was shown by Guo-Strain [33]. For the Cauchy problem in the whole space, the large-time behavior of classical solutions in the situation of both cutoff and non-cutoff potentials were studied by Duan-Strain [10] and Duan-Liu-Yang-Zhao [7], respectively. And recently, Duan-Lei-Yang-Zhao [5] investigated the the time decay rates of the cutoff Vlasov-Maxwell-Boltzmann system with very soft potential.
However, for the relativistic Vlasov-Maxwell-Boltzmann system, to our best, much less is known. Recently, Guo-Strain [21] studied the global classical existence of the Vlasov-Maxwell-Boltzmann system with "hard ball" condition. Compared with the non-relativistic Vlasov-Maxwell-Boltzmann system, the main difficulty is created by the collision "cross section" and the post collision momentum. Due to the complexity and singularity of both the "cross section" σ(g, θ) and the post collision momentum, it is very hard to study the global classical existence of the system with general "cross section". More precisely, unlike the non-relativistic Vlasov-Maxwell-Boltzmann system for the non-hard potential [7,5], one can not directly take momentum derivatives to the collision operators, which must be considered in the investigation of the classical solutions. It should be pointed out that Guo and Strain [21] used two alternative forms of the collision operator to eliminate the singularities caused by the momentum derivatives. The main purpose of the paper is to extend the case of "hard ball" model considered by  to the short range interactions and to obtain the time decay rates of the solutions.
The proof of Theorem 1.1 is based on an interplay of the velocity weighted energy method developed in [19,34,4,6,7,8], and also some techniques for obtaining the optimal temporal decay rates used in [41,42,43,44,40]. Unlike the "hard ball" model studied by , the dissipation of the linearized relativistic Boltzmann collision operator for the short range interaction is weaker in the sense that it is degenerate in the large-velocity domain, which is much similar to the linearized non-relativistic Boltzmann operator with non-hard potentials [34]. To overcome this problem, as it is shown in [19,34,4,6,7,8], one solvable way is to introduce the velocity-weight when performing the energy estimates, so as to balance the possible velocity-growth coming from the transport term of the original Boltzmann equation. Nevertheless, compared to the non-relativistic Vlasov-Maxwell-Boltzmann system with non-hard potentials [7,5], one of the main contribution of the paper is that we do not impose any weight to the highest order derivatives of the solution, since the bad term can be controlled by C E L ∞ |β|=N ∂ β f 2 and E L ∞ enjoys the time decay rates (1 + t) −3/2 . This phenomenon can be regarded as a kind of the relativistic effects and do not take place in the case of the non-relativistic Vlasov-Maxwell-Boltzmann system with non-hard potentials where there is always a growth in velocity. Thanks to this observation, we can avoid the trouble coming from the regularity-loss of the solution, cf. [7,5,25]. Another important idea of the paper is that we design a new time weighted energy norm X(t) and apply some crucial Sobolev inequalities (see (4.4)) to deduce the time decay rates of the every order derivative of the solution.
The rest of the paper is arranged as follows. In Section 2, we carry out the weighted estimates on Γ and L. In Section 3, we obtain the time-decay property of the linearized homogeneous system. In Section 4, we shall prove series of lemmas to obtain the closed estimate on X(t)-norm so as to conclude the proof of Theorem 1.1 basing on the standard continuity argument.
Notations. Throughout this paper, C denotes some generic positive (generally large) constant and λ denote some generic positive (generally small) constant, where C and λ may take different values in different places. A B means that there is a generic constant C > 0 such that A ≤ CB. A ∼B means A B and A B. We use L 2 to denote the usual Hilbert spaces L 2 = L 2 x,p or L 2 x with the norm · , and use (·, ·) and ·, · to denote the inner product over L 2 x,p and L 2 p , respectively. The mixed velocity-space Lebesgue space will be used to record spatial and velocity derivatives, respectively. And The length of α is |α| = α 1 + α 2 + α 3 and similar for |β|.

2.
Weighted estimates on Γ and L. This section is concerned with the weighted estimates on Γ and L with respect to the weight w (p). Many of these estimates are similar to those in [21].
In light of (1.8), for scalar functions g 1 , g 2 and h, we use the following notations , one can see that the above operators T , L , ν, and K have the following expressions: With the above notations, recalling (1.14) and (1.15), it is straightforward to see 2.1. Weighted estimates on Γ. In order to make the weighted estimates on (1.16), particularly on L and Γ, we will split the desired estimate into two different cases. Those cases correspond to the following two different integration regions To make our presentation more clear, we also introduce the smooth test function By the splitting we split T (g 1 , g 2 ) = T 2,A + T 1,A using the definitions of T 1 and T 2 as (2.1)

SHUANGQIAN LIU AND QINGHUA XIAO
By these important decomposition, we will deduce the main estimate: This lemma will follow directly from the later lemmas below.
2.1.1. Estimates in the Glassey-Strauss frame T 2,A . To avoid taking derivatives for the singular factor of |ω · (p −q)| inside B(p, q, ω) for T 2,A , we introduce the following change of variables q → u (for fixed p) as in [21]: According to [21], one also have We next express T 2,A as Then we have where the sum is over β 0 + β 1 + β 2 = β, and Also ∂ p,q β0 denotes the mixed partial derivatives with respect to variables p and q, µ β1 β2 and µ βi (i = 0, 2) are the terms which result from applying the chain ruler to the post-collisional momentum (p , q ) and momentum q respectively. Here µ β1 β2 and µ β2 contain the sum of products of high order p-momentum derivatives of smooth functions (p , q ) and q. The next step is to reverse the change of variables q → u, after the change of variables u → q: Proof. The proof of this lemma can be done in the same way as Lemma 1 in [21]. We omit the details for brevity.
Corresponding to Lemma 2 in [21], one has Lemma 2.3. On the set A, it holds that Moreover, from the above estimate one can deduce the following uniform upper bound Center of momentum frame. In this subsection, we turn to prove the estimates for T 1,A . By a change of variable p − q → u and using the product ruler as well as a reverse change of variables, we have Here µ β1 β2 is the collection of sums of product of momentum derivatives of p and q , from (1.10), which result from the chain rule of differentiation. Again the sum is over β 0 + β 1 + β 2 = β. As in [21], one has: With the estimates above, we show that Lemma 2.5. For any |β| ≥ 0, we have the uniform estimate: Moreover, for − |β| ≥ 0, one has Proof. The first estimate (2.2) follows immediately from Lemma 2.4. Now we show that the second estimate holds. From (2.2), one has the upper bound of The second term clearly has the desired upper bound by using the Cauchy-Schwartz inequality. For the first term, notice that the expressions of p 0 , q 0 in (1.11) implies . We now use the Cauchy-Schwartz inequality to obtain the upper bound Once again the pre-post change of variables from (1.15) yields the desired estimate. This completes the proof of Lemma 2.5.
With Lemma 2.1 in hand, it is quite easy to show that Lemma 2.6. Let N ≥ 9, |α| + |β| ≤ N . Then, it holds that for any smooth function ζ(p) exponentially decay in p, 2.2. Weighted estimates on L. In this subsection, we deduce some weighted estimates on the linearized collision operator L with respect to the weight w (p). we now give the basic estimates for ν and K whose exact expressions are given at the begging of this section.
Lemma 2.7. Under the condition of (1.8), it holds that Moreover, for |β| ≥ 1, we have Proof. We only prove (2.4), the estimate for (2.3) is similar to that of Lemma 3.1 in [36]. Notice that On the other hand, we have from (2.1) that This completes the proof of Lemma 2.7.
With Lemma 2.7 in hand, we can obtain Lemma 2.8. Let |β| ≥ 1. For any small η > 0, it holds that By the same arguments used in Proposition 8 of [21], we obtain Lemma 2.9. Let |β| ≥ 0. For any η > 0, there exists C η > 0 such that Basing on Lemmas 2.8 and 2.9, one has Lemma 2.10. For any η > 0, there exists C η > 0 such that where |β| ≥ 1 and l − |β| ≥ 0. For |β| = 0, one also has | w 2 L (h), h | |w h| 2 ν − C|h| 2 ν . Therefore, from the definitions of the operators L and L, we can obtain 3. Linearized analysis. Consider the Cauchy problem on the linearized relativistic Vlasov-Maxwell-Boltzmann system with a source: initial data [f 0 , E 0 , B 0 ] satisfy the compatibility condition and the source term S is assumed to satisfy To consider the solution to the Cauchy problem (3.1), for simplicity, we denote Then the solution to the Cauchy problem (3.1) can be formally denoted by where A(t) is the linear solution operator for the Cauchy problem on the linearized homogeneous system corresponding to (3.1) in the case S = 0.
3.1. Macro structure. Before continuing our investigation of the system (3.1), we introduce the notation for some integrals as follows Recalling the definition of Pf , we see that Taking velocity integrations of (3.1) 1 with respect to the velocity moments Define the high-order moment functions Θ(f ± ) = (Θ ij (f ± )) 3×3 and where A 1 and A 2 satisfy respectively. Further taking velocity integrations of the first equation of (3.1) 1 with respect to the above high-order moments one has where Here we have used (3.3).
In particular, for the nonlinear system (3.1), the non-homogeneous source S = [S + (t, x, p), S − (t, x, p)]takes the form of Then, it is straightforward to compute from integration by parts that  0 . Then, for any multiindex α with |α| = m and l 1 > m+ 3 2 , l 1 ≥ j, the first part U 1 of the solution to the linearized homogeneous system satisfies for any t ≥ 0, and the second part of the solution U 2 corresponding to the solution of the linearized homogeneous system with vanishing initial data satisfies

(3.5)
Proof. Basing on the analysis in subsection 3.1, we can apply the same arguments in [4] to have for any l ≥ 0 that ∂ t E l (Û 1 (t, k)) + λD l (Û 1 (t, k)) 0, (3.6) whereÛ 1 (t, k) denotes the Fourier transform of U 1 with respect to x and k is the corresponding variable after Fourier transform. E l (Û 1 (t, k)) and D l (Û 1 (t, k)) are functionals given by Here the constants λ 0 , λ 1 , λ 2 > 0 are sufficiently small and E int (t, k) is a time- From the definitions of E l (Û 1 (t, k)) and D l (Û 1 (t, k)) , we have D l (Û 1 (t, k))χ |k|≤1 |k| 2 E l−1 (Û 1 (t, k))χ |k|≤1 |k| 2 µ l−1f Then for l 1 > m + 3 2 , noticing the fact The estimates above imply for l 1 > m + 3 2 . For the case |k| ≥ 1, in a similar way, one has Then we can obtain (3.11) For the term I 1 , we have the upper bound As for the term I 2 , it is straightforward to see that Plugging the estimates of I 1 , I 2 into (3.11) gives the desired estimate (3.5). This ends the proof of Lemma 3.1.

4.
Global a priori estimates. In this section, we are going to prove Theorem 1.1, the main result of the paper. The key point is to deduce the uniform-in-time a priori estimates on solutions to the relativistic Vlasov-Maxwell-Boltzmann system where the nonlinear term S = [S + , S − ] is given by For this purpose, let (f, E, B) be a smooth solution to (4.1) over the time interval 0 ≤ t ≤ T with initial data (f 0 , E 0 , B 0 ) for some 0 < T ≤ ∞, and further suppose that (f, E, B) satisfies where X(t) is given in (1.19) and the constant δ > 0 is sufficiently small. What we want to do in the following is to deduce some a priori estimates on (f (t, x, p), E(t, x), B(t, x)) based on the a priori assumption (4.3). To make the presentation easy to follow, we divide this section into several subsections and the first one is concerned with the macro dissipation of the relativistic Vlasov-Maxwell-Boltzmann system.

Macro dissipation.
We first define the macro dissipation D N,mac (t) by With the above macro structure of the system (4.1) in hand, we have Proof. Basing on the previous works [10] and [11], it is a quite standard process to obtain the above estimates. We hence omit the details for brevity.

4.2.
Uniform spatial energy estimate. In this subsection, we derive the basic energy estimates on the spatial derivatives in E N,l (t). Now we state our main result in this subsection as follows.

Lemma 4.2.
There exist suitably small positive constants κ 1 , κ 2 and κ 3 such that (4.5) Proof. The proof is divided into three steps.
Step 1. Spatial energy estimates without any weight. From (4.1), it is straightforward to establish the energy identy: Then Lemma 2.10 implies where I 3 = |α|≤N |(∂ α S, ∂ α f )|. We claim that We first consider the estimate of I 3 corresponding to Γ(f, f ) in S. By using Lemma 2.6, it is straightforward to see that it is bounded up to a generic constant by E N, (t)D N, (t). For the zero-order term related to the electromagnetic field, one has For the term related to (4.8) Now we estimate I 3,1 and I 3,2 as follows. For the term I 3,1 , since from which and the Sobolev inequality in R 3 it follows that N, (t). Similarly, I 3,2 can be bounded by N, (t)D N, (t). Combining the estimates of I 3,1 , I 3,2 and (4.8), we arrive at N, (t)D N, (t). In a similar way, one has Putting the above estimates into I 3 , we see that the bound (4.7) is valid.
Step 2. Higher order spatial energy estimates with weight. Now we turn to do the weighted energy estimate on ∂ α f with 1 ≤ |α| ≤ N − 1. From (4.1) and Lemma 2.10, one has where I 4 = 1≤|α|≤N −1 ∂ α S, w 2 −|α| ∂ α f . We now prove that Similar to the estimate of I 3 , by using Lemma 2.6, the corresponding weighted estimate of Γ(f, f ) can be bounded up to a generic constant by E Next, we treat the estimate related to ζ(E +p × B) · ∇ p f . Noticing that for |α − α 1 | ≥ 2. We get from Hölder's inequality and Sobolev's inequality that Then (4.10) follows from the above estimates.
Step 3. Zero energy estimates on the micro component with weight.
With Lemmas 4.2 and 4.3 in hand, we make a proper linear combination of (4.5) and (4.13) Recalling the definitions of E N, (t) and X(t) and using the smallness assumption of X(t) in (4.3), we can obtain d dt 4.4. Decay of pure spatial derivatives. In this part, basing on the definition of X(t) and the estimates in the previous subsections, we will derive the time-decay of pure spatial derivatives ∂ α (f, E, B) with |α| ≤ N − 1. Before this, we cite the following calculus inequalities in the Sobolev spaces, which will be used frequently.
The main result of this subsection can be stated as follows: Proof. Recalling the definition of r m , we first prove that To verify (4.23), we will deduce the decay rates of the N * −th and zeroth order derivatives of solution explicitly then obtain the corresponding decays rates of the other ones by the method of interpolation. For this, we start from the following mild form which denotes the solution U = [f, E, B] to the Cauchy problem on system (3.1) with the nonlinear term S given in (4.2) and initial data U 0 = [f 0 , E 0 , B 0 ]. Then for m = N * , it follows from Lemma 3.1 that 2 Y 0 + I 5,1 + I 5,2 + I 5,3 .

(4.24)
Here we have taken and denoted In view of Lemmas 2.6, (4.2) and (4.21), we use the definition of X(t) to obtain Continuing, we set which yields Therefore, we can further bound I 5,3 by for N ≥ 9 and N * = N −3

3
. Plugging the estimates of I 5,1 , I 5,2 and I 5,3 into (4.24) gives Applying (4.25) and (4.26), we get from the interpolation inequality that for 1 ≤ m ≤ N * , It remains now to prove (4.22) in the case of N * + 1 ≤ m ≤ N − 1. To do this, we first establish the following estimate: In fact, going back to (4.1), the direct energy estimate on ∂ α f with |α| = N − 1 gives and |α|=N −1 Moreover, one has Combining the above estimates, we have Multiplying this inequality by (1 + t) yields We integrate this inequality over [0, t] for 0 ≤ t ≤ T and use (4.21) to obtain (4.28). Once (4.28) is obtained, we then get from (4.25) that where 0 < < 1 and 0 ≤ m ≤ N * − 1.
Proof. For |α| = m with 0 ≤ m ≤ N * − 1, apply ∂ α of (4.11) and (4.1) 2 and take the direct energy estimate to have ( . Let 0 < ρ < 1, define the low-velocity domain and the corresponding high-velocity domain: Then it follows that We get from the above inequality, (4.21) and Lemma 4.5 that for m ≤ N * − 1. Plugging (4.38) into (4.37), we arrive at which gives the desired estimate (4.30). This completes the proof of Lemma 4.6.
Since Y 0 is sufficiently small, (1.20) holds true. The global existence follows further from the local existence (cf. [32,21]) and the continuity argument in the usual way. This completes the proof of Theorem 1.1.