Macroscopic regularity for the relativistic Boltzmann equation with initial singularities

In this paper, it is proved that the macroscopic parts of the relativistic Boltzmann equation will be continuous, even though the macroscopic components are discontinuity initially. The Lorentz transformation plays an important role to prove the continuity of nonlinear term.


Introduction. The relativistic Boltzmann equation is written as
where F (t, x, p) is a distribution function for the gas particles at time t > 0, position x ∈ Ω = R 3 or T 3 , and particle velocity p ∈ R 3 . The collision operator C(F, F ) takes the following bilinear form Here the translation rate W (p, q|p , q ) is given by W (p, q|p , q ) = c 2 sσ(g, θ)δ 4 (p µ + q µ − p µ − q µ ), where σ(g, θ)is the scattering kernel which measures the interactions between particles and Dirac function δ 4 is the delta function of four variables. The constant c is the light speed, we nomalize c to be 1 for simplicity of presentation. The relativistic momentum of a particle is denoted by p µ ,µ = 0, 1, 2, 3. We raise and lower the indices with the Minkowski metric p µ = g µν p ν , where g µν = diag(−1, 1, 1, 1). The signature of the metric is (−, +, +, +). For p ∈ R 3 , we write p µ = (p 0 , p), where p 0 = |p| 2 + 1 represents the energy of a relativistic particle with velocity p. Using Einstein convection of implicit summation over repeated indices, then the Lorentz inner product is given by The streaming term of relativistic Boltzmann equation is given by Then the Boltzmann equation (1.1)becomes with collison operator Q(F, F ) = 1 p0 C(F, F ). Andp is the normalized particle velocity which is given byp The steady solutions of this model are the well known Jüttner solution, also known as the relativistic Maxwellian, i.e., where K 2 (·) is the Bessel function K 2 (z) = z 2 2 ∞ 1 e −zt (t 2 − 1) 3 2 dt, T is the temperature and k B is the Boltzmann constant. Throughout this paper, we normalize all the physical constants to be one, including the speed of light. Then the normalized relativistic Maxwellian becomes Now, we define the quantity s, which is the square of the energy in the "center of momentum" system, p + q = 0, as s = s(p µ , q µ ) = −(p µ + q µ )(p µ + q µ ) = −2(p µ q µ − 1).
(2) For hard potentials, where γ > −2, a ∈ [0, 2 + γ], b ∈ [0, min{4, 4 + γ}). We impose the relativistic Boltzmann equation (1) with an initial condition There are many mathematical investigations in the Boltzmann equation and relativistic Boltzmann equation. In 1940, Lichnerowicz-Marrot [16] derived the relativistic Boltzmann equation which is a fundamental model for relativistic particles whose speed is comparable to the speed of light. In 1992, Dudyński and Ekiel-Jeżewska [4] obtained the global existence of the Diperna-Lions renormalized solution of the relativistic Boltzmann equation by using their causality results [5,6] for the case of large initial data. Dudyński and Ekiel-Jeżewska [8,7] studied the linearized relativistic Boltzmann equation.
For small initial data, there are lots of results on the existence and uniqueness of global solutions to the relativistic Boltzmann equation. The local existence and uniqueness were firstly investigated by Bichteler [1] in the L ∞ framework under smallness conditions on the initial data. 1n 1993, Glassey and Strauss [10] proved the global existence of smooth solution on the torus for the relativistic Boltzmann equation, and in the meanwhile obtained the exponential decay rate was also obtained. It is noted that they [10] considered only the hard potential cases. In 1995, it was further extended to the Cauchy problem [11]. In 2006, Hsiao and Yu [14] reduced the restrictions on the cross-section of [10], but still fell into the case of hard potentials. In 2006, Glassey [12] constructed a global continuious mild solution to relativistic Boltzmann equation near vacuum. In 2010, Strain [17] proved the global existence of L ∞ -mild solution to the relativistic Boltzmann equation in n torus for the soft potentials. In 2016, by using L 1 x L ∞ p L ∞ x,p method, Wang [19] obtained the global existence of L ∞ -mild solution to relativistic Boltzmann equation even though the initial data may have large amplitude, which greatly extended the result [17].
It is an interesting problem to consider the macroscopic regularity of the Boltzmann equation. For the case with angular cut-off, it is believed that the initial singularities propagate in time since the Boltzmann equation is hyperbolic. This property was proved by Boudin-Desvillettes [2] with propagation of Sobolev H 1 25 singularity in the case near vacuum, later by Duan-Li-Yang [3] in the case near global Maxwellian. In fact, the famous averaging lemma [13] shows that R 3 F (t, x, p)ϕ(p)dp ∈ H 1 2 (t, x) for any smooth compact support function ϕ(p). This indicates that the macroscopic components like density, momentum and total energy probably have H 1 2 (t, x) regularity. However more regularity is not known from the average lemma. Recently, Huang-Wang [15] proved that the macroscopic parts of the Boltzmann equation are continous for any positive time even though it is discontinuous at the initial time.
The purpose of this paper is to investigate the regularity of macroscopic quantities of solutions to the relativistic Boltzmann equation (1) with angular cut-off even initial singularities are contained. For this, we firstly define a weight function We remark that we have to choose the exponential weight (14) but not some polynomial weight. This is mainly due to that the polynomial weight is not compatible with Lorentz transformation. For any fixed (t, x) ∈ (0, +∞) × Ω, we assume the initial data F 0 satisfies wF 0 L ∞ x,p < +∞ and We note that there is a large class of initial data satisfying (15). For example, we can choose where |P (p)| ≤ C(1 + |p|) k holds for some positive constant k ≥ 0, which is allowed to be discontinuous for p ∈ R 3 , ρ 0 (x) may also be discontinuous in x ∈ R 3 and satisfies One can check that F 0 (x, v) in (16) satisfies the condition (15), see appendix for details. The solutions considered in this paper are in the following space X: Near vacuum, F (t, x, p) ≥ 0 for a.e. (t, x, p) ∈ (0, +∞) × R 3 × R 3 , w(·)F (·, ·, ·) ∈ L m m−1 (0, T ; L 2 x,p ) and w(·)F (·, ·, ·) ∈ L m (0, T ; L ∞ x,p ) for m > 2.
One can check that X is not empty. For example, in the case of near relativistic Maxwellian, the global L ∞ -mild solutions constructed in [17,19] indeed belong to the space X. Especially, the solution of [19] allows large amplitude initial data. For the sake of completeness, we shall write down the results in [19] in appendix. Then our main results are as follows: Theorem 1.1. For any given 0 < α 1, let the initial data F 0 satisfis (15). Let F (t) ∈ X be the mild solution to the relativistic Boltzmann equation (4), (13), then the macroscopic components of solutions F (t) are continuous functions of (x, t) ∈ Ω × (0, +∞).
Moreover, if the initial data F 0 further satisfies where t 1 and T are any given times with 0 < t 1 < T < +∞, then the macroscopic components of solutions F (t) are uniformly continuous functions of (x, t) ∈ Ω × [t 1 , T ].
A few remarks are in order.
Remark 2. It is noted that the solutions constructed in [17,19], belong to X. Moreover, the solutions of [17,19] allow the initial macroscopic components to be discontinuous. If their initial data further satisfy (15), then the macroscopic quantities of these solutions are continuous in (x, t). Thus, Theorems 1.1 are selfcontained in this sense.
Remark 3. We have to use the exponential weight function w(p) = e αp0 , 0 < α 1 but not a polynomial weight. Indeed, if we use the polynomial weight w(p) = p β 0 , β > 0, then we will meet difficulty when we utilize Lorentz transformation to treat the nonlinear term Q + (F, F ). Especially, we can not bound the key quantity A(p, q) in (52).
It is noted that the relativistic Boltzmann equation is essentially transport, the solution propagates along the direction of normalized velocityp. Thus the integral with respect to p can be translated to x by changing variables. The continuity of macroscopic components is then derived from the continuity of translations on L 2 x,p . Indeed, the changing of variable is the main difficulty in the relativistic Boltzmann equation due to the complicated form of collision kernel. To realize the changing of variable, one has to use the Lorentz transformation, and this is why we need the exponential weight (14). Notations. Throughout this paper, C denotes a generic positive constant which may depend on a, b, γ, α and may vary from line to line. C(t) denotes the generic positive continuous function depending on time t > 0 and γ which also may vary from line to line. · L 2 denotes the standard L 2 (Ω x ×R 3 p )-norm, and · L ∞ denotes the L ∞ (Ω x × R 3 p )-norm.

YAN YONG AND WEIYUAN ZOU
2. Proof of the main Theorem. Let F (t, x, p) be the mild solution of the relativistic Boltzmann equation (4), then for (t, x, p) ∈ (0, +∞) × Ω × R 3 , one has where 2.1. Proof of Theorem 1.1 for Ω = R 3 . Part I. Continuity of Macroscopic Components: That is, for any ε > 0, we need to prove that there exists χ > 0, which may depend on (x, t) and ε, such that if |η| ≤ χ and |δ| ≤ χ, We assume η ≥ 0 without loss of generality and denote t η . = t + η for notation simplicity. We divide the proof into two parts. The proof is based on the initial condition (15) and the continuity of translations on L l x,p . Firstly, we introduce a lemma in [19], which will be used frequently later.

MACROSCOPIC REGULARITY FOR THE RELATIVISTIC BOLTZMANN EQUATION 951
We firstly take N large such that C ≤ ε 32 , for. On the other hand, from (15), there exists χ 1 > 0 depending only on (t, x) and ε such that if η + |δ| ≤ χ 1 , it holds that For H 12 , it follows from (22) that where we have used the fact that F ∈ X and the Lemma 2.1. Hence there exists For H 13 , it is noted that A direct calculation shows that For H 132 , one has For H 133 , it follows from Lemma 2.1 that From (11), (12), we know γ > −2, b ∈ [0, min{4, 4 + γ}), thus it holds that Using the change of variable y = x −p(t η − τ ) and Lemma 3.1 in the appendix, one can further bounded the above as where For H 1332 , it is more complicated. By the definition of X, there exists a smooth compact support function F ε (·, ·, ·) such that where C 1 will be chosen later. Using (33), we have where we have used the fact that and C 1 is chosen so large that (33) holds. Thus, from (28)-(34), there exists χ 3 > 0 depending on (t, x) and ε such that if |δ| + η ≤ χ 3 , it holds that Therefore, it follows from (23), (25), (27) and (36) that Estimation on R 3 I 2 (t, x, p)dp: It is noted that x, p)dp For H 21 , it follows from (21) that Using Lemma 2.1, we have and we have used the fact that Thus, it follows from (22) and (40) that It is straightforward to obtain that and On the other hand, for any fixed λ > 0 and N , it follows from Lemma 2.1 that For H 212 , we have We only consider H 2121 , since H 2122 can be treated similarly. It is noted that Noting (41), a direct calculation shows that It follows from (30)that To estimate H 21213 , we need to use the changing of variables to translate the continuity of macroscopic into the continuity of translations in L 2 x,p . But it is difficult to realize this process since the collision kernel is very complicated in the relativistic Boltzmann equation. And it is in this part that we need the exponential velocity weight w(p) but not a polynomial weight.
= |p|, it follows from (56) that Using the above relations, one obtains that where we have used which is due to k < 3 2 , (59) and Lemma 3.5. Combining (39)-(49) and (60), there exists χ 4 > 0 depending only on (t, x) and ε such that if |δ| + η ≤ χ 4 , it holds that It remains to estimate H 22 . We first divide it into several parts.

YAN YONG AND WEIYUAN ZOU
By the same arguments as in (24), one has that For the second term on the right hand side of (72), it follows from (20) that there exists χ 1 > 0 depending only on t −1 1 , T and ε such that if η + |δ| ≤ χ 1 , one has that which, together with (72), yields that On the other hand, by the same arguments as in (26)-(36), one can prove that there exists χ 2 > 0 depending only on t −1 1 , T and ε such that if η + |δ| ≤ χ 2 , it holds that Finally, by similar arguments as in (38)-(69), one can prove that there exists χ 3 > 0 depending only on t −1 1 , T and ε such that if |δ| + η ≤ χ 3 , it holds that Hence, taking χ = min{χ 1 , χ 2 , χ 3 }, it follows from (73), (74) and (75) that Thus R 3 F (t, x, p)dp is uniformly continuous in (x, t) ∈ R 3 × [t 1 , T ]. By similar arguments, one can prove that R 3 pF (t, x, p)dp and R 3 p 0 F (t, x, p)dp are also uniformly continuous in (x, t) ∈ R 3 × [t 1 , T ]. Therefore we proved that the macroscopic components of relativistic Boltzmann equation are uniformly continuous in (x, t) ∈ Ω × [t 1 , T ].

2.2.
Proof of Theorem 1.1 for Ω = T 3 . For Ω = T 3 , most of the proof is similar to the case of Ω = R 3 . Here we only point out some differences. Firstly, in the period case, one should pay attention when using change variable x → y := x −p(t η − τ ). For example, we consider the change of variable in H 1331 : On the other had, to bound H 1332 , one can choose a smooth function F ε (t, x, p) (may not has compact support) which is periodic in variable x such that t 0 w(F (τ, ·, ·) − F ε (τ, ·, ·)) m m−1 Then, using (78) and by similar arguments in the case of Ω = R 3 , one can prove the continuity of macroscopic components of solution to the relativistic Boltzmann equation.
3. Appendix A. Proof. where Then we have that Lemma 3.2. The example given in (16) satisfies the condition (15) and (20).
Proof. Using (17), it is obvious that wF 0 L ∞ ≤ C < ∞. Next we prove the second part of (15). Given any N > 0, we only need to prove that for any small ε > 0 there exists χ > 0 depending on t, ε such that if |η| + |δ| ≤ χ, it holds that For any fixed t > 0, it is noted that We assume η ≥ 0 for simplicity of presentation. By changing y = x −p(t + η) and using Lemma 3.1, we have where we have used the continuity of translations on L l . Thus there exists χ 1 > 0 depending only on t > 0 and ε > 0 such that if η + |δ| ≤ χ 1 , it holds that For L 2 , it is straightforward to get that |ρ 0 (x −pt −pη) − ρ 0 (x −pt)|e − p 0 4 dp.
Thus the example given in (16) also satisfies the condition (20).
4. Appendix B. We introduce the existence of L ∞ -mild solutions to relativistic Boltzmann equation near relativistic Maxwellian. Recently, Wang [19] obtained the global L ∞ -mild solution to the relativistic Boltzmann equation for a class of large initial data. To introduce the result of [19], we denote It follows by a direct calculation that E(F (t)) ≥ 0 for all t ≥ 0. Note, in particular, that E(F 0 ) ≥ 0 holds true for any function F 0 (x, p) ≥ 0.
For Ω = T 3 , from L ∞ -estimate (85), it is easy to know that F (t, x, p) ∈ X in the case near relativistic Maxwellian. On the other hand, for Ω = R 3 , using (85) and the energy estimates, one can get easily that where C depends only on the initial data. Thus we also have F (t, x, p) ∈ X in the case Ω = R 3 . It is noted that the solution constructed in Proposition 1 allows initial macroscopic singularities.