Asymptotic stability of the relativistic Maxwellian

Solutions of the relativistic Boltzmann equation are studied for all initial data which are periodic in the space variable and near equilibrium. An equilibrium is a relativistic Maxwellian distribution of momenta. Under appropriate conditions on the scattering kernel, this equilibrium is asymptotically stable in a variety of function spaces. Introduction Although the classical (non-relativistic) Boltzmann equation has been heavily studied, the relativistic version has received scant attention. The relativistic Boltzmann euqation is V-?XF=-C(F,F). (RB\ Here the dot represents the Lorentz inner product (H ) of 4-vectors, v=(vl9v29v3)9 V=(v0,vl9v29v^9 X=(xQ9 xl9 x2, x3), x=(xl9 x29 x3\ xQ=—t and C(F, F) is the collision integral. Normalizing the speed of light c=l and the particle mass m=l9 we have K-F=1 or v0=\/l + \ v \ 2 For our purposes it is convenient to separate the time and space variables and to divide (RB\ by v0 to obtain 8tF+v-rxF=Q(F9F) (RB) where Q(F, F)=VQ C(F, F) and Communicated by H. Araki, July 22, 1992. 1991 Mathematics Subject Classification: 82 A45 * Dept. of Mathematics, Indiana University, Bloomington, IN 47405, USA ** Dept. of Mathematics, Brown University, Providence, RI 02912, USA Research supported in part by National Science Foundation grants DMS 90-23864 and DMS 90-23196 and by ARO grant DAAL-3-90-G0012. 302 ROBERT T. GLASSEY AND WALTER A. STRAUSS (0.1) . ^ \/ l+M The collision integral has the form Q(F.F)(v) = ~ \\\ d(U*-l)d(U'*-l)d(V'*-l) 2vQ J J J -U'-V) sa(s, 0) [F(u') F(v')~F(u) F(v)] d* Ud* U'd* V (0.2) where U=U*U=ul~ u, u=ulriil+ul, d is the delta function in one variable, d^ is the delta function in four variables, and all the F are evaluated at the same space-time point (t, x). Furthermore a(s, 0) is called the differential cross section or the scattering kernel; it is a function of variables s and 6 which will be defined later. The delta functions express the conservation of momentum and energy: u'+v'=u+v, (0.3) Of course, the 12-fold integral (0.2) defining Q may be reduced to a 5-fold integral by carrying out the delta function integrations (see Section 1). A relativistic Maxwellian is characterized as a particle distribution /JL(V) which minimizes the entropy subject to constant mass, momentum and energy. It is an equilibrium solution of (RB\ since Q(#, /0=0, d it has the form fi(v) = exp (a+b-v-c^/ l+v ) (0.5) where a, 6 and c> \b\ are five parameters (constants). We consider a solution F(t, x, v) of (RB\ which has period 2?r in each x variable and satisfies an initial condition F(Q, x,v)=F0(x9v). We shall assume that the initial distribution FQ(X, v) is close to a Maxwellian ju(v). The 5 parameters a, 6, c are chosen so that F° and ju have the same total mass, energy and momentum: 0 = ( ( GF°-JH) dxdv = || v(F°-v) dxdv = i( VT+v* (F°-j>t) dxdv (0.5a) where the integration is over x^£=(Q, Inf and v^R. (See Appendix I). The following is a special case of our main result. Theorem Qe Let a(s, 6) satisfy (1.16)-(1.18) below. Let F°(x, v) be a nonnegative continuous function which has period lit in each x-variable. Assume that ju(v) is a Maxwellian (0.5) and CQ is a positive constant such that RELATIVISTIC MAXWELLIAN 303 sup \F\x, v)-#(v)\ — (3+/?) is fixed. If c0 is sufficiently small, then there exist constants cl and /z>0 and a unique global continuous solution F(t, x, v) of(RB\for Q Q is a scalar function of v which represents the dissipation. In order to prove dissipation on the operator level in the sense of spectral theory, some compactness property of the solutions of (0.8) is required. Some compactness in the v variable follows from the form of K, while some compactness in the x variable follows from the fact that -^-averages of solutions of transport equations tend to be .x-smoothing. The precise condition we use is a kind of relative compactness of operators, called ^-smoothing. To a certain extent we follow the abstract approach that Shizuta [20] applied to the classical Boltzmann equation. For background on the relativistic equation we mention the books of Stewart [21] and deGroot et. al [7]. There are some analyses of (RB\ by Cercignani and Majorana [6]. The linearized relativistic equation (0.8) is solved by 304 ROBERT T. GLASSEY AND WALTER A. STRAUSS Dudynski and Ekiel-Jezewska [9], In § 1 we explicitly write the equation and derive the collision invariants and entropy inequality. Then we linearize the equation, and we formulate the function spaces Y and the main theorem. In §2 we essentially show that the (nonlinear) collision integral is bounded on Y. The relativistic energy plays an important role here. In §3 we estimate the kernel of the linearized collision operator. The relativistic estimates, which are related to those of Dudynski et. al. [9], are quite distinct from the classical ones of Grad and others. The smoothing property of the linearized equation is proved in §4, following in part the approach of Shizuta [20]. Finally in §5 we solve the nonlinear problem by iteration by making use of the exponential decay of the linearized problem. In particular, we get a classical solution of (RB\ where each term is at least a continuous function. In a succeeding paper we shall use the methods developed here to solve a relativistic Vlasov-Boltzmann system near equilibrium. We thank R. Caflisch and A. M. Anile for providing us with several references on the mathematical theory of the Boltzmann equation. § 1, Formulation We begin by defining the remaining variables in the collision integral (0.2). We define s = (U+ V) = (uQ+vQ) \u+v\ = 2uQvQ-2u°v+u 2 Q~\u\ +v0-\v\ 2 4g = _(C/-F) = -(uQ-v0) +\u-v\ = 2u0vQ-2u*v-u 2 J+\u\ -vt+\v\ = 2(v l+\u\ vl+\v\-u-v-l) = s-4, (1.2) and . (1.3) (V-U) Furthermore, we define the M011er velocity as the scalar VM given by RELATWISTIC MAXWELLIAN 305