Some Estimates on the Boltzmann Collision Operator.pdf

Some Estimates on the Boltzmann CollisionOperatorJens StruckmeierDepartment of MathematicsUniversity of KaiserslauternGermanyAbstractThe paper presents some new estimates on the gain term of the Boltzmann collisionoperator. For Maxwellian molecules, it is shown that the L1{norm of the gain termcan be bounded in terms of the L1 and L1{norm of the density function f . In thecase of more general collision kernels, like the hard{sphere interaction potential, thegain term is estimated pointwise by the L1{norm of the density function and the lossterm of the Boltzmann collision operator.1 IntroductionThe Boltzmann equation for monoatomic gases given in the formft + vrxf = Q(f) (1:1)is a nonlinear transport equation which describes the time evolution of a rareed gas. In(1.1), the collision operator Q(f) can be expressed in the formQ(f) = Q+(f) fL(f)with L(f) = ZIR3 ZS2+ k(kv vk; n)f(v)dndv ;Q+(f) = ZIR3 ZS2+ k(kv vk; n)f(v0)f(v0)dndvHere, k(kv vk; n) denotes some appropiate collision kernel and the postcollisional ve-locities are given by the collision transformation (written explicitely in the next section).The models for the collision kernel used in the following are of the formk(kv vk; n) = kv vk cos 1 ; (1:2)1 where (v v; n) = cos 1.In the following two sections we derive two new estimates for the gain term Q+(f),which are obtained without introducing any kind of truncation. In Section 2 we considerthe case of Maxwellian molecules and prove the estimatekQ+(f)k1  Ckfk1kfk1 ;which is based on an appropiate transformation of Q+ in the center of mass system. Themore general form of the collision kernel as given in (1.2) is discussed in Section 3. Here,it is obvious that Q+ is { in general { unbounded in L1 (see Remark 3.4) and we provea pointwise estimate of Q+ in the formQ+(f)(v)  Ckfk1L(f)(v) 8 v 2 IR3 :2 Estimate for Maxwellian MoleculesIn this section we are concerned with the gain term Q+(f) of the Boltzmann collisionoperator for Maxwellian