11:00 am Wednesday, October 19, 2005
Mathematical Physics: Self-similar solutions of the Boltzmann-type equations for Maxwell models by A.V.Bobylev (Karlstad University, Sweden; and Keldish Institute, Moscow.) in RLM 12.166
We present a review of recent results obtained jointly with Carlo Cercignani and Irene Gamba. We consider the spatially homogeneous non-linear Boltzman equation (BE) for elastic and inelastic Maxwell models and some generalizations of this equation. We study the existence of self-similar solutions of the form f(v,t)= A(t)F(vexp(-at)), a=const. and address issues of large time asymptotics for some classes of the initial data. Moreover, we constructed two such solutions for the classical (elastic) BE in explicit form and prove the existence of such solutions for any a>0 and that they are asymptotic states for some classes of the initial data having infinite second moment (energy). A by-product we found sufficient conditions under which the Krook-Wu conjecture ('77) is true and large time asymptotics is described by BKW mode. In case of inelastic Boltzmann equation, we proved a recent Ernst-Brito conjecture (2002): any solution having at t=0 the finite moment of an order p>2 has the self-similar large time asymptotics (this result was obtained joinly also with G.Toscani), whose self-similar solution has a power-like high energy tail for all values of the restitution coefficient. Some very recent related results are also discussed. Submitted by
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