On the multi-modal approximate solutions of non-linear Boltzmann equation (Q2761547)

This paper deals with the multi-modal approximate solutions of the nonlinear integro-differential Boltzmann equation \(D(f)=Q(f,f)\), where \(D(f)=\partial f/\partial t+V \partial f/\partial X\), NEWLINE\[NEWLINEQ(f,f)={d^2\over 2}\int_{R^3} dV_1\int_{\Sigma} d\alpha |(V-V_1,\alpha)|[f(t,V',X)f(t,V_1',X)- f(t,V,X)f(t,V_1,X)],NEWLINE\]NEWLINE \(d\) is the molecule diameter; \(t\) is the time; \(X=(X^1,X^2,X^3)\in \mathbb{R}^3\) are the particle coordinates; \(V=(V^1,V^2,V^3)\in \mathbb{R}^3\) is the particle velocity; \(f(t,V,X)\) is the unknown function of the particles distribution; \(\Sigma\) is the unit sphere in \(\mathbb{R}^3\); \(V', V_1'\) are the particles velocities after collision. The author studies the properties of a multi-modal approximate solution of the considered Boltzmann equation in the form \(f(t,v,x)=\sum_{i=1}^{m}\phi_{i}(t,x)M_{i}(v)\), \(M_{i}(v)=\rho_{i}(\beta_{i}/\pi)^{3/2}e^{-\beta_{i}(v-\bar v_{i})^2}\), \(i=1,\ldots,m\). The parameters of the Maxwellians \(M_{i}(v)\) are the following: \(\rho_{i}>0\), \(i=1,\ldots,m\) are the densities; \(\beta_{i}={1\over 2T_{i}}\), \(i=1,\ldots,m\) are the inverse temperatures; \(\bar v_{i}\in \mathbb{R}^3\), \(i=1,\ldots,m\) are the mass velocities. The goal of this paper is to find classes of distributions such that the residual NEWLINE\[NEWLINE\Delta=\sup_{(x,t)} \int_{R^3} dV|D(f)-Q(f,f)|,\quad \text{or} \quad \Delta_1=\int_{R^1} dt\int_{R^3} dx\int_{R^3} dv |D(f)-Q(f,f)|NEWLINE\]NEWLINE converges to zero.