Self-similar Asymptotics for a Modified Maxwell–Boltzmann Equation in Systems Subject to Deformations

Abstract

In this paper we study a generalized class of Maxwell–Boltzmann equations which in addition to the usual collision term contains a linear deformation term described by a matrix A. This class of equations arises, for instance, from the analysis of homoenergetic solutions for the Boltzmann equation considered by many authors since 1950s. Our main goal is to study a large time asymptotics of solutions under assumption of smallness of the matrix A. The main result of this paper is that for sufficiently small norm of A any non-negative solution with finite second moment tends to a self-similar solution of relatively simple form for large values of time. We also prove that the higher order moments of the self-similar profile are finite under further smallness condition on the matrix A.

Appendix A: Derivation of the matrix equation for B(t)

To obtain the equation for the second moments (cf. (6.11)) we compute first the left hand side of (2.6) acting over functions ${\varphi }$ with the form (6.10). Then it becomes

$$\begin{aligned}&-\frac{1}{2} \sum _{j,\ell =1}^{d} \partial _t b_{j,\ell }(t) \ k_j k_{\ell } -\frac{1}{2}\sum _{j}^{d} \sum _{r,s=1}^{d} \left( A_{r,j}b_{r,s}(t) \ k_j k_{s}+A_{s,j}b_{r,s}(t) \ k_j k_{r} \right) \nonumber \\&\quad =-\frac{1}{2} \sum _{j,\ell =1}^{d} \partial _t b_{j,\ell }(t) \ k_j k_{\ell } -\frac{1}{2} \sum _{r=1}^{d} \sum _{j,\ell =1}^{d}\left( A_{r,j}b_{\ell ,r}(t) + A_{r,\ell }b_{j,r}(t) \right) k_j k_{\ell } \end{aligned}$$

(11.1)

where we have used the symmetry of $B=(b_{j,\ell })_{j,\ell =1}^{d}$. The right hand side of (2.6) gives

$$\begin{aligned} \Gamma [{\varphi }](k)-{\varphi }(k)&= -\frac{1}{4} \int _{S^{d-1}} dn \ g(\hat{k}\cdot n) \left[ \sum _{j,\ell =1}^{d}b_{j,\ell }(t) \left( k_{j}k_{\ell }+|k|^2 n_{j}n_{\ell }\right) \right] \nonumber \\&\quad +\frac{1}{2}\sum _{j,\ell =1}^{d} b_{j,\ell }(t) \ k_j k_{\ell } + O(\vert k\vert ^p)\nonumber \\&=\frac{1}{2} \sum _{j,\ell =1}^{d}b_{j,\ell }(t) \int _{S^{d-1}} dn \ g(\hat{k}\cdot n) \left[ \frac{1}{2} k_{j}k_{\ell } -\frac{1}{2} \vert k \vert ^2 n_j n_\ell \right] +O(\vert k\vert ^p)\nonumber \\&=\frac{1}{4} \sum _{j,\ell =1}^{d}b_{j,\ell }(t) T_{j,\ell }+O(\vert k\vert ^p) \end{aligned}$$

(11.2)

where

$$\begin{aligned} T_{j,\ell }:= \int _{S^{d-1}} dn \ g(\hat{k}\cdot n) \big ( k_{j}k_{\ell } - \vert k \vert ^2 n_j n_\ell \big ). \end{aligned}$$

(11.3)

Notice that $T_{j,\ell }$ is a traceless isotropic tensor depending on k. Therefore, it has the form $T_{j,\ell }= C_0\big (k_{j}k_{\ell } -\frac{\vert k \vert ^2}{ d} \delta _ {j,\ell }\big )$ for some $C_0\in {\mathbb {R}}$.

Computing the tensor for the particular choice of the vector $k=(1,0,\ldots )$ we obtain

$$\begin{aligned} C_0\left( \frac{d-1}{d}\right) = \int _{S^{d-1}} dn \ g(\hat{k}\cdot n) \big ( 1- (\hat{k}\cdot n)^2 \big ) :=q, \end{aligned}$$

(11.4)

hence

$$\begin{aligned} T_{j,\ell }= \left( \frac{d}{d-1}\right) q \big ( k_{j}k_{\ell } - \frac{1}{d} \vert k \vert ^2 \delta _{j,\ell } \big ). \end{aligned}$$

Thus (11.2) becomes

$$\begin{aligned} \Gamma [{\varphi }](k)-{\varphi }(k)= \frac{q d}{4(d-1)} \sum _{j,\ell =1}^{d}b_{j,\ell }(t) \big ( k_{j}k_{\ell } - \frac{1}{d} \vert k \vert ^2 \delta _{j,\ell } \big )+O(\vert k\vert ^p). \end{aligned}$$

(11.5)

Thus, combining (11.1) with (11.5), we obtain

$$\begin{aligned}&\partial _t b_{j,\ell }(t) + \sum _{r=1}^{d} \left( A_{r,j}b_{\ell ,r}(t) + A_{r,\ell }b_{j,r}(t) \right) =- \frac{q d}{2(d-1)} b_{j,\ell }(t) + \frac{q}{2(d-1)} {{\,\mathrm{Tr}\,}}(B) \delta _{j,\ell } \end{aligned}$$

that can be rewritten in matrix form as

$$\begin{aligned} \partial _t B+ \big (BA +(BA)^T\big )+\frac{q d}{2(d-1)} \left( B - \frac{{{\,\mathrm{Tr}\,}}(B)}{d}I\right) =0 \end{aligned}$$

(11.6)

where $M^T$ denotes the transposed of the matrix M.