On existence of a global solution to the initial-boundary value problem for the Boltzmann equation
A. Sakabekov
Abstract:
For the initial-boundary value problem
\begin{gather*}
\frac{\partial f}{\partial t}+\biggl(v,\frac{\partial f}{\partial x}\biggr)=I(f,f), \quad (t,x,v)\in(0,T]\times G\times R_3^v,
\\
f(t,x,v)|_{t=0}=f^0(x,v), \quad (x,v)\in G\times R_3^v,
\\
f(t,x_{\partial G},v)=g(t,x_{\partial G},v), \quad (v,n_{\partial G})<0,
\end{gather*}
the existence of a global solution in the space
$C([0,T]$;
$L^1(G\times R_3^v))$ $\forall\,T<\infty$ is proved under the conditions
\begin{gather*}
f^0\in L^1\bigl(G\times\mathbf{R}_3^v\bigr), \quad f^0\ge0,
\\
\int_{G\times\mathbf{R}_3^v}f^0(1+|v|^2+|{\ln f^0}|)\,dx\,dv<\infty, \quad |v|g\in C\bigl([0,T];\,L^1\bigl(\partial G\times\mathbf{R}_3^-\bigr)\bigr), \quad g\ge0,
\\
\sup_{t\in[0,T]}\int_{\partial G\times\mathbf{R}_3^-}|v|g(1+|v|^2+|{\ln g}|)\,dx\,dv<\infty, \quad B\in L^1\bigl(S_2\times\mathbf{R}_3^v\bigr), \quad B\ge0,
\end{gather*}
where
$G$ is a bounded convex domain in
$\mathbf{R}_3^x$ with boundary
$\partial G$;
$B$, the collision cross section;
$S_2$, the unit sphere; and $\mathbf{R}_3^-=\bigl\{v\in\mathbf{R}_3^v:(v,n_{\partial G})<0\bigr\}$.
UDC:
517.958 Received: 10.10.1991
Fulltext:
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