Boundary value problem for the Burgers system

N. N. Frolov

Matematičeskie zametki, Tome 62 (1997) no. 6, pp. 921-932

Résumé

We consider the boundary value problem $$ \begin{gathered} \operatorname{div}(\rho V)=0,\qquad\rho|_{\Gamma_1}=\rho_0, \\ \rho(V,\nabla V)=\nu\Delta V,\qquad V|_\Gamma=V^0 \end{gathered} $$ for a vector function $V=(V_1,V_2)$ and a scalar function $\rho\ge0$ in a rectangular domain $\Omega\subset\mathbb R^2$ with boundary $\Gamma$. Here $$ \Gamma_1=\{x\in\Gamma: (V^0,n)0\},\qquad V_1^0|_\Gamma>0,\qquad\nu=\operatorname{const}>0. $$ We prove that this problem is solvable in Hölder classes.