Boundary value problem for the Burgers system

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.