Remarks on the damped stationary Euler equations (Q808326)

The following boundary value problem is investigated in a bounded simply connected plane domain \(\Omega\) with smooth boundary \(\Gamma\) \[ \epsilon \bar u+R(\bar u\cdot \nabla)\bar u+\nabla p+{\bar \phi}=f\text{ in } \Omega, \] \[ div \bar u=0,\quad \frac{\partial \phi_ 2}{\partial x_ 1}-\frac{\partial \phi_ 1}{\partial x_ 2}=u_ 2,\quad div {\bar \phi}=0\text{ in } \Omega;\quad \bar u\cdot \bar n|_{\Gamma}=0,\quad {\bar \phi}\cdot \bar n|_{\Gamma}=0. \] Here \(\bar u=(u_ 1,u_ 2)\), \({\bar \phi}=(\phi_ 1,\phi_ 2)\), \(\bar n\) is the unit outward normal to \(\Gamma\), \(\epsilon >0\), \(R\geq 0\) are given constants, \(\bar f\) is a given vector field. The author concludes on the base on results for elliptic systems the existence of solution of the problem for any \(\bar f\in H^ 1(\Omega)\) such that \(\bar u\in H^ 1(\Omega)\), \(\phi \in H^ 2(\Omega)\), \(p\in W^{1,q}(\Omega)\), \(q\in (1,2)\). The case of nonhomogeneous boundary condition \(\bar u\cdot \bar n|_{\Gamma}=g\) is considered. The unique solvability of the problem in classes \(H^ k\) are proved for vector fields \(\bar f\) with sufficiently small norms by the linearization method. It is asserted (without proof) that this result holds true for the three-dimensional case. There are some misprints in the text of the paper.