Solvability of a free noncompact boundary problem for a stationary Navier-Stokes system. I (Q803418)

The auxiliary results which have been used by the first author [Litov. Mat. Sb. 28, No. 2, 315--333 (1988; Zbl 0657.76030)] are proved. Namely, in the plane domain \(\Omega\) bounded in the lines \(S_ 1=\{\bar x\in {\mathbb R}^ 2:\) \(x_ 2=0\}\), \(S_ 2=\{\bar x\in {\mathbb R}^ 2:\) \(x_ 1<0\), \(x_ 2=h>0\}\), \(\Gamma =\{\bar x\in {\mathbb R}^ 2:\) \(x_ 1=x(s)\), \(x_ 2=y(s)\}\) the following boundary value problem is considered \[ - \nu\nabla^ 2\bar v+ (\bar v\cdot\nabla)\bar v+ \nabla p= -\nabla G,\;\nabla\cdot\bar v=0,\;\bar x\in \Omega, \] \[ \bar v|_{S_ 1\cup S_ 2}=0,\;\bar v\cdot \bar n|_{\Gamma}=0,\;{\bar \tau}\cdot T(\bar v)\bar n|_{\Gamma}=0, \] where \(\nu=\)const\(>0\) is the viscosity coefficient, \(\bar n\) and \({\bar \tau}\) are the normal and the tangential unit vectors to \(\Gamma\) respectively, \(T(\bar v)\) is the tensor with components \(T_{ij}=\partial v_ i/\partial x_ j+\partial v_ j/\partial x_ i\) \((i,j=1,2)\), \(G=g(-x_ 1\sin \alpha +x_ 2\cos \alpha)\), \(g=\)const\(>0\), \(\alpha=\)const\(>0\), \(\nabla=(\partial /\partial x_ 1,\partial /\partial x_ 2).\) The solvability of this problem in weighted Hölder spaces is proved and the estimates of solutions are established. The solvability of analogous boundary value problem for the Stokes system is proved too.