On the smoothness of generalized solutions of the Dirichlet problem for the Navier-Stokes equation in non-smooth two-dimensional domains (Q1825363)

The author considers the following equation (derived from the Navier- Stokes equations) in a nonsmooth domain of two-dimensional space: \[ \nu \Delta \Delta u+u_{x_ 1}\Delta u_{x_ 2}-u_{x_ 2}\Delta u_{x_ 1}=f,\quad \nu =const, \] with boundary conditions \(u=0\), grad u\(=0\). Under some geometric assumptions on the domain the following a priori estimates for the generalized solutions are proved: \[ \| | x|^{-b}| D^ 2u| \|_{L^ 2}+\| | x|^{- b-1}| Du| \|_{L^ 2}+\| | x|^{-b- 2}u\|_{L^ 2}\leq C\| f\|_{L^ 2},\quad | u(x)| \leq C| x|^{1+b}\| f\|_{L^ 2}, \] for suitable constants, \(b>0\), \(C>0\).