On the two-dimensional initial boundary value problem for the Navier-Stokes equations with discontinuous boundary data

Abstract

We consider the initial boundary value problem for the Navier-Stokes equations with boundary conditions\(\vec v|\partial \Omega = \vec a\). We assume that\(\vec a\) may have jump discontinuities at finitely many points ξ1;. . .,ξm of the boundary ϖΩ of a bounded domain Ω ⊂ ℝ2. We prove that this problem has a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a class of vector fields with infinite energy integral. The case of a moving boundary is also considered. Bibliography: 11 titles.