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

We consider initial-boundary value problem for the Navier–Stokes equations with boundary conditions $\overrightarrow{v}\bigm|_{x\in\partial\Omega}=\overrightarrow{a}$ assuming that $\overrightarrow{a}$ may have jump discontinuities at a finite number of points $\xi_1,\dots,\xi_m$ of the boundary $\partial\Omega$ of a bounded domain $\Omega\subset\mathbb{R}^2$. It is proved that this problem possesses a unique generalized solution in a finite time interval or for small initial and boundary data. The solution is found in a certain class of vector fields with an infinite energy integral. The case of moving boundary is also considered.