On a problem for the Navier-Stokes equations with the infinite Dirichlet integral (Q1779066)

The author studies incompressible Navier-Stokes equations in a two-dimensional pipe-like infinite domain \(\Omega\) with an inlet and outlet at infinity. The problem is considered with slip boundary conditions involving non-zero friction, and with a compatibility condition on the total flux. First, under some restrictions on the curvature of the boundary, the author proves the existence of global-in-time unique weak solution. This is done by means of the Galerkin method. Then, assuming additionally that the vorticity belongs to \(L_\infty(\Omega)\), the author arrives at new \(L_\infty\)-bounds on the vorticity. The proof is based on the Poincaré inequality, Schauder estimates, and the maximum principle for a reformulation of the problem. Finally, some results are obtained for large data (including the fluxes at infinity) with infinite Dirichlet integral.