We assume that $𝕧$ is a weak solution to the non-steady Navier-Stokes initial-boundary value problem that satisfies the strong energy inequality in its domain and the Prodi-Serrin integrability condition in the neighborhood of the boundary. We show the consequences for the regularity of $𝕧$ near the boundary and the connection with the interior regularity of an associated pressure and the time derivative of $𝕧$.

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s > 2 to obtain local solutions for any initial data, and global solutions for small initial data.

We investigate the inviscid limit for the stationary Navier-Stokes equations in a two dimensional bounded domain with slip boundary conditions admitting nontrivial inflow across the boundary. We analyze admissible regularity of the boundary necessary to obtain convergence to a solution of the Euler system. The main result says that the boundary of the domain must be at least C²-piecewise smooth with possible interior angles between regular components less than π.

We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain $\Omega$, which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves ${\Gamma }_{-}$ and ${\Gamma }_{+}$ (lower and upper parts of $\partial \Omega$), the Dirichlet boundary conditions on ${\Gamma }_{\mathrm{in}}$ (the inflow) and ${\Gamma }_0$ (boundary of the profile) and an artificial “do nothing”-type boundary condition...

We formulate a boundary value problem for the Navier-Stokes equations with prescribed u·n, curl u·n and alternatively (∂u/∂n)·n or curl²u·n on the boundary. We deal with the question of existence of a steady weak solution.

We study the motion of a viscous incompressible fluid filling the whole three-dimensional space exterior to a rigid body, that is rotating with constant angular velocity ω, under the action of external force f. By using a frame attached to the body, the equations are reduced to (1.1) in a fixed exterior domain D. Given f = divF with $F\in BUC(ℝ;L_{3/2,\infty }\left(D\right))$, we consider this problem in D × ℝ and prove that there exists a unique solution $u\in BUC(ℝ;L_{3,\infty }\left(D\right))$ when F and |ω| are sufficiently small. If, in particular, the external force for...

Corresponding to the wellposedness result [2] for the classical 3-D Navier-Stokes equations $\left(N{S}_{\nu }\right)$ with initial data in the scaling invariant Besov space, ${ℬ}_{p,\infty }^{-1+\frac{3}{p}},$ here we consider a similar problem for the 3-D anisotropic Navier-Stokes equations $\left(AN{S}_{\nu }\right),$ where the vertical viscosity is zero. In order to do so, we first introduce the Besov-Sobolev type spaces, ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}}$ and ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}}\left(T\right).$ Then with initial data in the scaling invariant space ${ℬ}_4^{-\frac{1}{2},\frac{1}{2}},$ we prove the global wellposedness for $\left(AN{S}_{\nu }\right)$ provided the norm of initial data is small enough compared...

This paper solves the scalar Oseen equation, a linearized form of the Navier-Stokes equation. Because the fundamental solution has anisotropic properties, the problem is set in a Sobolev space with isotropic and anisotropic weights. We establish some existence results and regularities in $L^p$ theory.

We prove that strong solutions of the Boussinesq equations in 2D and 3D can be extended as analytic functions of complex time. As a consequence we obtain the backward uniqueness of solutions.