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 $𝕧$.

We consider the homogeneous time-dependent Oseen system in the whole space ${ℝ}^3$. The initial data is assumed to behave as $O\left(\right|x{|}^{-1-ϵ})$, and its gradient as $O\left(\right|x{|}^{-3/2-ϵ})$, when $\left|x\right|$ tends to infinity, where $ϵ$ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $\left|u(x,t)\right|=O\left(\right|x{|}^{-1})$, and its spatial gradient ${\nabla }_{x}u$ decreases with the rate ${\left|x\right|}^{-3/2}$, for $\left|x\right|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible...

The Navier-Stokes system is studied on a family of domains with rough boundaries formed by oscillating riblets. Assuming the complete slip boundary conditions we identify the limit system, in particular, we show that the limit velocity field satisfies boundary conditions of a mixed type depending on the characteristic direction of the riblets.

We prove the existence of a weak solution and of a strong solution (locally in time) of the equations which govern the motion of viscous incompressible non-homogeneous fluids. Then we discuss the decay problem.

This paper is devoted to the study of smooth flows of density-dependent fluids in ${ℝ}^N$ or in the torus ${𝕋}^N$. We aim at extending several classical results for the standard Euler or Navier-Stokes equations, to this new framework.Existence and uniqueness is stated on a time interval independent of the viscosity $\mu$ when $\mu$ goes to $0$. A blow-up criterion involving the norm of vorticity in $L^1(0,T;L^{\infty })$ is also proved. Besides, we show that if the density-dependent Euler equations have a smooth solution on a given time...

We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty }$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

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...

In this paper we prove global existence and uniqueness for solutions of the 3-dimensional Navier-Stokes equations with small initial data in spaces which are Hδi in the i-th direction, δ1 + δ2 + δ3 = 1/2, -1/2 < δi < 1/2 and in a space which is L2 in the first two directions and B2,11/2 in the third direction, where H and B denote the usual homogeneous Sobolev and Besov spaces.

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...

The existence of a periodic solution of a nonlinear equation $z^{\text{'}}+A_{0}z+B_{0}z=F$ is proved. The theory developed may be used to prove the existence of a periodic solution of the variational formulation of the Navier-Stokes equations or the equations of magnetohydrodynamics. The proof of the main existence theorem is based on Rothe method in combination with the Galerkin method, using the Brouwer fixed point theorem.

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.