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.

On prouve l’unicité des solutions du système de Navier-Stokes incompressible dans $𝒞([0,T);L^d{\left(\Omega \right)}^d)$, où $\Omega$ est un domaine lipschitzien borné de ${ℝ}^d$ ($d\ge 3$).

The main result of this paper is the proof of uniqueness for mild solutions of the Navier-Stokes equations in L3(R3). This result is extended as well to some Morrey-Campanato spaces.

Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2\left({ℝ}^d\right)$. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2((0,T);{\dot{X}}_1{\left({ℝ}^d\right)}^d)$ where ${\dot{X}}_1\left({ℝ}^d\right)$ is the multiplier space, then we have $u=v$.

Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis...

We prove the existence of weak solutions to the Navier-Stokes equations describing the motion of a fluid in a Y-shaped domain.

Consider the flow of a viscous, incompressible fluid past a rotating obstacle with velocity at infinity parallel to the axis of rotation. After a coordinate transform in order to reduce the problem to a Navier-Stokes system on a fixed exterior domain and a subsequent linearization we are led to a modified Oseen system with two additional terms one of which is not subordinate to the Laplacean. In this paper we describe two different approaches to this problem in the whole space case. One of them...

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in ${ℝ}^N$ with $N\ge 2$. We address the question of well-posedness for large and small initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where $u_0\in B_{p,r}^{\frac{N}{p}-1}$ with $1\le p\<+\infty ,1\le r\le +\infty$). This improves the classical analysis where $u_0$ is considered belonging in $B_{p,1}^{\frac{N}{p}-1}$ such that the velocity $u$ remains...

In this paper, we consider the interaction between a rigid body and an incompressible, homogeneous, viscous fluid. This fluid-solid system is assumed to fill the whole space ${ℝ}^d$, $d=2$ or $3$. The equations for the fluid are the classical Navier-Stokes equations whereas the motion of the rigid body is governed by the standard conservation laws of linear and angular momentum. The time variation of the fluid domain (due to the motion of the rigid body) is not known a priori, so we deal with a free boundary...