Chérif Amrouche

Pointwise decay of solutions and of higher derivatives to Navier-Stokes equations

In this paper we study the space-time asymptotic behavior of the solutions, and their derivatives, to the incompressible Navier--Stokes equations in dimension

Lp-theory for vector potentials and sobolev's inequalities for vector fields: Application to the stokes equations with pressure boundary conditions

2\le n \le 5

The Neumann problem in the half-space

. Using moment estimates...

Weak solutions for the exterior Stokes problem in weighted Sobolev spaces

In a three dimensional bounded possibly multiply-connected domain, we give gradient and higher order estimates of vector fields via div and curl in Lp theory. Then, we prove the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u × n = g × n and π = π0 on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions. Our proofs are based on obtaining Inf − Sup conditions that play a fundamental role. We give a variant of the Stokes system with these boundary conditions, in the case where the compatibility condition is not verified. Finally, we give two Helmholtz decompositions that consist of two kinds of boundary conditions such as u * n and u × n on Γ.

ON THE OSEEN PROBLEM IN THREE-DIMENSIONAL EXTERIOR DOMAINS

We study here the nonhomogeneous Neumann problem in the half-space RN+ with N⩾2. We give in Lp theory, with 1<p<∞, a basic existence and regularity results in weighted Sobolev spaces. To cite this article: C. Amrouche, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 151–156.

New estimates for the div, curl, grad operators and elliptic problems with L-1-data in the half-space

We extend here some existence and uniqueness results for the exterior Stokes problem in weighted Sobolev spaces. We also study the regularity of the solutions (u, π) and prove optimal a priori estimates for the solutions with ⊇ 2 u, ⊇π ∈ L p . The influence of some compatibility conditions on the behaviour at infinity of the solution is finally studied and leads to new asymptotic expansions.

FROM STRONG TO VERY WEAK SOLUTIONS TO THE STOKES SYSTEM WITH NAVIER BOUNDARY CONDITIONS IN THE HALF-SPACE

In this paper, we prove existence and uniqueness results for the Oseen problem in exterior domains of ℝ3. To prescribe the growth or decay of functions at infinity, we set the problem in weighted Sobolev spaces. The analysis relies on a Lp-theory for any real p such that 1 < p < ∞.

Weighted estimates for the Oseen problem in R 3

Abstract In this Note, we study some properties of the div, curl, grad operators and elliptic problems in the half-space. We consider data in weighted Sobolev spaces and in L 1 .

Isotropically and Anisotropically Weighted Sobolev Spaces for the Oseen Equation

We consider the Stokes problem with slip-type boundary conditions in the half-space

On the Regularity and Decay of the Weak Solutions to the Steady-State Navier-Stokes Equations in Exterior Domains

\mathbb{R}^n_+