arXiv subjects

Cherif Amrouche

Publications and source records attributed to Cherif Amrouche.

5 recordsLinked to original sources

arXiv2023

The Dirichlet problem for the Laplacian in Lipschitz domain. Abstract

The main purpose of this paper is to address some questions concerning boundary value problems related to the Laplacian and bi-Laplacian operators, set in the framework of classical $H^s$ Sobolev spaces on a bounded Lipschitz domain of R^N. These questions are not new and a lot of work has been done in this direction by many authors using various techniques since the 80's. If for regular domains almost every thing is elucidated, it is not the case for Lipschitz ones and for $s$ of the form $s = k + 1/2$, with $k$ integer. It is well known that this framework is delicate. Even in these cases many results are well established but sometimes not satisfactory. Several questions remain posed. Our main goal through this work is on one hand to give some improvements to the theory and on another one by using techniques which do not require too intricate calculations. We also tried to obtain maximal regularity for the solutions and as far as we can optimality of the results.

math.AP↗

arXiv2019

Turbulent flows as generalized Kelvin-Voigt materials: modeling and analysis

We model a 3D turbulent fluid, evolving toward a statistical equilibrium, by adding to the equations for the mean field $(v, p)$ a term like $-α\nabla\cdot(\ell(x) D v_t)$. This is of the Kelvin-Voigt form, where the Prandtl mixing length $\ell$ is not constant and vanishes at the solid walls. We get estimates for velocity $v$ in $L^\infty_t H^1_x \cap W^{1,2}_t H^{1/2}_x$, that allow us to prove the existence and uniqueness of a regular-weak solutions $(v, p)$ to the resulting system, for a given fixed eddy viscosity. We then prove a structural compactness result that highlights the robustness of the model. This allows us to pass to the limit in the quadratic source term in the equation for the turbulent kinetic energy $k$, which yields the existence of a weak solution to the corresponding Reynolds Averaged Navier-Stokes system satisfied by $(v, p, k)$.

math.AP↗

arXiv2018

Stokes and Navier-Stokes equations with Navier boundary condition

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $Ω\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(Ω)$ and $\mathbf{W}^{2,p}(Ω)$ for all $1<p<\infty$ considering minimal regularity on the friction coefficient $α$. Moreover, we deduce uniform estimates on the solution with respect to $α$ which enables us to analyze the behavior of the solution when $α\rightarrow \infty$.

math.AP↗

arXiv2018

Uniform $W^{1,p}$ estimate for elliptic operator with Robin boundary condition in $\mathcal{C}^1$ domain

We consider the Robin boundary value problem $\mathrm{div} (A \nabla u) = \mathrm{div} \mathbf{f}+F$ in $Ω$, $\mathcal{C}^1$ domain, with $(A \nabla u - \mathbf{f})\cdot \mathbf{n} + αu = g$ on $Γ$, where the matrix $A$ belongs to $VMO (\mathbb{R}^3) $, and discover the uniform estimates on $\|u\|_{W^{1,p}(Ω)}$, with $1 < p < \infty$, independent on $α$. At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse Hölder inequality. This estimates show that the solution of Robin problem converges strongly to the solution of Dirichlet (resp. Neumann) problem in corresponding spaces when the parameter $α$ tends to $\infty$ (resp. $0$).

math.AP↗

arXiv2018

Semigroup theory for the Stokes operator with Navier boundary condition on $L^p$ spaces

We consider the incompressible Navier-Stokes equations in a bounded domain with $\mathcal{C}^{1,1}$ boundary, completed with slip boundary condition. Apart from studying the general semigroup theory related to the Stokes operator with Navier boundary condition where the slip coefficient $α$ is a non-smooth scalar function, our main goal is to obtain estimate on the solutions, independent of $α$. We show that for $α$ large, the weak and strong solutions of both the linear and non-linear system are bounded uniformly with respect to $α$. This justifies mathematically that the solution of the Navier-Stokes problem with slip condition converges in the energy space to the solution of the Navier-Stokes with no-slip boundary condition as $α\to \infty$.

math.AP↗