# Cherif AmroucheUniversité de Pau et des Pays de l'Adour | UPPA · Laboratory of Mathematics and Applications

Cherif Amrouche

PhD in Mathematics

## About

156

Publications

14,327

Reads

**How we measure 'reads'**

A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more

3,074

Citations

## Publications

Publications (156)

We are considering a steady-state turbulent Reynolds-Averaged Navier–Stokes (RANS) one-equation model, that couples the equation for the velocity-pressure mean field with the equation for the turbulent kinetic energy. Eddy viscosities vanish at the boundary, characterized by terms like \(d(x, \Gamma )^\eta \) and \(d(x, \Gamma )^\beta \), where \(0...

In this paper, we study a linear elliptic system involving Navier‐type boundary conditions or tangential components in a three‐dimensional bounded domain, possibly multiply connected and with boundary possibly nonconnected. We establish an appropriate Inf‐Sup condition and then we prove the existence and uniqueness of the generalized solutions in L...

The purpose of this chapter is to study a system modeling a magnetohydrodynamic (MHD) flow. More precisely, we study the existence and regularity of solutions for a stationary MHD problem coupling an equation of polymer aqueous solution with Maxwell equation. First, we are interested in the Stokes problem and then in the existence of solutions for...

In this course, we are interested to the resolution of elliptical equations with various types of boundary conditions: Dirichlet, Neumann or Fourier-Robin, Navier or Navier type conditions. We will study the existence, the uniqueness, and the regularity of the solutions. This regularity will depend, as we will see, on data of the problem and, in pa...

We are concerned here with results of existence and regularity of solutions to a stationary MHD problem, with a possibly multi-connected 2D domain associated with Navier-type boundary conditions. The RHS f, corresponding to the external forces, is in Lp(Ω), with p≥2, without any restriction on its norm. Two main difficulties are encountered in solv...

Geymonat extended Gurtin’s result on Beltrami’s completeness to [Formula: see text]-case. The first objective of this paper is to generalize Geymonat’s result to [Formula: see text]-case with tangential and normal boundary conditions. Maggiani et al. proposed an extension of Beltrami’s-type decomposition for symmetric matrix fields in [Formula: see...

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 t...

In this paper we study the resolvent of wave operators on open and bounded Lipschitz domains of \({\mathbb {R}}^N\) with Dirichlet or Neumann boundary conditions. We give results on existence and estimates of the resolvent for the real and complex cases.

We study the stationary Stokes and Navier-Stokes equations with nonhomogeneous Navier boundary conditions in a bounded domain Ω⊂R3 of class C1,1. We prove the existence and uniqueness of weak and strong solutions in W1,p(Ω) and W2,p(Ω) for all 1<p<∞, considering minimal regularity on the friction coefficient α. Moreover, we deduce uniform estimates...

As well-known, De Rham's Theorem is a classical way to characterize vector fields as the gradient of the scalar fields, it is a tool of great importance in the theory of fluids mechanic. The first aim of this paper is to provide a useful rotational version of this theorem to establish several results on boundary value problems in the field of elect...

We consider the incompressible Navier–Stokes equations in a bounded domain with \(\mathcal {C}^{1,1}\) boundary, completed with slip boundary condition. We study the general semigroup theory in Lp-spaces related to the Stokes operator with Navier boundary condition where the slip coefficient α is a non-smooth scalar function. It is shown that the s...

In this paper we study the regularity for the homogenous and nonhomogenous Helmholtz equations on open and bounded Lipschitz domains of $\mathbb{R}^N$ with Dirichlet or Neumann boundary conditions. We give results on existence and regularity estimates of the solutions of the real and complex Helmholtz equations.

We perform a new modeling procedure for a 3D turbulent fluid, evolving towards a statistical equilibrium. This will result to add to the equations for the mean field (v, p) the term −α∇ · (ℓ(x)Dvt), which is of the Kelvin-Voigt form, where the Prandtl mixing length ℓ = ℓ(x) is not constant and vanishes at the solid walls. We get estimates for mean...

We consider the Robin boundary value problem \({\mathrm {div}}\,(A\nabla u) = {\mathrm {div}}\,\varvec{f}+F\) in \(\Omega \), a \(C^1\) domain, with \((A\nabla u - \varvec{f})\cdot {\varvec{n}}+ \alpha u = g\) on \(\Gamma \), where the matrix A belongs to \(VMO ({\mathbb {R}}^3) \), and discover the uniform estimates on \(\Vert u\Vert _{W^{1,p}(\Om...

In this paper, we study a model of magnetohydrodynamics problem and prove the existence of weak solution to the stationary magnethohydrodynamic system in a three dimensional bounded domain Ω of class C1,1. To our knowledge, all previous works consider the domain Ω simply-connected. Our proof is based on some weak estimates concerning vectors potent...

In a three-dimensional bounded possibly multiply connected domain, we prove the existence, uniqueness and regularity of some vector potentials, associated with a divergence-free function and satisfying mixed boundary conditions. For such a construction, the fundamental tool is the characterization of the kernel which is related to the topology of t...

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 $-\alpha \nabla\cdot(\ell(x) D v_t)$. 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...

In this paper, we study the stationary Stokes and Navier–Stokes equations with non-homogeneous Navier boundary condition in a bounded domain Ω⊂R³ of class C1,1 from the viewpoint of the behavior of solutions with respect to the friction coefficient α. We first prove the existence of a unique weak solution (and strong) in W1,p(Ω) (and W2,p(Ω)) to th...

The aim of this work is twofold: proving the existence of solution (u,π)∈H¹(Ω)×L²(Ω) in bounded domains of R² and the whole plane for the Oseen problem (O) for solenoidal vector fields v in L²(Ω), and analyzing the same problem in bounded domains of Rⁿ for n=2,3 when h=0, g=0 and the solenoidal field v belongs to Ls(Ω) for s<n.

We consider the stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain of class with a possibly disconnected boundary. We prove the existence of weak solutions in , strong solutions in and very weak solutions in of the stationary Boussinesq system by assuming that the fluxes of the velocity are sufficien...

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 $\alpha$ is a non-smooth scalar function, our main goal is to...

We study $W^{1,p}$-estimates of inhomogeneous second order elliptic operator of divergence form with Robin boundary condition in $\mathcal{C}^1$ domain. For any $p>2$, we prove that a weak reverse H\"{o}lder inequality holds which in turn provides the $W^{1,p}$-estimates for solutions with Robin boundary condition, independent of $\alpha$. As a res...

We study the stationary Stokes and Navier-Stokes equations with non-homogeneous Navier boundary condition in a bounded domain $\Omega\subset\mathbb{R}^{3}$ of class $\mathcal{C}^{1,1}$. We prove the existence, uniqueness of weak and strong solutions in $\mathbf{W}^{1,p}(\Omega)$ and $\mathbf{W}^{2,p}(\Omega)$ for all $1<p<\infty$ considering minima...

Maximal $L^p$-$L^q$ regularity is proved for the strong, weak and very weak solutions of the inhomogeneous Stokes problem with Navier-type boundary conditions in a bounded domain $\Omega$, not necessarily simply connected. This extends previous results of the authors (2017).

In this article we consider the Stokes problem with Navier-type boundary conditions on a domain $\Omega$, not necessarily simply connected. Since under these conditions the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stok...

In this paper, we prove the analyticity of the semi-group generated by the Stokes operator with a boundary condition involving the pressure. This allows us to obtain weak and strong solutions for the time-dependent Stokes problem with the corresponding boundary condition.

A classical stationary Boussinesq system with non-homogeneous Dirichlet boundary conditions in a bounded domain Ω.

In this paper, we study the Navier–Stokes equations with Navier boundary conditions in a bounded domain of . We prove the existence of weak solution in W1,p(Ω) × Lp(Ω) for by using fixed point theorem for the case . Copyright © 2015 John Wiley & Sons, Ltd.

The very weak solution for the Stokes, Oseen and Navier–Stokes equations has been studied by several authors in the last decades in domains of . The authors studied the Oseen and Navier–Stokes problems assuming a solenoidal convective velocity in a bounded domain of class for v ∈ L s (Ω) for s ≥3 in some previous papers. The results for the Navier–...

Let Ω be a domain in , i.e., a bounded and connected open subset of with a Lipschitz-continuous boundary ∂Ω, the set Ω being locally on the same side of ∂Ω. A fundamental lemma, due to Jacques-Louis Lions, provides a characterization of the space , as the space of all distributions on Ω whose gradient is in the space . This lemma, which provides in...

In this work, we study the linearized Navier-Stokes equations in R3, the Oseen equations. We are interested in the existence and the uniqueness of generalized and strong solutions in Lp-theory which makes analysis more difficult. Our approach rests on the use of weighted Sobolev spaces.

Soit Ω un ouvert borné et connesce de RNRN de frontière ∂Ω lipschitzienne, l'ensemble Ω étant localement du même côté de ∂Ω . On montre dans cette Note qu'une caractérisation fondamentale de l'espace L2(Ω)L2(Ω) due à Jacques-Louis Lions est en fait équivalente à un certain nombre d'autres propriétés. L'une des clés pour établir ces équivalences est...

In this work, we study the linearized Navier–Stokes equations in an exterior domain of R3R3 at the steady state, that is, the Oseen equations. We are interested in the existence and the uniqueness of weak, strong and very weak solutions in LpLp-theory which makes our work more difficult. Our analysis is based on the principle that linear exterior p...

This work was intended as an attempt at studying stationary Stokes and Navier–Stokes problem with Navier boundary conditions (1.3). We wish to investigate some results of existence, uniqueness and regularity of solutions in Hilbert case and in LpLp-theory.

We consider the Navier-Stokes equations with pressure boundary conditions in the case of a bounded open set, connected of class C 1;1 of ℝ3. We prove existence of solution by using a fixed point theorem over the type-Oseen problem. This result was studied in [5] in the Hilbertian case. In our study we give the Lp-theory for 1 < p < ∞.

In this article, we solve the Stokes problem in an exterior domain of
$\mathbb{R}^{3}$, with non-standard boundary conditions.
Our approach uses weighted Sobolev spaces to
prove the existence, uniqueness of weak and strong solutions.
This work is based on the vector potentials studied in [7]
for exterior domains, and in [1] for bounded domains.
Thi...

This article addresses some theoretical questions related to the choice of boundary conditions, which are essential for modelling and numerical computing in mathematical fluids mechanics. Unlike the standard choice of the well known non slip boundary conditions, we emphasize three selected sets of slip conditions, and particularly stress on the int...

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 s...

This work is dedicated to the resolution of a singular equation set in the half-space, with a diffusion coefficient that blows up on the boundary. More precisely, for a datum g: ℝ+
3 → ℝ, our problem involves seeking u: ℝ+
3 → ℝ as a formal solution to
We give existence and uniqueness results of weak and strong solutions in suitable weighted space...

Motived by the boundary values problem solved by correctors in the asymptotic analysis of singular perturbation of the domain, we consider Navier equations of linear elasticity in the half-space. We present a general theory of existence and uniqueness in the LpLp setting: we consider the weak solutions, the strong solutions and also very weak solut...

In this paper, we study the div-curl-grad operators and some elliptic problems in the half-space ℝn+, with n≥2. We consider data in weighted Sobolev spaces and in L1.

The existence and the uniqueness of very weak solutions of Stokes system are well known in the classical Sobolev spaces W m,p (Ω) when Ω is bounded (see [C. Amrouche and M. Á. Rodriguez-Bellido, Arch. Ration. Mech. Anal. 199, No. 2, 597–651 (2011; Zbl 1229.35164)]). When Ω is an exterior domain, a similar approach would fall (in particular because...

Motived by the boundary values problem solved by correctors in the asymptotic analysis of singular perturbation of the domain, we consider Navier equations of linear elasticity in the half-space. We present a general theory of existence and uniqueness in the L p setting: we consider the weak solutions, the strong solutions and also very weak soluti...

In a three-dimensional bounded possibly multiply-connected domain of class C1,1, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u⋅n=g and curlu×n=h×n or u×n=g×n and π=π0 on the boundary Γ. We prove the existence and uniqueness of weak, strong and very weak solutions corresponding to each boundary condit...

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 L1.

In a three-dimensional bounded possibly multiply-connected domain, we prove the existence and uniqueness of vector potentials in Lp-theory, associated with a divergence-free function and satisfying some boundary conditions. We also present some results concerning scalar potentials and weak vector potentials. Furthermore, various Sobolev-type inequa...

In this paper, we study the div-curl-grad operators and some elliptic problems in the whole space Rn and in the half-space R+n, with n⩾2. We consider data in weighted Sobolev spaces and in L1.

The concept of very weak solution introduced by Giga (Math Z 178:287–329, 1981) for the Stokes equations has hardly been studied in recent years for either the Navier–Stokes
equations or the Navier–Stokes type equations. We treat the stationary Stokes, Oseen and Navier–Stokes systems in the case
of a bounded open set, connected of class C1,1{\mathc...

We prove continuity properties for the Oseen potential. As a consequence, we show some new properties on solutions of the
Oseen equations. The study relies on weighted Sobolev spaces in order to control the behavior of functions at infinity.
KeywordsRiesz potentials–Oseen potentials–Poisson’s equations–Weighted spaces

The stationary Oseen equations are studied in R3 in its general form, that is, with a non-constant divergenceless function on the convective term. We prove existence, uniqueness and regularity results in weighted Sobolev spaces. From this new ap- proach, we also state existence, uniqueness and regularity results for the generalized Oseen model whic...

The purpose of this work is to show a broad framework in which the theory of very weak solutions for the Dirichlet stationary problem for the Laplace and Stokes equations in bounded domains of R n , n ≥ 2, could be developed. Broad in the sense of giving the more general spaces in which data can be taken in order to obtain a very weak solution and...

In a possibly multiply-connected three dimensional bounded domain, we prove in the L p theory the existence and uniqueness of vector potentials, associated with a divergence-free function and satisfying non homogeneous boundary conditions. Furthermore, we consider the stationary Stokes equations with nonstandard boundary conditions of the form u ·n...

We prove continuity properties for Riesz and Oseen potentials. As a consequence, we show some new properties on solutions of Poisson's and Oseen equations. The study relies on weighted Sobolev spaces in order to control the behavior of functions at infinity.

This contribution is devoted to the Oseen equations, a linearized form of the Navier-Stokes equations. We give here some results concerning the scalar Oseen operator and we prove Hardy inequalities concerning functions in Sobolev spaces with anisotropic weights that appear in the investigation of the Oseen equations.

This paper is devoted to some mathematical questions related to the stationary
Navier-Stokes problem in three-dimensional exterior domains. Our approach is
based on a combination of properties of Oseen problems in $\mathbb{R}^3$ and in
exterior domains of $\mathbb{R}^3$.

We study the existence of very weak solutions regularity for the Stokes, Oseen and Navier-Stokes system when non-smooth Dirichlet boundary data for the velocity are considered in domains of class C 1,1. In the Navier-Stokes case, the results will be valid for external forces non necessarily small. Regularity results for more regular data will be al...

We consider the stationary Oseen and Navier-Stokes equations in a bounded domain of class $C^{1,1}$ of $R^3$. Here we give a new and simpler proof of the existence of very weak solutions $(u, q) \in L^p(\Omega) × W^{−1,p}(\Omega)$ corresponding to boundary data in $W^{−1/p,p}(\Gamma)$. These solutions are obtained without imposing smallness assumpt...

In this paper, we study the Stokes system in the half-space $R^n+$, with n >= 2. We consider data and give solutions which live in weighted Sobolev spaces, for a whole scale of weights. We start to study the kernels of the biharmonic and Stokes operators. After the central case of the generalized solutions, we are interested in strong solutions and...

The concept of very weak solution introduced by Giga [9] for the Stokes equations has been intensively studied in the last years for the Navier-Stokes equations. However, a more rigorous study about the existence of traces for non regular vector fields is necessary in order to make a precise extension of the Stokes result to the Navier-Stokes case....

In this paper, we present several results concerning vector potentials and scalar potentials with data in Sobolev spaces with negative exponents, in a not necessarily simply-connected, three-dimensional domain. We then apply these results to Poincaré's theorem and to Korn's inequality.

We deal with the hydrostatic Stokes approximation with non homogeneous Dirichlet boundary conditions. After having investigated the homogeneous case, we build a lifting operator of boundary values related to the divergence operator, and solve the non homogeneous problem in a cylindrical type domain.

Purpose
The purpose of this paper is to analyse the convective heat transfer of an unsteady pulsed, laminar, incompressible flow in axisymmetric tubes with periodic sections. The flow is supposed to be developing dynamically and thermally from the duct inlet. The wall is heated at constant and uniform temperature.
Design/methodology/approach
The p...

The purpose of this work is to solve the exterior Stokes problem in the half-space \({\mathbb{R}{^n_+}}\) . We study the existence and the uniqueness of generalized solutions in weighted L
p
theory with 1 < p < ∞. Moreover, we consider the case of strong solutions and very weak solutions. This paper extends the studies done in Alliot, Amrouche (Mat...

The purpose of this work is to solve exterior problems in the half-space for the Laplace operator. We give existence and unicity results in weighted L(p)'s theory with 1 < p < infinity. This paper extends the studies done in [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator, an appr...

We consider the Stokes problem with slip-type boundary conditions in the half-space ℝ + n , with n⩾2. The weighted Sobolev spaces yield the functional framework. We first study generalized and strong solutions and then the case with very low regularity of data on the boundary. We apply the method of decomposition introduced in our previous work [J....

The purpose of this work is to solve the Stokes problem with a Diri-chlet boundary condition in a perturbed half-space and in an aperture domain, two unbounded geometries with noncompact boundaries. We study the exis-tence and the uniqueness of generalized solutions in weighted L p 's theory with 1 < p < ∞. We study too the case of strong solutions...

In this Note, we study the characterization of the kernel of the Laplace operator with Dirichlet boundary conditions in exterior domains. We consider data in weighted Sobolev spaces. (Comptes Rendus de l Académie des Sciences - France- Series I - Mathematics)

This paper is devoted to some mathematical questions related to the 3-dimensional stationary Navier-Stokes. Our approach is based on a combination of properties of Oseen problems in $\R^3$.

The aim of this article is to study a nonlinear system modeling a Non-Newtonian fluid of polymer aqueous solutions. We are interested here in the existence of weak solutions for the stationary problem in a bounded plane domain or in two-dimensional exterior domain. Due to the third order of derivatives in the non-linear term, it’s difficult to obta...

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.

In this paper, we study the Stokes system in the half-space $\R^N_+$, with $N\geq 2$. We give existence and uniqueness results in weighted Sobolev spaces. After the central case of the generalized solutions, we are interested in strong solutions and symmetricaly in very weak solutions by means of a duality argument.

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of R-n. This paper extends this study to the resolution of a...

The Oseen equations are obtained by linearizing the Navier-Stokes equations around a nonzero constant vector which is the velocity at infinity. We are interested with the study of the scalar problem corresponding to the anisotropic operator$−\Delta + \frac{\partial}{\partial x_1}$. The Marcinkiewicz interpolation’s theorem and the Sobolev embedding...

We deal with the existence of the material derivative of the Laplace equation with the Neumann boundary condition in the half space. We consider two different perturbations of domains to get the existence of weak Gateaux material derivative and the existence of Fréchet material derivatives.

In this Note, we present several results concerning vector potentials and scalar potentials in a bounded, not necessarily simply-connected, three-dimensional domain. In particular, we consider singular potentials corresponding to data in negative order Sobolev spaces. We also give some applications to Poincaré's theorem and to Korn's inequality. To...

In this paper, we are interested in some aspects of the biharmonic equation in the half-space $\R^N_+$, with $N\geq 2$. We study the regularity of generalized solutions in weighted Sobolev spaces, then we consider the question of singular boundary conditions. To finish, we envisage other sorts of boundary conditions.

This paper is devoted to a scalar model of the Oseen equations, a linearized form of the Navier–Stokes equations. To control
the behavior of functions at infinity, the problem is set in weighted Sobolev spaces including anisotropic weights. In a first
step, some weighted Poincaré-type inequalities are obtained. In a second step, we establish existe...

In this paper we solve the stationary Oseen equations in \({\mathbb{R}}^{3}\) . The behavior of the solutions at infinity is described by setting the problem in weighted Sobolev spaces including anisotropic weights. The study is based on a L
p
theory for 1 < p < ∞.