arXiv subjects
Publications and source records attributed to Jamel Benameur.
In this paper, we establish a new result for the Laplace problem with exponential Robin boundary conditions posed on the unit disk in $\R^2$. More precisely, we prove the existence and uniqueness of a solution under suitable smallness assumptions on the boundary data. Our approach relies on an iterative method combined with periodic Sobolev embedding results.
In \cite{CJ1} M. Jaoua et al. studied the linear approximation of Robin problem on $Ω$ an open bounded domain of $\R^d$, and they given some important results. In this paper, we study a nonlinear approximation of an elliptic problem with a nonlinear Robin boundary condition in a domain of $\R^2$. We prove the existence and uniqueness of solution by an iterative construction method with admissible condition on $\partialΩ$.
In \cite{MRSC1} the authors proved some asymptotic results for the global solution of critical Quasi-geostrophic equation with a condition on the decay of $\widehat{θ_0}$ near at zero. In this paper, we prove that this condition is not necessary. Fourier analysis and standard techniques are used.
We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with the nonlinear exponential damping term $a (e^{b |u|^{\bf 2}}-1)u$, ($a,b>0$) studied by the second author in \cite{J1}.
In this paper we study the anisotropic incompressible Navier-Stokes equations with a logarithm damping $α\log(e+|u|^2)|u|^2u$ in $H^{0.1}$, where we used new methods, new tools and Fourier analysis.
In this paper, we focus on the two-dimensional surface quasi-geostrophic equation with fractional horizontal dissipation and vertical thermal diffusion which represents a general case of the classical surface quasi-geostrophic equation. On the one hand, we will show the local existence and uniqueness of the solution in Sobolev space $H^{2-2α}(\mathbb{R}^2)\cap H^{2-2β}(\mathbb{R}^2)$, which is the critical space in the classical case. Furthermore, we will demonstrate that the solution is global even when the initial data is very small. Finally, we will study the asymptotic representation of our global solution in infinity.
In \cite{CJ}, the authors show that the Cauchy problem of the Navier-Stokes equations with damping $α|u|^{β-1}u(α>0,\;β\geq1)$ has global weak solutions in $L^2(\R^3)$. In this paper, we prove the uniqueness, the continuity in $L^2$ for $β>3$, also the large time decay is proved for $β\geq\frac{10}3$. Fourier analysis and standard techniques are used.
We study the uniqueness, the continuity in $L^2$ and the large time decay for the Leray solutions of the $3D$ incompressible Navier-Stokes equations with nonlinear exponential damping term $a (e^{b |u|^{\bf 4}}-1)u$, ($a,b>0$).
In \cite{YZ}, the author proved the global existence of the two-dimensional anisotropic quasi-geostrophic equations with condition on the parameters $α,$ $β$ in the Sobolev spaces $H^s( \R^2)$; $s\geq 2$. In this paper, we show that this equations has a global solution in the spaces $H^s(\R^2)$, where $\max\{2-2α,2-2β\}< s<2$, with additional condition over $α$ and $β$. The proof is based on the Gevrey-class regularity of the solution in neighborhood of zero.
In this paper we prove the existence and uniqueness of strong solution of the incompressible Navier-Stokes equations with damping $α(e^{β|u|^2}-1)u$.
In this paper we prove the global existence of incompressible Navier-Stokes equations with damping $α(e^{β|u|^2}-1)u$, where we use Friedrich method and some new tools. The delicate problem in the construction of a global solution, is the passage to the limit in exponential nonlinear term. To solve this problem, we use a polynomial approximation of the damping part and a new type of interpolation between $L^\infty(\mathbb{R}^+,L^2(\mathbb{R}^3))$ and the space of functions $f$ such that $(e^{β|f|^2}-1)|f|^2\in L^1(\mathbb{R}^3)$. Fourier analysis and standard techniques are used.
In this paper we prove, if $θ\in C([0,\infty),H^{2-2α}(\mathbb R^2))$ is a global solution of supercritical surface Quasi-Geostrophic equation with small initial data, then $\|θ(t)\|_{H^{2-2α}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
In this paper we show the global existence for critical dissipative quasi-geostrophic equations if $\|\widehat{θ^0}\|_{L^1}$ is small enough; among others we prove the analyticity of such a solution. If in addition the initial condition verifies $|D|^{-δ}θ^0\in L^1(\mathbb R^2)$ with $0<δ<1$, then the solution remains regular and $\lim_{t\rightarrow\infty}t^δ\|\widehatθ(t)\|_{L^1}=0$. Fourier analysis and standard techniques are used.
In this paper we study the incompressible Navier-Stokes equations in $L^2(\mathbb R^3)\cap\mathcal X^{-1}(\mathbb R^3)$. In the global existence case, we establish that if the solution $u$ is in the space $C(\mathbb R^+,L^2\cap\mathcal X^{-1})$, then for $σ>-3/2$ the decay of $\|u(t)\|_{\mathcal X^σ}$ is at least of the order of $t^{-\frac{σ+\frac{3}{2}}{2}}$. Fourier analysis and standard techniques are used.
In this paper, we prove two results about the blow up criterion of the three-dimensional incompressible Navier-Stokes equation in the sobolev space $\dot H^{5/2}$. The first one improves the result of \cite{CZ}. The second deals with the relationship of the blow up in $\dot H^{5/2}$ and some critical spaces. Fourier analysis and standard techniques are used.
In \cite{JAMH1}, we prove the well posedness of the quasi-geostrophic equation $(QG)_α\;,1/2<α\leq 1$, in the space introduced by Z. Lei and F. Lin in \cite{ZY1}. In this chapter we discuss the long time behaviour. Mainly, we prove that $\|θ\|_{{\mathcal X}^{1-2α}}$ decays to zero as time goes to infinity.
In this paper we prove, if $u$ is a global solution to Navier-Stokes equations in the Sobolev-Gevrey spaces $H^1_{a,σ}(\mathbb R^3)$, then $\|u(t)\|_{H^1_{a,σ}}$ decays to zero as time goes to infinity. Fourier analysis is used.
In this paper, we prove that there exists a unique global solution of $3D$ Navier-Stokes equation if $\exp(a|D|^{1/σ})u^0\in{\mathcal{X}}^{-1}(\mathbb R^3)$ and $\|u^0\|_{\mathcal{X}^{-1}}<ν$. Moreover, we will show that $\|\exp(a|D|^{1/σ}) u(t)\|_{\mathcal{X}^{-1}}$ goes to zero if the time $t$ goes to infinity.