arXiv subjects
Publications and source records attributed to Jamel Benameur.
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.
In \cite{JB1}, Benameur proved a blow-up result of the non regular solution of $(NSE)$ in the Sobolev-Gevrey spaces. In this paper we improve this result, precisely we give an exponential type explosion in Sobolev-Gevrey spaces with less regularity on the initial condition. Fourier analysis is used.
In this paper, we study the sub-critical dissipative quasi-geostrophic equations $({\bf S}_α)$. We prove that there exists a unique local-in-time solution for any large initial data $θ_0$ in the space ${\bf{\mathcal X}}^{1-2α}(\mathbb R^2)$ defined by (\ref{ec}). Moreover we show that $({\bf S}_α)$ has a global solution in time if the norms of the initial data in ${\bf{\mathcal X}}^{1-2α}(\mathbb R^2)$ are bounded by $1/4$. Also, we prove a blow-up criterion of the non global solution of $({\bf S}_α)$.
In this paper we prove, if $u\in\mathcal C([0,\infty),{\bf {\mathcal X}^{-1}}(\mathbb R^3))$ is global solution of 3D Navier-Stokes equations, then $\|u(t)\|_{{\bf {\mathcal X}^{-1}}}$ decays to zero as time goes to infinity. Fourier analysis and standard techniques are used.
We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also obtain a new energy estimate. Other results in this direction and with different methods can be found in \cite{C4}.
In this paper, we study the convergence of strong solutions of a Magneto-Hydro-Dynamic system. On the torus ${\mathbb{T}}^3$, the proof is based on Schochet's methods, whereas in the case of the whole space ${\bf \mathbb{R}^3}$, we use Strichartz's type estimates and a product law's $2D\times3D$.