Searcharxiv⌕ Search

arXiv subjects

Slim Ibrahim

Publications and source records attributed to Slim Ibrahim.

At least 55 records · Page 3Linked to original sources
arXiv2015

The continuous transition of Hamiltonian vector fields through manifolds of constant curvature

We ask whether Hamiltonian vector fields defined on spaces of constant Gaussian curvature $κ$ (spheres, for $κ>0$, and hyperbolic spheres, for $κ<0$), pass continuously through the value $κ=0$ if the potential functions $U_κ, κ\in\mathbb R$, that define them satisfy the property $\lim_{κ\to 0}U_κ=U_0$, where $U_0$ corresponds to the Euclidean case. We prove that the answer to this question is positive, both in the 2- and 3-dimensional cases, which are of physical interest, and then apply our conclusions to the gravitational $N$-body problem.

math.DS↗
arXiv2015

The Vlasov-Poisson System for Stellar Dynamics in Spaces of Constant Curvature

We obtain a natural extension of the Vlasov-Poisson system for stellar dynamics to spaces of constant Gaussian curvature $κ\ne 0$: the unit sphere $\mathbb S^2$, for $κ>0$, and the unit hyperbolic sphere $\mathbb H^2$, for $κ<0$. These equations can be easily generalized to higher dimensions. When the particles move on a geodesic, the system reduces to a 1-dimensional problem that is more singular than the classical analogue of the Vlasov-Poisson system. In the analysis of this reduced model, we study the well-posedness of the problem and derive Penrose-type conditions for linear stability around homogeneous solutions in the sense of Landau damping.

math.AP↗
arXiv2015

Energy decay for linear dissipative wave equations in exterior domains

In earlier works, we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turns out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.

math.AP↗
arXiv2013

Scattering for the critical 2-D NLS with exponential growth

In this article, we establish in the radial framework the $H^1$-scattering for the critical 2-D nonlinear Schrödinger equation with exponential growth. Our strategy relies on both the a priori estimate derived in \cite{CGT, PV} and the characterization of the lack of compactness of the Sobolev embedding of $H_{rad}^1(\R^2)$ into the critical Orlicz space ${\cL}(\R^2)$ settled in \cite{BMM}. The radial setting, and particularly the fact that we deal with bounded functions far away from the origin, occurs in a crucial way in our approach.

math.AP↗
arXiv2012

Finite-time Blowup for the Inviscid Primitive Equations of Oceanic and Atmospheric Dynamics

In an earlier work we have shown the global (for all initial data and all time) well-posedness of strong solutions to the three-dimensional viscous primitive equations of large scale oceanic and atmospheric dynamics. In this paper we show that for certain class of initial data the corresponding smooth solutions of the inviscid (non-viscous) primitive equations blow up in finite time. Specifically, we consider the three-dimensional inviscid primitive equations in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-normal flow boundary conditions on the solid, top and bottom, boundaries. For certain class of initial data we reduce this system into the two-dimensional system of primitive equations in an infinite horizontal strip with the same type of boundary conditions; and then show that for specific sub-class of initial data the corresponding smooth solutions of the reduced inviscid two-dimensional system develop singularities in finite time.

math.AP↗
arXiv2012

On the wellposedness of the Navier-Stokes-Maxwell system

We study the local and global wellposedness of a full system of Magneto-Hydro-Dynamic equations. The system is a coupling of the forced (Lorentz force) incompressible Navier-Stokes equations with the Maxwell equations through Ohm's law for the current. We show the local existence of mild solutions for arbitrarily large data in a space similar to the scale invariant spaces classically used for Navier-Stokes. These solutions are global if the initial data are small enough.

math.AP↗
arXiv2011

Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state.

math.AP↗
arXiv2011

Threshold solutions in the case of mass-shift for the critical Kline-Gordon equation

We study global dynamics for the focusing nonlinear Klein-Gordon equation with the energy-critical nonlinearity in two or higher dimensions when the energy equals the threshold given by the ground state of a mass-shifted equation, and prove that the solutions are divided into scattering and blowup. In short, the Kenig-Merle scattering/blowup dichotomy extends to the threshold energy in the case of mass-shift for the critical nonlinear Klein-Gordon equation.

math.AP↗
arXiv2011

Trudinger-Moser inequality on the whole plane with the exact growth condition

Trudinger-Moser inequality is a substitute to the (forbidden) critical Sobolev embedding, namely the case where the scaling corresponds to $L^\infty$. It is well known that the original form of the inequality with the sharp exponent (proved by Moser) fails on the whole plane, but a few modified versions are available. We prove a precised version of the latter, giving necessary and sufficient conditions for the boundedness, as well as for the compactness, in terms of the growth and decay of the nonlinear function. It is tightly related to the ground state of the nonlinear Schrodinger equation (or the nonlinear Klein-Gordon equation), for which the range of the time phase (or the mass constant) as well as the energy is given by the best constant of the inequality.

math.AP↗
arXiv2011

Long-time solvability of the Navier-Stokes-Boussinesq equations with almost periodic initial large data

We investigate large time existence of solutions of the Navier-Stokes-Boussinesq equations with spatially almost periodic large data when the density stratification is sufficiently large. In 1996, Kimura and Herring \cite{KH} examined numerical simulations to show a stabilizing effect due to the stratification. They observed scattered two-dimensional pancake-shaped vortex patches lying almost in the horizontal plane. Our result is a mathematical justification of the presence of such two-dimensional pancakes. To show the existence of solutions for large times, we use $\ell^1$-norm of amplitudes. Existence for large times is then proven using techniques of fast singular oscillating limits and bootstrapping argument from a global-in-time unique solution of the system of limit equations.

math.AP↗
arXiv2011

Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data

Existence of local-in-time unique solution and loss of smoothness of full Magnet-Hydro-Dynamics system (MHD) is considered for periodic initial data. The result is proven using Fujita-Kato's method in $\ell^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwells's equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.

math.AP↗
arXiv2010

Well posedness and unconditional non uniqueness for a 2D semilinear heat equation

We investigate the initial value problem for a semilinear heat equation with exponential-growth nonlinearity in two space dimension. First, we prove the local existence and unconditional uniqueness of solutions in the Sobolev space $H^1(\R^2)$. The uniqueness part is non trivial although it follows Brezis-Cazenave's proof \cite{Br} in the case of monomial nonlinearity in dimension $d\geq3$. Next, %Following Caffarelli-Vasseur \cite{cv}, we show that in the defocusing case our solution is bounded, and therefore exists for all time. In the focusing case, we prove that any solution with negative energy blows up in finite time. Lastly, we show that the unconditional result is lost once we slightly enlarge the Sobolev space $H^1(\R^2)$. The proof consists in constructing a singular stationnary solution that will gain some regularity when it serves as initial data in the heat equation. The Orlicz space appears to be appropriate for this result since, in this case, the potential term is only an integrable function.

math.AP↗
arXiv2010

Scattering threshold for the focusing nonlinear Klein-Gordon equation

We show scattering versus blow-up dichotomy below the ground state energy for the focusing nonlinear Klein-Gordon equation, in the spirit of Kenig-Merle for the $H^1$ critical wave and Schrödinger equations. Our result includes the $H^1$ critical case, where the threshold is given by the ground state for the massless equation, and the 2D square-exponential case, where the mass for the ground state may be modified, depending on the constant in the sharp Trudinger-Moser inequality. The main difficulty is lack of scaling invariance both in the linear and nonlinear terms.

math.AP↗
arXiv2009

Well and ill-posedness issues for energy supercritical waves

We investigate the initial value problem for some energy supercritical semilinear wave equations. We establish local existence in suitable spaces with continuous flow. We also obtain some ill-posedness/weak ill-posedness results. The proof uses the finite speed of propagation and a quantitative study of the associated ODE. It does not require any scaling invariance of the equation.

math.AP↗