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Slim Ibrahim

Publications and source records attributed to Slim Ibrahim.

At least 37 records · Page 2Linked to original sources
arXiv2021

Numerical investigation on the Hill's type lunar problem with homogeneous potential

We consider the planar Hill's lunar problem with a homogeneous gravitational potential. The investigation of the system is twofold. First, the starting conditions of the trajectories are classified into three classes, that is bounded, escaping, and collisional. Second, we study the no-return property of the Lagrange point $L_2$ and we observe that the escaping trajectories are scattered exponentially. Moreover, it is seen that in the supercritical case, with $α\geq 2$, the basin boundaries are smooth. On the other hand, in the subcritical case, with $α< 2$ the boundaries between the different types of basins exhibit fractal properties.

nlin.CD↗
arXiv2021

Transverse stability of line soliton and characterization of ground state for wave guide Schrödinger equations

In this paper, we study the transverse stability of the line Schrödinger soliton under a full wave guide Schrödinger flow on a cylindrical domain $\mathbb R\times\mathbb T$. When the nonlinearity is of power type $|ψ|^{p-1}ψ$ with $p>1$, we show that there exists a critical frequency $ω_{p} >0$ such that the line standing wave is stable for $0<ω< ω_{p}$ and unstable for $ω> ω_{p}$. Furthermore, we characterize the ground state of the wave guide Schrödinger equation. More precisely, we prove that there exists $ω_{*} \in (0, ω_{p}]$ such that the ground states coincide with the line standing waves for $ω\in (0, ω_{*}]$ and are different from the line standing waves for $ω\in (ω_{*}, \infty)$.

math.AP↗
arXiv2020

Global existence and singularity of the Hill's type lunar problem with strong potential

We characterize the fate of the solutions of Hill's type lunar problem using the ideas of ground states from PDE. In particular, the relative equilibrium will be defined as the ground state, which satisfies some crucial energetic variational properties in our analysis. We study the dynamics of the solutions below, at, and (slightly) above the ground state energy threshold.

math.DS↗
arXiv2020

Finite-time Blowup and Ill-posedness in Sobolev Spaces of the Inviscid Primitive Equations with Rotation

Large scale dynamics of the oceans and the atmosphere are governed by the primitive equations (PEs). It is well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces. On the other hand, the inviscid PEs without rotation is known to be ill-posed in Sobolev spaces, and its smooth solutions can form singularity in finite time. In this paper, we extend the above results in the presence of rotation. First, we construct finite-time blowup solutions to the inviscid PEs with rotation, and establish that the inviscid PEs with rotation is ill-posed in Sobolev spaces in the sense that its perturbation around a certain steady state background flow is both linearly and nonlinearly ill-posed in Sobolev spaces. Its linear instability is of the Kelvin-Helmholtz type similar to the one appears in the context of vortex sheets problem. This implies that the inviscid PEs is also linearly ill-posed in Gevrey class of order $s > 1$, and suggests that a suitable space for the well-posedness is Gevrey class of order $s = 1$, which is exactly the space of analytic functions.

math.AP↗
arXiv2019

On pseudospectral bound for non-selfadjoint operators and its application to stability of Kolmogorov flows

We study the stability of the Kolmogorov flows which are stationary solutions to the two-dimensional Navier-Stokes equations in the presence of the shear external force. We establish the linear stability estimate when the viscosity coefficient $ν$ is sufficiently small, where the enhanced dissipation is rigorously verified in the time scale $O(ν^{-\frac12})$ for solutions to the linearized problem, which has been numerically conjectured and is much shorter than the usual viscous time scale $O(ν^{-1})$. Our approach is based on the detailed analysis for the resolvent problem. We also provide the abstract framework which is applicable to the resolvent estimate for the Kolmogorov flows.

math.AP↗
arXiv2019

Ground State Solutions of the Complex Gross Pitaevskii Equation Associated to Exciton-Polariton Bose-Einstein Condensates

We investigate the existence of ground state solutions of a Gross-Pitaevskii equation modeling the dynamics of pumped Bose Einstein condensates (BEC). The main interest in such BEC comes from its important nature as macroscopic quantum system, constituting an excellent alternative to the classical condensates which are hard to realize because of the very low temperature required. Nevertheless, the Gross Pitaevskii equation governing the new condensates presents some mathematical challenges due to the presence of the pumping and damping terms. Following a self-contained approach, we prove the existence of ground state solutions of this equation under suitable assumptions: This is equivalent to say that condensation occurs in these situations. We also solve the Cauchy problem of the nonlinear Schroedinger equation and prove some corresponding laws.

math.AP↗
arXiv2019

Uniqueness and nondegeneracy of ground states to nonlinear scalar field equations involving the Sobolev critical exponent in their nonlinearities for high frequencies

The study of the uniqueness and nondegeneracy of ground state solutions to semilinear elliptic equations is of great importance because of the resulting energy landscape and its implications for the various dynamics. In [AIKN3], semilinear elliptic equations with combined power-type nonlinearities involving the Sobolev critical exponent are studied. There, it is shown that if the dimension is four or higher, and the frequency is sufficiently small, then the positive radial ground state is unique and nondegenerate. In this paper, we extend these results to the case of high frequencies when the dimension is five and higher. After suitably rescaling the equation, we demonstrate that the main behavior of the solutions is given by the Sobolev critical part for which the ground states are explicit, and their degeneracy is well characterized. Our result is a key step towards the study of the different dynamics of solutions of the corresponding nonlinear Schrödinger and Klein-Gordon equations with energies above the energy of the ground state. Our restriction on the dimension is mainly due to the existence of resonances in dimension three and four.

math.AP↗
arXiv2019

Non-uniqueness for an energy-critical heat equation on $\mathbb{R}^2$

We construct a singular solution of a stationary nonlinear Schrödinger equation on $\mathbb{R}^2$ with square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of strong solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.

math.AP↗
arXiv2019

Sharp threshold nonlinearity for maximizing the Trudinger-Moser inequalities

We study existence of maximizer for the Trudinger-Moser inequality with general nonlinearity of the critical growth on $R^2$, as well as on the disk. We derive a very sharp threshold nonlinearity between the existence and the non-existence in each case, in asymptotic expansions with respect to growth and decay of the function. The expansions are explicit, using Apéry's constant. We also obtain an asymptotic expansion for the exponential radial Sobolev inequality on $R^2$.

math.AP↗
arXiv2019

Global Existence and Singularity of the N-body Problem with Strong Force

We use the idea of ground states and excited states in nonlinear dispersive equations (e.g. Klein-Gordon and Schrödinger equations) to characterize solutions in the N-body problem with strong force under some energy constraints. Indeed, relative equilibria of the N-body problem play a similar role as solitons in PDE. We introduce the ground state and excited energy for the N-body problem. {We are able to give a conditional dichotomy of the global existence and singularity below the excited energy in Theorem \ref{thm:dichotomy}, the proof of which seems original and simple. This dichotomy is given by the sign of a threshold function $K_ω$}. The characterization for the two-body problem in this new perspective is non-conditional and it resembles the results in PDE nicely. For $N\geq3$, we will give some refinements of the characterization, in particular, we examine the situation where there are infinitely transitions for the sign of $K_ω$.

math.CA↗
arXiv2018

Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem

Consider the thin-film equation $h_t + \left(h h_{yyy}\right)_y = 0$ with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation $h_t - (h^m)_{yy} = 0$, where $m > 1$ is a free parameter. Both porous-medium and thin-film equation degenerate as $h \searrow 0$, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves $h = \frac V 6 x^3 + νx^2$ for $x \ge 0$, where $x = y-V t$ and $V, ν\ge 0$ are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in $x$ in a right-neighborhood of the contact line $x = 0$. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as $x \searrow 0$ and $x \to \infty$, the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture.

math.AP↗
arXiv2018

Linear instability and nondegeneracy of ground state for combined power-type nonlinear scalar field equations with the Sobolev critical exponent and large frequency parameter

We consider combined power-type nonlinear scalar field equations with the Sobolev critical exponent. In \cite{AIKN3}, it was shown that if the frequency parameter is sufficiently small, then the positive ground state is nondegenerate and linearly unstable, together with an application to a study of global dynamics for nonlinear Schrödinger equations. In this paper, we prove the nondegeneracy and linear instability of the ground state frequency for sufficiently large frequency parameters. Moreover, we show that the derivative of the mass of ground state with respect to the frequency is negative.

math.AP↗
arXiv2018

Remarks on solitary waves and Cauchy problem for a Half-wave-Schrödinger equations

In this paper, we study the solitary wave and the Cauchy problem for Half-wave-Schrödinger equations in the plane. First, we show the existence and orbital stability of the ground states. Secondly, we prove that traveling waves exist and converge to zero as the velocity tends to $1$. Finally, we solve the Cauchy problem for initial data in $L^{2}_{x}H^{s}_{y}(\mathbb{R}^{2})$, with $s>\frac{1}{2}$.

math.AP↗
arXiv2018

Construction of type II blowup solutions for the 1-corotational energy supercritical wave maps

We consider the energy supercritical wave maps from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear wave equation $$\partial_t^2 u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$λ(t) \sim c_u(T-t)^\frac{\ell}γ, \quad \ell \in \mathbb{N}^*, \;\; \ell > γ= γ(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski for the energy supercritical nonlinear Schrödinger equation, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem.

math.AP↗
arXiv2017

Global well posedness for a two-fluid model

We study a two fluid system which models the motion of a charged fluid with Rayleigh friction, and in the presence of an electro-magnetic field satisfying Maxwell's equations. We study the well-posdness of the system in both space dimensions two and three. Regardless of the size of the initial data, we first prove the global well-posedness of the Cauchy problem when the space dimension is two. However, in space dimension three, we construct global weak-solutions à la Leray, and we prove the local well-posedness of Kato-type solutions. These solutions turn out to be global when the initial data are sufficiently small. Our results extend Giga-Yoshida (1984) ones to the space dimension two, and improve them in terms of requiring less regularity on the velocity fields.

math.AP↗
arXiv2017

On the stability of type II blowup for the 1-corotational energy supercritical harmonic heat flow

We consider the energy supercritical harmonic heat flow from $\mathbb{R}^d$ into the $d$-sphere $\mathbb{S}^d$ with $d \geq 7$. Under an additional assumption of 1-corotational symmetry, the problem reduces to the one dimensional semilinear heat equation $$\partial_t u = \partial^2_r u + \frac{(d-1)}{r}\partial_r u - \frac{(d-1)}{2r^2}\sin(2u).$$ We construct for this equation a family of $\mathcal{C}^{\infty}$ solutions which blow up in finite time via concentration of the universal profile $$u(r,t) \sim Q\left(\frac{r}{λ(t)}\right),$$ where $Q$ is the stationary solution of the equation and the speed is given by the quantized rates $$λ(t) \sim c_u(T-t)^\frac{\ell}γ, \quad \ell \in \mathbb{N}^*, \;\; 2\ell > γ= γ(d) \in (1,2].$$ The construction relies on two arguments: the reduction of the problem to a finite-dimensional one thanks to a robust universal energy method and modulation techniques developed by Merle, Raphaël and Rodnianski [Camb. Jour. Math, 3(4):439-617, 2015] for the energy supercritical nonlinear Schrödinger equation and by Raphaël and Schweyer [Anal. PDE, 7(8):1713-1805, 2014] for the energy critical harmonic heat flow, then we proceed by contradiction to solve the finite-dimensional problem and conclude using the Brouwer fixed point theorem. Moreover, our constructed solutions are in fact $(\ell - 1)$ codimension stable under perturbations of the initial data. As a consequence, the case $\ell = 1$ corresponds to a stable type II blowup regime.

math.AP↗