arXiv subjects

Slim Ibrahim

Publications and source records attributed to Slim Ibrahim.

At least 19 recordsLinked to original sources

arXiv2026

Well-posedness and stability of the self-similar profile for a thin-film equation with gravity

We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem in mass-Lagrangian coordinates and exploit the underlying gradient-flow structure of the equation with respect to a weighted $L^2$ inner product, where the weight is given by the self-similar source-type profile. This framework allows us to establish a coercivity result for the Hessian (the linearization around the self-similar solution) in a suitably weighted inner product. As a consequence, we prove the convergence of perturbations toward the self-similar profile at an algebraic rate of order $t^{-\frac 1 5}$, in arbitrary scales of weighted Sobolev norms. The analysis relies on maximal-regularity estimates for the linearized evolution, combined with appropriate estimates for the nonlinear terms. Notably, beyond perturbative regimes and in contrast to previous results for the thin-film equation (convergence to the Smyth-Hill profile) or the porous-medium equation (convergence to the Barenblatt-Pattle solution), our analysis does not rely on an explicit (algebraic) representation of the self-similar profile. Instead, it is based solely on a systematic use of the ordinary differential equation satisfied by the self-similar solution, together with a careful analysis of its boundary asymptotics. As a result, we expect that the approach developed here can serve as a flexible toolbox for the study of more general classes of equations and for the stability analysis of special solutions in future work.

math.AP↗

arXiv2025

Phase transition thresholds and chiral magnetic fields of general degree

We study a variational problem for the Landau--Lifshitz energy with Dzyaloshinskii--Moriya interactions arising in 2D micromagnetics, focusing on the Bogomol'nyi regime. We first determine the minimal energy for arbitrary topological degree, thereby revealing two types of phase transitions consistent with physical observations. In addition, we prove the uniqueness of the energy minimizer in degrees $0$ and $-1$, and nonexistence of minimizers for all other degrees. Finally, we show that the homogeneous state remains stable even beyond the threshold at which the skyrmion loses stability, and we uncover a new stability transition driven by the Zeeman energy.

math.AP↗

arXiv2025

Global well-posedness for intermediate NLS with nonvanishing conditions at infinity

The intermediate nonlinear Schr\"odinger equation (INLS) describes the dynamics of the envelope of weakly nonlinear internal waves in a stratified fluid of finite depth. While the INLS equation is known to admit dark soliton solutions, these solutions possess nonvanishing boundary conditions at spatial infinity and therefore fall outside the scope of existing well-posedness frameworks. This paper establishes the local and global well-posedness of a generalized INLS equation in Zhidkov-type spaces tailored to these nonvanishing boundary conditions. Furthermore, we rigorously justify the deep-water limit, proving that solutions of the generalized INLS converge to those of the generalized Calogero-Moser (CM) derivative NLS equation in Zhidkov-type spaces. Our well-posedness theory relies on the modified energy method combined with frequency envelopes, marking the first application of these techniques to Zhidkov-type spaces.

math.AP↗

arXiv2025

Two-Point Vortex Confinement in a simply connected domain

This paper investigates the vortex confinement property of the two-point vortex system in a planar domain. We compute the time over which initial point vortices around a stable stationary point remain within a slightly larger ball. In particular, we show that this concentration persists indefinitely regardless of the vorticity strengths. In the borderline of the stability condition, we show that this time becomes a power law, if in addition, one relaxes the size of the stability ball.

math-ph↗

arXiv2025

Classification and qualitative properties of positive solutions to double-power nonlinear stationary Schr\"{o}dinger equations

In this paper, we investigate positive radial solutions to double-power nonlinear stationary Schrodinger equations in three space dimensions. It is now known that the non-uniqueness of H^{1}-positive solutions can occur in three dimensions when the frequency is sufficiently small. Under suitable conditions, in addition to the ground state solution (whose L^{\infty} norm vanishes as the frequency tends to zero), there exists another positive solution that minimizes a different constrained variational problem, with an L^{\infty} norm diverging as the frequency tends to zero (see Theorem 1.4). We classify all positive solutions with small frequency into two categories: the ground state and the Aubin-Talenti type solution. As a consequence, we establish the multiplicity of positive solutions. Finally, we also examine the non-degeneracy and Morse index of each positive solution.

math.AP↗

arXiv2025

On the Profile of Singularity Formation for the Incompressible Hydrostatic Boussinesq system

The primitive equations (PEs) model planetary large-scale oceanic and atmospheric dynamics. While it has been shown that there are smooth solutions to the inviscid PEs (also called the hydrostatic Euler equations) with constant temperature (isothermal) that develop stable singularities in finite time, the effect of non-constant temperature on the singularity formation has not been established yet. This paper studies the stability of singularity formation for non-constant temperature in two scenarios: when there is no diffusion in the temperature, or when a vertical diffusivity is added to the temperature dynamics. For both scenarios, our results indicate that the variation of temperature affects neither the formation of singularity, nor its stability, in the velocity field, respectively.

math.AP↗

arXiv2025

Global perturbation of isolated equivariant chiral skyrmions from the Bogomol'nyi case

Isolated skyrmion solutions to the two-dimensional Landau-Lifshitz equation with Dzyaloshinskii-Moriya interaction, Zeeman term, and easy-plane anisotropy of various strengths are studied. In the full range of parameter values for which the energy is a positive variation of the Bogomol'nyi case, we construct solutions to the corresponding Euler-Lagrange equation and analyze their qualitative properties, including monotonicity, exponential decay, and stability. Our analysis is global and non-perturbative. Moreover, we derive precise estimates qualifying the difference between these solutions and those in the Bogomol'nyi regime. A key ingredient of our approach is a novel resolvent estimate for the linearized operator, which remains uniform with respect to additional implicit potentials arising in the problem.

math.AP↗

arXiv2023

The relativistic Vlasov-Maxwell system: Local smooth solvability for weak topologies

This article is devoted to the Relativistic Vlasov-Maxwell system in space dimension three. We prove the local smooth solvability for weak topologies (and its long time version for small data). This result is derived from a representation formula decoding how the momentum spreads, and showing that the domain of influence in momentum is controlled by mild information. We do so by developing a Radon Fourier analysis on the RVM system, leading to the study of a class of singular weighted integrals. In the end, we implement our method to construct smooth solutions to the RVM system in the regime of dense, hot and strongly magnetized plasmas. This is done by investigating the stability properties near a class of approximate solutions.

math.AP↗

arXiv2022

Phase transition threshold and stability of magnetic skyrmions

We examine the stability of vortex-like configuration of magnetization in magnetic materials, so-called the magnetic skyrmion. These correspond to critical points of the Landau-Lifshitz energy with the Dzyaloshinskii-Moriya (DM) interactions. In an earlier work of the D\"oring and Melcher, it is known that the skyrmion is a ground state when the coefficient of the DM term is small. In this paper, we prove that there is an explicit critical value of the coefficient above which the skyrmion is unstable, while stable below this threshold. Moreover, we show that in the unstable regime, the infimum of energy is not bounded below, by giving an explicit counterexample with a sort of helical configuration. This mathematically explains the occurrence of phase transition observed in some experiments.

math.AP↗

arXiv2022

Refined probabilistic local well-posedness for a cubic Schr\"odinger half-wave equation

We obtain probabilistic local well-posedness in quasilinear regimes for the Schr\"odinger half-wave equation with a cubic nonlinearity. We need to use a refined ansatz because of the lack of probabilistic smoothing in the Picard's iterations, which is due to the high-low-low frequency interactions. The proof is an adaptation of the method of Bringmann on the derivative nonlinear wave equation to Schr\"odinger-type equations. In addition, we discuss ill-posedness results for this equation.

math.AP↗

arXiv2021

Stable Singularity Formation for the Inviscid Primitive Equations

The primitive equations (PEs) model large scale dynamics of the oceans and the atmosphere. While it is by now well-known that the three-dimensional viscous PEs is globally well-posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blowup mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic in time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameters family of non-smooth profiles, and is stable only by smoother perturbations.

math.AP↗

arXiv2021

Non-existence of ground states and gap of variational problems for combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions

In this paper, we consider minimization problems related to the combined power-type nonlinear scalar field equations involving the Sobolev critical exponent in three space dimensions. In four and higher space dimensions, it is known that for any frequency and any power of the subcritical nonlinearity, there exists a ground state. In contrast to those cases, when the space dimension is three and the subcritical power is three or less, we can show that there exists a threshold frequency, above which no ground state exists, and below which the ground state exists. Furthermore, we prove the difference between two typical variational problems used to characterize the ground states.

math.AP↗

arXiv2021

Uniform Lifetime for Classical Solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell System

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function $f(t,\cdot)$. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field $ x \mapsto \epsilon^{-1} \mathbf{B}_e(x)$, where $\epsilon$ is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields $(E,B) $ are supposed to be small. In the non-magnetized setting, local $ C^1 $-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since $ \epsilon^{-1} $ is large, standard results predict that the lifetime $T_\epsilon$ of solutions may shrink to zero when $ \epsilon $ goes to $ 0 $. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound ($0<T<T_\epsilon$) on the lifetime of solutions and uniform Sup-Norm estimates. A bootstrap argument allows us to show $f$ remains at a distance $\epsilon$ from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.

math.AP↗

arXiv2021

Pitchfork bifurcation at line solitons for nonlinear Schr\"{o}dinger equations on the product space $\mathbb{R} \times \mathbb{T}$

In this paper, we study the bifurcation problem from a line soliton for a stationary nonlinear Schr\"{o}dinger equation on the product space $\mathbb{R} \times \mathbb{T}$. We extend earlier results to a larger class of the nonlinearity in the equation. The salient point of our analysis relies on a lower bound of solution to the ``auxiliary equation'' and then on the application of the Crandall-Rabinowitz argument

math.AP↗

arXiv2021

Transverse stability of line soliton and characterization of ground state for wave guide Schr\"{o}dinger equations

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

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

On the effect of rotation on the life-span of analytic solutions to the $3D$ inviscid primitive equations

We study the effect of the rotation on the life-span of solutions to the $3D$ hydrostatic Euler equations with rotation and the inviscid Primitive equations (PEs) on the torus. The space of analytic functions appears to be the natural space to study the initial value problem for the inviscid PEs with general initial data, as they have been recently shown to exhibit Kelvin-Helmholtz type instability. First, for a short interval of time that is independent of the rate of rotation $|\Omega|$, we establish the local well-posedness of the inviscid PEs in the space of analytic functions. In addition, thanks to a fine analysis of the barotropic and baroclinic modes decomposition, we establish two results about the long time existence of solutions. (i) Independently of $|\Omega|$, we show that the life-span of the solution tends to infinity as the analytic norm of the initial baroclinic mode goes to zero. Moreover, we show in this case that the solution of the $3D$ inviscid PEs converges to the solution of the limit system, which is governed by the $2D$ Euler equations. (ii) We show that the life-span of the solution can be prolonged unboundedly with $|\Omega|\rightarrow \infty$, which is the main result of this paper. This is established for "well-prepared" initial data, namely, when only the Sobolev norm (but not the analytic norm) of the baroclinic mode is small enough, depending on $|\Omega|$. Furthermore, for large $|\Omega|$ and "well-prepared" initial data, we show that the solution to the $3D$ inviscid PEs is approximated by the solution to a simple limit resonant system with the same initial data.

math.AP↗

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↗