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Haroune Houamed

Publications and source records attributed to Haroune Houamed.

At least 19 recordsLinked to original sources
arXiv2026

Optimal Time-Decay of global solutions to the Navier-Stokes-Maxwell system

We prove the existence and uniqueness of global-in-time solutions to the Navier--Stokes--Maxwell (NSM) system for small initial data in the critical Fujita--Kato space $\dot H^{\frac{d}{2}-1}(\mathbb R^d)$, in any dimension $d\geq 3$. Under the additional assumption that the initial data belong to the Besov space $\dot B^{-\frac{d}{2}}_{2,\infty}(\mathbb R^d)$, we also establish the optimal decay rate $t^{-\frac{s}{2}-\frac{d}{4}}$ at infinity of the solution in $\dot H^s(\mathbb R^d)$, for any $s\in\left(-\frac{d}{2},\frac{d}{2}-1\right]$. This is achieved by constructing a Lyapunov functional that is equivalent to the pointwise energy on the Fourier side, and by combining it with an adaptation of the Fourier splitting method to establish its optimal decay rate. All the analysis carried out here - from the global existence theory to the study of the large-time behavior - is performed within a framework that is uniform with respect to the speed of light $c\in(0,\infty)$. In particular, in the non-relativistic limit $c\to\infty$, this allows us to recover the same results for the corresponding limiting magnetohydrodynamic system.

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arXiv2026

Strong Ill-Posedness in critical and subcritical regimes for the Hunter-Saxton equation

We study the Hunter--Saxton equation on the real line and its global dissipative solution in the energy space $ L^\infty(\mathbb R)\cap \dot H^1(\mathbb R). $ We prove strong ill-posedness through instantaneous failure of Sobolev regularity at and below the Lipschitz threshold. More precisely, for every $s\in(1,\nicefrac32]$, we construct $ u_0\in L^\infty(\mathbb R)\cap\dot H^1(\mathbb R)\cap\dot H^s(\mathbb R) $ whose unique global dissipative solution satisfies $u\notin C([0,T]; \dot H^s (\mathbb R))$, for every $T>0$. The constructions differ substantially in the subcritical and critical regimes. For $1<s<\nicefrac32$, we superpose rescaled, localized bubbles with increasingly negative slopes; subcritical scaling preserves $\dot H^s$-summability, while the explicit characteristic formula produces norm inflation. At $s=\nicefrac32$, where scaling yields no smallness, we use logarithmically distributed compactly supported multiscale profiles whose breaking times converge to zero. A one-sided localization principle and an almost-orthogonality estimate then transfer the inflation of individual profiles to the full dissipative solution.

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arXiv2026

Singular traveling waves for the Euler-Poisson system

We consider the Euler-Poisson system for ions where the electrons are given by a Maxwell-Boltzmann distribution, and we investigate the existence of one-dimensional periodic traveling waves. More precisely, we first establish the existence of a smooth global branch of bifurcation emanating from a constant equilibrium. We then construct a singular traveling wave emerging as the limiting profile at the end of the global curve of bifurcation. Our analysis accommodates a wide class of pressure laws and provides a comprehensive characterization of both smooth and singular traveling waves. A central difficulty in this model arises from the exponential nonlinearity, induced by the nonlocal Poisson-Boltzmann equation, which prevents any explicit representation of the electron field in terms of the ion density. This poses significant obstacles compared to previous studies on related models, where such explicit formulas were crucial for global bifurcation arguments.

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arXiv2025

Sharp Strong Convergence in Ideal Flows

We investigate the strong convergence of weak solutions to the two-dimensional Quasi-Geostrophic Shallow-Water (QGSW) equation as the inverse Rossby radius tends to zero. In this limit, we recover the Yudovich solution of the incompressible Euler equations. We prove that the vorticity convergence holds in $L^\infty_t L^p_x$, for any finite integrability exponent $p<\infty$. This extends to the case $p=\infty$ provided that the initial vorticities are continuous and converge uniformly. We also discuss the sharpness of this limit by demonstrating that the continuity assumption on the initial data is necessary for the endpoint convergence in $L^\infty_{t,x}$. The proof of the strong convergence relies on the {\em Extrapolation Compactness} method, recently introduced by Arsénio and the first author to address similar stability questions for the Euler equations. The approach begins with establishing the convergence in a lower regularity space, at first. Then, in a later step, the convergence to Yudovich's vorticity of Euler equations in Lebesgue spaces comes as a consequence of a careful analysis of the evanescence of specific high Fourier modes of the QGSW vorticity. A central challenge arises from the absence of a velocity formulation for QGSW, which we overcome by employing advanced tools from Littlewood Paley theory in endpoint settings. The sharpness of the convergence in the endpoint $L^\infty_{t,x}$ case is obtained in the context of vortex patches, drawing insights from key findings on uniformly rotating and stationary solutions of active scalar equations.

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arXiv2025

Ill-Posedness of the Incompressible Euler--Maxwell Equations in the Yudovich Class

It was shown recently by Arsénio and the author that the two-dimensional incompressible Euler--Maxwell system is globally well-posed in the Yudovich class, provided that the electromagnetic field enjoys appropriate conditions, including the Normal Structure. In this paper, we prove that this assumption is sharp, in the sense that the Euler--Maxwell system becomes ill-posed in the Yudovich class for initial data that do not obey the Normal Structure condition. The proof applies to both the whole plane and the two-dimensional torus, and holds for any value of the speed of light $c\in (0,\infty)$. This is achieved by expanding the magnetic field around a horizontal background and showing that the Lorentz force can be decomposed into two parts: the first is in the form of a singular operator acting on the vorticity, and the second, a "remainder", is of lower order when analyzed in a specific time regime.

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arXiv2024

Global unique solutions to the planar inhomogeneous Navier--Stokes--Maxwell equations

The evolution of an electrically conducting imcompressible fluid with nonconstant density can be described by a set of equations combining the continuity, momentum and Maxwell's equations; altogether known as the inhomogeneous Navier--Stokes--Maxwell system. In this paper, we focus on the global well-posedness of these equations in two dimensions. Specifically, we are able to prove the existence of global energy solutions, provided that the initial velocity field belongs to the Besov space $\dot{B}^{r}_{p,1}(\mathbb{R}^2)$, with $r=-1+\frac{2}{p}$, for some $p\in (1,2)$, while the initial electromagnetic field enjoys some $H^s(\mathbb{R}^2)$ Sobolev regularity, for some $s \geq 2-\frac{2}{p} \in (0,1)$, and whenever the initial fluid density is bounded pointwise and close to a nonnegative constant. Moreover, if it is assumed that $s>\frac{1}{2}$, then the solution is shown to be unique in the class of all energy solutions. It is to be emphasized that the solutions constructed here are global and uniformly bounded with respect to the speed of light $c\in (0,\infty)$. This important fact allows us to derive the inhomogeneous MHD system as the speed of light tends to infinity.

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arXiv2024

Damped Strichartz estimates and the incompressible Euler--Maxwell system

Euler--Maxwell systems describe the dynamics of inviscid plasmas. In this work, we consider an incompressible two-dimensional version of such systems and prove the existence and uniqueness of global weak solutions, uniformly with respect to the speed of light $c\in (c_0,\infty)$, for some threshold value $c_0>0$ depending only on the initial data. In particular, the condition $c>c_0$ ensures that the velocity of the plasma nowhere exceeds the speed of light and allows us to analyze the singular regime $c\to\infty$. The functional setting for the fluid velocity lies in the framework of Yudovich's solutions of the two-dimensional Euler equations, whereas the analysis of the electromagnetic field hinges upon the refined interactions between the damping and dispersive phenomena in Maxwell's equations in the whole space. This analysis is enabled by the new development of a robust abstract method allowing us to incorporate the damping effect into a variety of existing estimates. The use of this method is illustrated by the derivation of damped Strichartz estimates (including endpoint cases) for several dispersive systems (including the wave and Schrödinger equations), as well as damped maximal regularity estimates for the heat equation. The ensuing damped Strichartz estimates supersede previously existing results on the same systems.

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arXiv2024

Convergence rate for a regularized scalar conservation law

This work revisits a recent finding by the first author concerning the local convergence of a regularized scalar conservation law. We significantly improve the original statement by establishing a global convergence result within the Lebesgue spaces $L^\infty_{\mathrm{loc}}(\mathbb{R}^+;L^p(\mathbb{R}))$, for any $p \in [1,\infty)$, as the regularization parameter $\ell$ approaches zero. Notably, we demonstrate that this stability result is accompanied by a quantifiable rate of convergence. A key insight in our proof lies in the observation that the fluctuations of the solutions remain under control in low regularity spaces, allowing for a potential quantification of their behavior in the limit as $\ell\to 0$. This is achieved through a careful asymptotic analysis of the perturbative terms in the regularized equation, which, in our view, constitutes a pivotal contribution to the core findings of this paper.

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arXiv2024

Dynamic Behavior of a Multi-Layer Quasi-Geostrophic Model: Weak and Time-Periodic Solutions

The quasi-geostrophic two-layer (QS2L) system models the dynamic evolution of two interconnected potential vorticities, each is governed by an active scalar equation. These vorticities are linked through a distinctive combination of their respective stream functions, which can be loosely characterized as a parameterized blend of both Euler and shallow-water stream functions. In this article, we study (QS2L) in two directions: First, we prove the existence and uniqueness of global weak solutions in the class of Yudovich, that is when the initial vorticities are only bounded and Lebesgue-integrable. The uniqueness is obtained as a consequence of a stability analysis of the flow-maps associated with the two vorticities. This approach replaces the relative energy method and allows us to surmount the absence of a velocity formulation for (QS2L). Second, we show how to construct $m$-fold time-periodic solutions bifurcating from two arbitrary distinct initial discs rotating with the same angular velocity. This is achieved provided that the number of symmetry $m$ is large enough, or for any symmetry $m\in \mathbb{N}^*$ as long as one of the initial radii of the discs does not belong to some set that contains, at most, a finite number of elements. Due to its multi-layer structure, it is essential to emphasize that the bifurcation diagram exhibits a two-dimensional pattern. Upon analysis, it reveals some similarities with the scheme accomplished for the doubly connected V-states of the Euler and shallow-water equations. However, the coupling between the equations gives rise to several difficulties in various stages of the proof when applying Crandall-Rabinowitz's Theorem. To address this challenge, we conduct a careful analysis of the coupling between the kernels associated with the Euler and shallow-water equations.

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arXiv2024

Uniformly rotating vortices for the lake equation

We investigate the existence of time-periodic vortex patch solutions, in both simply and doubly-connected cases, for the two-dimensional lake equation where the depth function of the lake is assumed to be non-degenerate and radial. The proofs employ bifurcation techniques, where the most challenging steps are related to the regularity study of some nonlinear functionals and the spectral analysis of their linearized operators around Rankine type vortices. The main difficulties stem from the roughness and the implicit form of the Green function connecting the fluid vorticity and its stream function. We handle in part these issues by exploring the asymptotic structure of the solutions to the associated elliptic problem. As to the distribution of the spectrum, it is tackled by a fixed-point argument through a perturbative approach.

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arXiv2023

Stability analysis of two-dimensional ideal flows with applications to viscous fluids and plasmas

We are interested in the stability analysis of two-dimensional incompressible inviscid fluids. Specifically, we revisit a recent result on the stability of Yudovich's solutions to the incompressible Euler equations in $L^\infty([0,T];H^1)$ by providing a new approach to its proof based on the idea of compactness extrapolation and by extending it to the whole plane. This new method of proof is robust and, when applied to viscous models, leads to a remarkable logarithmic improvement on the rate of convergence in the vanishing viscosity limit of two-dimensional fluids. Loosely speaking, this logarithmic gain is the result of the fact that, in appropriate high-regularity settings, the smoothness of solutions to the Euler equations at times $t\in [0,T)$ is strictly higher than their regularity at time $t=T$. This ``memory effect'' seems to be a general principle which is not exclusive to fluid mechanics. It is therefore likely to be observed in other setting and deserves further investigation. Finally, we also apply the stability results on Euler systems to the study of two-dimensional ideal plasmas and establish their convergence, in strong topologies, to solutions of magnetohydrodynamic systems, when the speed of light tends to infinity. The crux of this asymptotic analysis relies on a fine understanding of Maxwell's system.

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arXiv2023

Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics

This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier--Stokes--Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light $c\in (c_0, \infty)$, for some threshold $c_0>0$ depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime $c\rightarrow \infty$ and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting. The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler--Maxwell system to exploit the dissipative--dispersive structure of Maxwell's system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime $c\to\infty$ allows us to derive a robust nonlinear energy estimate which holds uniformly in $c$. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system. This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as $c\to\infty$.

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arXiv2021

Rigidity aspects of singular patches in stratified flows

We explore the local well-posedness theory for the 2d inviscid Boussinesq system when the vorticity is given by a singular patch. We give a significant improvement of \cite{Hassainia-Hmidi} by replacing their compatibility assumption on the density with a constraint on its platitude degree on the singular set. The second main contribution focuses on the same issue for the partial viscous Boussinesq system. We establish a uniform LWP theory with respect to the vanishing conductivity. This issue is much more delicate than the inviscid case and one should carefully deal with various difficulties related to the diffusion effects which tend to alter some local structures. The weak a priori estimates are not trivial and refined analysis on transport-diffusion equation subject to a logarithmic singular potential is required. Another difficulty stems from some commutators arising in the control of the co-normal regularity that we counterbalance in part by the maximal smoothing effects of transport-diffusion equation advected by a velocity field which scales slightly below the Lipschitz class.

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arXiv2021

Remarks on the global well-posedness of the axisymmetric Boussinesq system with rough initial data

This work concerns the global well-posedness problem for the 3D axisymmetric viscous Boussinesq system with critical rough initial data. More precisely, we aim to extending our recent result \cite{Hanachi-Houamed-Zerguine} to the case of initial data of measure type. To this end, we should first develop some notions of axisymmetric measures in a general context, then, in the spirit of \cite{Gallay-Sverak}, we prove the global wellposedness result provided that the atomic parts of the initial measures are small enough.

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arXiv2020

Well-posedness and long time behavior for the Electron Inertial Hall-MHD system in Besov and Kato-Herz spaces

In this paper, we study the wellposedeness of the Hall-magnetohydrodynamic system augmented by the effect of electron inertia. Our main result consists of generalising the wellposedness one in \cite{Zhao} from the Sobolev context to the general Besov spaces and Kato-Herz space, then we show that we can reduce the requared regularity of the magnetic field in the first result modulo an additional condition on the maximal time of existence. Finally, we show that the $\widehat{L}^p$ (and eventually the $L^p$) norm of the solution $(u,B,\nabla \times B)$ associated to an initial data in $ \widehat{B}^{\frac{3}{p}-1}_{p,\infty}(\mathbb{R}^3)$, is controled by $t^{-\frac{1}{2}(1-\frac{3}{p})}$, for all $p\in (3,\infty)$, which provides a polynomial decay to zero of the $\widehat{L}^p$ norm of the solution.

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arXiv2020

About some possible blow-up conditions for the 3-D Navier-Stokes equations

In this paper, we study some conditions related to the question of the possible blow-up of regular solutions to the 3D Navier-Stokes equations. In particular, up to a modification in a proof of a very recent result from \cite{Isab}, we prove that if one component of the velocity remains small enough in a sub-space of $\dot{H}^{\frac{1}{2}}$ "almost" scaling invariant, then the 3D Navier Stokes is globally wellposed. In a second time, we investigate the same question under some conditions on one component of the vorticity and unidirectional derivative of one component of the velocity in some critical Besov spaces of the form $L^p_T(\dot{B}_{2,\infty}^{α, \frac{2}{p}-\frac{1}{2}-α})$ or $L^p_T(\dot{B}_{q,\infty}^{ \frac{2}{p}+\frac{3}{q}-2})$.

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arXiv2020

Uniqueness result for the 3-D Navier-Stokes-Boussinesq Equations with Horizontal Dissipation

In this paper, for the 3-D Navier-Stokes-Boussinesq system with horizontal dissipation, where there is no smoothing effect on the vertical derivatives, we prove a uniqueness result of solutions $ (u,ρ)\in L^{\infty}_T\big( H^{0,s}\times H^{0,1-s}\big)$ with $ (\nabla_h u,\nabla_hρ)\in L^{2}_T\big( H^{0,s}\times H^{0,1-s}\big)$ and $s\in [1/2,1]$. As a consequence, we improve the conditions stated in the paper \cite{Miao} in order to obtain a global well-posedness result in the case of axisymmetric initial data.

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arXiv2020

On the global well-posedness of axisymmetric viscous Boussinesq system in critical Lebesgue spaces

The contribution of this paper will be focused on the global existence and uniqueness topic in three-dimensional case of the axisymmetric viscous Boussinesq system in critical Lebesgue spaces. We aim at deriving analogous results for the classical two-dimensional and three-dimensional axisymmetric Navier-Stokes equations recently obtained in \cite{Gallay,Gallay-Sverak}. Roughly speaking, we show essentially that if the initial data $(v_0,ρ_0)$ is axisymmetric and $(ω_0,ρ_0)$ belongs to the critical space $L^1(Ω)\times L^1(\mathbb{R}^3)$, with $ω_0$ is the initial vorticity associated to $v_0$ and $Ω=\{(r,z)\in\mathbb{R}^2:r>0\}$, then the viscous Boussinesq system has a unique global solution.

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