Searcharxiv⌕ Search

arXiv subjects

Hammadi Abidi

Publications and source records attributed to Hammadi Abidi.

14 recordsLinked to original sources
arXiv2026

Global well-posedness of 2-D incompressible anisitropic Navier-Stokes equations with variable density

We establish the global well-posedness for two-dimensional inhomogeneous, incompressible, anisotropic Navier-Stokes systems. Two specific models are analyzed: one with partial dissipation (referred to as (AINS)) and one with only horizontal dissipation (referred to as (HINS)), under the assumption that the initial density is bounded away from zero and infinity. For the (AINS) system posed in the whole plane $\mathbb{R}^2$, we prove the existence and uniqueness of global solutions for finite-energy initial data, employing time-weighted energy estimates and a duality argument. For the (HINS) system on the domain $\mathbb{T} \times \mathbb{R}$, global well-posedness is established for sufficiently small initial velocity and sufficiently small density variation. By exploiting the anisotropic dissipation structure, employing Poincaré-type inequalities to obtain exponential decay for the oscillatory part of the velocity field, and carefully balancing the growth of the density gradient, we overcome the principal analytical challenges.

math.AP↗
arXiv2025

Global well-posedness of 3-D density-dependent incompressible MHD equations with variable resistivity

In this paper, we investigate the global existence of weak solutions to 3-D inhomogeneous incompressible MHD equations with variable viscosity and resistivity, which is sufficiently close to $1$ in $L^\infty(\mathbb{R}^3),$ provided that the initial density is bounded from above and below by positive constants, and both the initial velocity and magnetic field are small enough in the critical space $\dot{H}^{\frac{1}{2}}(\mathbb{R}^3).$ Furthermore, if we assume in addition that the kinematic viscosity equals $1,$ and both the initial velocity and magnetic field belong to $\dot{B}^{\frac{1}{2}}_{2,1}(\mathbb{R}^3),$ we can also prove the uniqueness of such solution.

math.AP↗
arXiv2024

Global refined Fujita-Kato solution of 3-D inhomogeneous incompressible Navier-Stokes equations with large density

We investigate the global unique Fujita-Kato solution to the 3-D inhomogeneous incompressible Navier-Stokes equations with initial velocity $u_0$ being sufficiently small in critical spaces and with initial density being bounded from above and below. We first prove the global existence of Fujita-Kato solution to the system if we assume in addition that the initial velocity is in the critical Sobolev space. While under the additional assumptions that the initial velocity is in the critical Besov space and initial density is in a critical Besov space, we prove that the solutions are controlled by the norm of the initial data. Our results not only improve the smallness condition in the previous references for the initial velocity concerning the global Fujita-Kato solution of the system but also improve the exponential-in-time growth estimate for the solution in the paper [Abidi-Gui-Zhang, ARMA 2012] to be the uniform-in-time estimate.

math.AP↗
arXiv2023

On the global existence and uniqueness of solution to 2-D inhomogeneous incompressible Navier-Stokes equations in critical spaces

In this paper, we establish the global existence and uniqueness of solution to $2$-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.2} with initial data in the critical spaces. Precisely, under the assumption that the initial velocity $u_0$ in $L^2 \cap\dot B^{-1+\frac{2}{p}}_{p,1}$ and the initial density $ρ_0$ in $L^\infty$ and having a positive lower bound, which satisfies $1-ρ_0^{-1}\in \dot B^{\frac{2}λ}_{λ,2}\cap L^\infty,$ for $p\in[2,\infty[$ and $λ\in [1,\infty[$ with $\frac{1}{2}<\frac{1}{p}+\frac{1}λ\leq1,$ the system \eqref{1.2} has a global solution. The solution is unique if $p=2.$ With additional assumptions on the initial density in case $p>2,$ we can also prove the uniqueness of such solution. In particular, this result improves the previous work in \cite{AG2021} where $u_{0}$ belongs to $\dot{B}_{2,1}^{0}$ and $ρ_0^{-1}-1$ belongs to $\dot{ B}_{\frac{2}{\varepsilon},1}^{\varepsilon}$, and we also remove the assumption that the initial density is close enough to a positive constant in \cite{DW2023} yet with additional regularities on the initial density here.

math.AP↗
arXiv2015

On the global solution of 3-D MHD system with initial data near equilibrium

In this paper, we prove the global existence of smooth solutions to the three-dimensional incompressible magneto-hydrodynamical system with initial data close enough to the equilibrium state, $(e_3,0).$ Compared with the the previous works \cite{XLZMHD1, XZ15}, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is non-degenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in \cite{XLZMHD1, XZ15}. By using Frobenius Theorem and anisotropic Littlewood-Paley theory for the Lagrangian formulation of the system, we achieve the global $L^1$ in time Lipschwitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large time decay rate of the solution is also obtained.

math.AP↗
arXiv2015

Global well-posedness of helicoidal Euler equations

This paper deals with the global existence and uniqueness results for the three-dimensional incompressible Euler equations with a particular structure for initial data lying in critical spaces. In this case the BKM criterion is not known.

math.AP↗
arXiv2014

Global smooth axisymmetric solutions of 3-D Inhomogenenous incompressible Navier-Stokes system

In this paper, we investigate the global regularity to 3-D inhomogeneous incompressible Navier-Stokes system with axisymmetric initial data which does not have swirl component for the initial velocity. We first prove that the $L^\infty$ norm to the quotient of the inhomogeneity by $r,$ namely $a/r\eqdefa\bigl(1/\r-1\bigr)\bigl/r,$ controls the regularity of the solutions. Then we prove the global regularity of such solutions provided that the $L^\infty$ norm of $a_0/r$ is sufficiently small. Finally, with additional assumption that the initial velocity belongs to $L^p$ for some $p\in [1,2),$ we prove that the velocity field decays to zero with exactly the same rate as the classical Navier-Stokes system.

math.AP↗
arXiv2013

On the global well-posedness of 2-D density-dependent Navier-Stokes system with variable viscosity

Given solenoidal vector $u_0\in H^{-2\d}\cap H^1(\R^2),$ $\r_0-1\in L^2(\R^2),$ and $\r_0 \in L^\infty\cap\dot{W}^{1,r}(\R^2)$ with a positive lower bound for $\d\in (0,\f12)$ and $2<r<\f{2}{1-2\d},$ we prove that 2-D incompressible inhomogeneous Navier-Stokes system \eqref{1.1} has a unique global solution provided that the viscous coefficient $μ(\r_0)$ is close enough to 1 in the $L^\infty$ norm compared to the size of $\d$ and the norms of the initial data. With smoother initial data, we can prove the propagation of regularities for such solutions. Furthermore, for $1<p<4,$ if $(\r_0-1,u_0)$ belongs to the critical Besov spaces $\dB^{\f2p}_{p,1}(\R^2)\times \bigl(\dB^{-1+\f2p}_{p,1}\cap L^2(\R^2)\bigr)$ and the $\dB^{\f2p}_{p,1}(\R^2)$ norm of $\r_0-1$ is sufficiently small compared to the exponential of $\|u_0\|_{L^2}^2+\|u_0\|_{\dB^{-1+\f2p}_{p,1}},$ we prove the global well-posedness of \eqref{1.1} in the scaling invariant spaces. Finally for initial data in the almost critical Besov spaces, we prove the global well-posedness of \eqref{1.1} under the assumption that the $L^\infty$ norm of $\r_0-1$ is sufficiently small.

math.AP↗
arXiv2008

Global existence for the MHD system in critical spaces

In this article, we show that the magneto-hydrodynamic system (MHD) in $\R^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space $\dot B^{\frac{N}{p_1}}_{p_1,1}(\R^N)\times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N) \times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N)$ for all $1<p_2<+\infty$ and some $1<p_1\leq\frac{2N}{3}$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\leq2N$. We prove also that the constructed solution exist globally in time if the initial data are small enough. In particular, this allows us to work in the frame of Besov space with negative regularity indices and this fact is particularly important when the initial data are strong oscillating.

math.AP↗
arXiv2008

On the global well-posedness for the axisymmetric Euler equations

This paper deals with the global well-posedness of the 3D axisymmetric Euler equations for initial data lying in critical Besov spaces $B_{p,1}^{1+3/p}$. In this case the BKM criterion is not known to be valid and to circumvent this difficulty we use a new decomposition of the vorticity.

math.AP↗