Convergence rates for a rotating MHD system with evanescent viscosity
Univ Paris Est Creteil, Univ Gustave Eiffel, CNRS, LAMA UMR8050（巴黎东 Créteil大学、古斯塔夫·埃菲尔大学、法国国家科学研究中心、LAMA实验室）；Institut Camille Jordan, Université Claude-Bernard-Lyon-I（卡米尔·让研究所、克洛德·贝尔纳里昂第一大学；LAMA，巴黎东 Créteil大学）；LAMA, Univ Paris Est Creteil
In this article, we focus on a MHD model that takes into account the Coriolis force. In this case, the strength of the rotation is measured by the Rossby number $ε$ (positive). Under the assumption that the rotation is strong (that is, when $ε$ goes to zero) and the kinematic viscosity goes to zero as $ε$^$α$ ($α$ positive), we first establish the global convergence of weak solutions, and then prove the global existence and convergence of strong solutions. In particular, the Strichartz estimates used to obtain explicit convergence rates depend on the above parameter $α$ and lead us to identify an admissible interval for $α$.
The current paper establishes the global well-posedness issue for the full viscous MHD equations in the axisymmetric setting. Global solutions are obtained in critical Besov spaces uniformly to the viscosity when the resistivity is fixed in the spirit of \cite{AKH,Hassainia,hz0}. Furthermore, strong convergence in the resolution spaces with a rate of convergence is also studied.
We consider the two-dimensional MHD Boundary layer system without hydrodynamic viscosity, and establish the existence and uniqueness of solutions in Sobolev spaces under the assumption that the tangential component of magnetic fields dominates. This gives a complement to the previous works of Liu-Xie-Yang [Comm. Pure Appl. Math. 72 (2019)] and Liu-Wang-Xie-Yang [J. Funct. Anal. 279 (2020)], where the well-posedness theory was established for the MHD boundary layer systems with both viscosity and resistivity and with viscosity only, respectively. We use the pseudo-differential calculation, to o…
We consider an initial boundary problem for the planar MHD system under the general condition on the heat conductivity $κ$ that may depend on both the density $ρ$ and the temperature $θ$ satisfying $κ(ρ,θ)\geqκ_1 θ^{q}$ for some constants $κ_1>0$ and $q>0.$ Firstly, the global existence of strong solution for large initial data is obtained, and then the limit of the vanishing shear viscosity is justified. In addition, the $L^2$ convergence rate is obtained together with the estimation on the thickness of the boundary layer.
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. More important, to capture the behavior of the solutions at vanishing shear viscosity, both the boundary-layer thickness and the boundary-layer solution are discussed. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are ba…
In this paper, we are concerned with the global well-posedness of a tri-dimensional MHD system with only vertical viscosity in velocity equation for the large axisymmetric initial data. By making good use of the axisymmetric structure of flow and the maximal smoothing effect of vertical diffusion, we show that $\displaystyle\sup_{2\leq p<\infty}\int_0^t\frac{\|\partial_{z}u(τ)\|_{L^p}^{2}}{p^{3/4}}\,\mathrm{d}τ<\infty$. With this regularity for the vertical first derivative of velocity vector field, we further establish losing estimates for the anisotropy tri-dimensional MHD system to get the …
This paper is concerned with an initial and boundary value problem of the one-dimensional planar MHD equations for viscous, heat-conducting, compressible, ideal polytropic fluids with constant transport coefficients and large data. The vanishing shear viscosity limit is justified and the convergence rates are obtained. To capture the behavior of the solutions at small shear viscosity, we also discuss the boundary-layer thickness and the boundary-layer solution. As by-products, the global well-posedness of strong solutions with large data is established. The proofs are based on the global (unif…
In this paper, we we study boundary layer problems for the incompressible MHD systems in the presence of physical boundaries with the standard Dirichlet oundary conditions with small generic viscosity and diffusion coefficients. We identify a non-trivial class of initial data for which we can establish the uniform stability of the Prandtl's type boundary layers and prove rigorously that the solutions to the viscous and diffusive incompressible MHD systems converges strongly to the superposition of the solution to the ideal MHD systems with a Prandtl's type boundary layer corrector. One of the …
In this paper we are concerned with the global well-posedness for the compressible MHD equations with large data. We show that if the shear viscosity $μ$ is a positive constant and the bulk viscosity $λ$ is the power function of the density, that is, $λ(ρ)=ρ^β$ with $β>3$, then the two dimensional compressible MHD system with the periodic boundary conditions on the torus $\mathbb{T}^2$ have a unique global classical solution $(ρ, u,H)$. In this work we extended the results about compressible Navier-Stokes equations in \cite{Karzhikhov} to compressible MHD equations by applying several new tech…
On the Vanishing Viscosity and Magnetic Diffusion Limit for Incompressible Magnetohydrodynamics
Institute of Mathematics of the Academy of Sciences of the Czech Republic（捷克科学院数学研究所）；College of Sciences Yangzhou University（扬州大学理学院）；Hohai University（河海大学）
By observing a new relation between the magnetic pressure and the hydrodynamic pressure, global existence of classical solution to the full perfect MHD equations with large data is established, in particular including the case when all the viscosity, heat conductivity and diffusivity coefficients are constant. This can be viewed as an analog of the classical work by Kazhikhov-Shelukhin for the Navier-Stokes equations to the MHD equations. In addition, the vanishing shear viscosity limit is proved.
Extending our previous results for artificial viscosity systems, we show, under suitable spectral hypotheses, that shock wave solutions of compressible Navier--Stokes (cNS) and magnetohydrodynamics (MHD) equations undergo Hopf bifurcation to nearby time-periodic solutions. The main new difficulty associated with physical viscosity and the corresponding absence of parabolic smoothing is the need to show that the difference between nonlinear and linearized solution operators is quadratically small in $H^s$ for data in $H^s$. We accomplish this by a novel energy estimate carried out in Lagrangian…
We prove the non-uniqueness of weak solutions to 3D magnetohydrodynamic (MHD for short) equations. The constructed weak solutions do not conserve the magnetic helicity and can be close to any given smooth, divergence-free and mean-free velocity and magnetic fields. Furthermore, we prove that the weak solutions constructed by Beekie-Buckmaster-Vicol [2] for the ideal MHD can be obtained as a strong vanishing viscosity and resistivity limit of a sequence of weak solutions to MHD equations. This shows that, in contrast to the weak ideal limits, Taylor's conjecture does not hold along the vanishin…
In this paper, we prove that as the viscosity and resistivity go to zero, the solution of the Cauchy problem for the incompressible MHD equations converges to the solution of the ideal MHD equations in the same topology with the initial data. Our proof mainly depends on the method introduced by the paper \cite{G.L.Y} and the constructions of the incompressible MHD equations.
In this paper, we study the barotropic compressible magnetohydrodynamic equations with the shear viscosity being a positive constant and the bulk one being proportional to a power of the density in a general two-dimensional bounded simply connected domain. For initial density allowed to vanish, we prove that the initial-boundary-value problem of 2D compressible MHD system admits the global strong and weak solutions without any restrictions on the size of initial data provided the shear viscosity is a positive constant and bulk one is $λ=ρ^β$ with $β>4/3$. As we known, this is the first result …
In this paper, we consider the quantum MHD equations with both the viscosity coefficient and the magnetic diffusion coefficient are depend on the density. we prove the global existence of weak solutions and perform the lower planck limit in a 3-dimensional torus for large initial data. The global existence is shown by using Faedo-Galerkin method and weak compactness techniques for the adiabatic exponent $γ>1$.
We consider the evolutionary MHD systems, and study the the regularity and vanishing viscosity limit of the 3-D viscous system in a class of bounded domains with a slip boundary condition. We derive the convergence is in H^{2k+1}, for k>0, if the initial date holds some sufficient conditions.
In this paper, we study the MHD equations with small viscosity and resistivity coefficients, which may be different. This is a typical setting in high temperature plasmas. It was proved that the MHD equations are globally well-posed if the initial velocity is close to 0 and the initial magnetic field is close to a homogeneous magnetic field in the weighted Hölder space, where the closeness is independent of the dissipation coefficients.
This paper is concerned with the initial-boundary value problem to 2D magnetohydrodynamics-Boussinesq system with the temperature-dependent viscosity, thermal diffusivity and electrical conductivity. First, we establish the global weak solutions under the minimal initial assumption. Then by imposing higher regularity assumption on the initial data, we obtain the global strong solution with uniqueness. Moreover, the exponential decay estimate of the solution is obtained.
Global uniform regularity for the 3D compressible MHD equations near a background magnetic field
School of Mathematics, Sun Yat-sen University（中山大学数学学院）；Department of Applied Mathematics, The Hong Kong Polytechnic University（香港理工大学应用数学系）；Department of Mathematics, University of Notre Dame（圣母大学数学系）
In this paper, we study the global well-posedness of classical solutions to the 2D compressible MHD equations with large initial data and vacuum. With the assumption $μ=const.$ and $λ=ρ^β,~β>1$ (Vaǐgant-Kazhikhov Model) for the viscosity coefficients, the global existence and uniqueness of classical solutions to the initial value problem is established on the torus $\mathbb{T}^2$ and the whole space $\mathbb{R}^2$ (with vacuum or non-vacuum far fields). These results generalize the previous ones for the Vaǐgant-Kazhikhov model of compressible Navier-Stokes.
In this paper, we prove the global well-posedness for the three-dimensional magnetohydrodynamics (MHD) equations with zero viscosity and axisymmetric initial data. First, we analyze the problem corresponding to the Sobolev regularities $ H^s\times H^{s-2}$, with $ s>5/2$. Second, we address the same problem but for the Besov critical regularities $ B_{p,1}^{3/p+1}\times B_{1,p}^{3/p-1}$, $2\leq p\leq \infty$. This case turns out to be more subtle as the Beale-Kato-Majda criterion is not known to be valid for rough regularities.
We obtain the global well-posedness to the 3D incompressible magnetohydrodynamics (MHD) equations in Besov space with negative index of regularity. Particularly, we can get the global solutions for a new class of large initial data. As a byproduct, this result improves the corresponding result in \cite{HHW}. In addition, we also get the global result for this system in $\mathcalχ^{-1}(\R^3)$ originally developed in \cite{LL}. More precisely, we only assume that the norm of initial data is exactly smaller than the sum of viscosity and diffusivity parameters.
This paper is devoted to the analysis of a singular perturbation problem for a $2$-D incompressible MHD system with density variations and Coriolis force, in the limit of small Rossby numbers. Two regimes are considered. The first one is the quasi-homogeneous regime, where the densities are small perturbations around a constant state. The limit dynamics is identified as an incompressible homogeneous MHD system, coupled with an additional transport equation for the limit of the density variations. The second case is the fully non-homogeneous regime, where the densities vary around a general non…
We consider viscous free-boundary magnetohydrodynamics(MHD) under vacuum in $\mathbb{R}^3$, especially when vacuum magnetic field is identically zero. It is a central problem in mathematics to perform vanishing viscosity limit to get a solution of hyperbolic inviscid system. However, boundary layer behavior happens near the free-boundary, so existence time $T^{\varepsilon} \rightarrow 0$ as kinematic viscosity $\varepsilon\rightarrow 0$ in standard sobolev space. Inspired by \cite{NMFR1}, we use sobolev conormal space to derive uniform regularity in viscosity $\varepsilon$. Finally, we get a s…
In one-dimensional unbounded domains, we consider the equations of a planar compressible magnetohydrodynamic (MHD) flow with constant viscosity and heat conductivity. More precisely, we prove the global existence of strong solutions to the MHD equations with large initial data satisfying the same conditions as those of Kazhikhov's theory in bounded domains (Kazhikhov 1987 Boundary Value Problems for Equations of Mathematical Physics (Krasnoyarsk)). In particular, our result generalizes the Kazhikhov's theory for the initial boundary value problem in bounded domains to the unbounded case.
In our earlier work \cite{DLL}, we have shown the global well-posedness of strong solutions to the three-dimensional primitive equations with the magnetic field (PEM) on a thin domain. The heart of this paper is to provide a rigorous justification of the derivation of the PEM as the small aspect ratio limit of the incompressible three-dimensional scaled magnetohydrodynamics (SMHD) equations in the anisotropic horizontal viscosity and magnetic field regime. For the case of $H^1$-initial data case, we prove that global Leray-Hopf weak solutions of the three-dimensional SMHD equation strongly con…
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…
