3D MHD Equations Research Articles

An optimal regularity criterion for the 3D MHD equations in homogeneous Besov spaces

In this paper, we establish a new regularity criterion for weak solutions to the 3D incompressible MHD equations in terms of two pairs of (∂iui, ∂ibi) . More precisely, it is proved that the weak solution (u, b) is smooth on (0, T], provided that for some i, j ∈ {1, 2, 3} with i ≠ j, it holds that

Sharp time decay rates of H1 weak solutions for the 2D MHD equations with linear damping velocity

This study is concerned with the large time decay rates of the H1 weak solutions for the 2D magnetohydrodynamic equations with linear damping velocity. Due to the inconsistencies of the dissipative effects between linear velocity and Laplacian magnetic diffusion, neither the classic Fourier splitting method nor Kato’s method is available directly. By developing an alternative analysis argument together with an iterative process, we give a detailed description on the sharp upper decay rates for the velocity field and for the magnetic field , . Additionally, the optimal lower decay rates of the magnetic field are also presented.

A regularity criterion of the 3D MHD equations involving one velocity and one current density component in Lorentz space

In this paper, we study the regularity criterion of weak solutions to the three-dimensional (3D) MHD equations. It is proved that the solution $(u,b)$ becomes regular provided that one velocity and one current density component of the solution satisfy% \begin{equation} u_{3}\in L^{\frac{30\alpha }{7\alpha -45}}\left( 0,T;L^{\alpha ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq \alpha \leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2\beta }{2\beta -3}}\left( 0,T;L^{\beta ,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq \beta \leq \infty , \label{eq02} \end{equation}% which generalize some known results.

Global well-posedness and decay characterization of solutions to 3D MHD equations with Hall and ion-slip effects

In this paper, we establish a small data global well-posedness results for 3D incompressible MHD equations with Hall and ion-slip effects. Then, by using Fourier splitting method and iteration, we consider the decay rate on higher order derivative of solutions.

Ideal magnetohydrodynamic equations on a sphere and elliptic-hyperbolic property

This work contains the derivation and type analysis of the conical ideal magnetohydrodynamic equations. The 3D ideal MHD equations with Powell source terms, subject to the assumption that the solution is conically invariant, are projected onto a unit sphere using tools from tensor calculus. Conical flows provide valuable insight into supersonic and hypersonic flow past bodies, but are simpler to analyze and solve numerically. Previously, work has been done on conical inviscid flows governed by the Euler equations with great success. It is known that some flight regimes involve flows of ionized gases, and thus there is motivation to extend the study of conical flows to the case where the gas is electrically conducting. To the authors’ knowledge, the conical magnetohydrodynamic equations have never been derived, and so this paper is the first investigation of that system. Among the results, we show that conical flows for this case do exist mathematically and that the governing system of partial differential equations is of mixed type. Throughout the domain it can be either hyperbolic or elliptic depending on the solution.

Random Attractors for 2D Stochastic Hydrodynamical-Type Systems

We study the asymptotic behavior of solutions to a class of abstract nonlinear stochastic evolution equations with additive noise that covers numerous 2D hydrodynamical models, such as the 2D Navier–Stokes equations, 2D Boussinesq equations, 2D MHD equations, etc., and also some 3D models, like the 3D Leray 𝛼-model. We prove the existence of random attractors for the associated continuous random dynamical systems. We also establish the upper semicontinuity of the random attractors as the parameter tends to zero.

Global stability of large solutions to the 3D nonhomogeneous incompressible MHD equations

In this paper, we consider the 3D nonhomogeneous incompressible MHD equations. More precisely, we obtain that the large solutions to the 3D nonhomogeneous incompressible MHD equations in 3D bounded or unbounded domains are stable under the conditions, where the initial data satisfy small perturbations and the suitable condition is prescribed on the large solutions.

Global strong solutions to the 3D compressible non-isentropic MHD equations with zero resistivity

In this paper, we study the 3D compressible non-isentropic MHD equations with zero resistivity in a bounded domain. We first establish a regularity criterion for the strong solutions, and then, we combine it with an abstract bootstrap argument to prove the global well-posedness of the system under some assumptions on the viscosity coefficients and the initial data. Our results do not need the positivity of initial density, and thus, it may vanish in an open subset of the domain.

Local interior regularity for the 3D MHD equations in nonendpoint borderline Lorentz space

<abstract> <p>We prove local regularity condition for a suitable weak solution to 3D MHD equations. Precisely, if a solution satisfies $u, b \in L^{\infty}(-(\frac{4}{3})^2, 0;L^{3, q}(B_{\frac{3}{4}}))$, $q\in (3, \infty)$ in Lorentz space, then $(u, b)$ is Hölder continuous in the closure of the set $Q_{\frac{1}{2}}$.</p> </abstract>

The Energy Conservations and Lower Bounds for Possible Singular Solutions to the 3D Incompressible MHD Equations

In this note, we give a new proof to the energy conservation for the weak solutions of the incompressible 3D MHD equations. Moreover, we give the lower bounds for possible singular solutions to the incompressible 3D MHD equations.

On the local well-posedness of strong solutions to 3D MHD equations with Hall and ion-slip effects

In this paper, through maximal regularity estimates for the Stokes equation, we study the local well-posedness of strong solutions to the 3D MHD equations with Hall and ion-slip effects under the condition that the initial data $$(u_0,B_0 )\in H^{\frac{1}{2}+\varepsilon }({\mathbb {R}}^3)\times H^2({\mathbb {R}}^3) $$ and any $$\varepsilon \in (0,1)$$. This is an improvement of the recent results given by Fan et al. (Z Angew Math Phys, 2015).

On Regularity for the 3D MHD Equations via One Directional Derivative of the Pressure

This work establishes a new regularity criterion for the 3D incompressible MHD equations in term of one directional derivative of the pressure (i.e., \(\partial _{3}P\)) on framework of the anisotropic Lebesgue spaces. More precisely, it is proved that for \(T>0\), if \(\partial _{3}P\in L^{\beta }(0,T; L^{\alpha }(\mathbb {R}^{2}_{x_{1}x_{2}};L^{\gamma }(\mathbb {R}_{x_{3}})))\) with \(\frac{2}{\beta }+\frac{1}{\gamma }+\frac{2}{\alpha }=k\in [2,3)\) and \(\frac{3}{k}\le \gamma \le \alpha \le \frac{1}{k-2},\) then the corresponding solution (u, b) to the 3D MHD equations is regular on [0, T].

A class large solution of the 2D MHD equations with velocity and magnetic damping

In this paper, we construct a class global large solution to the two-dimensional magnetohydrodynamics equations with damp terms whose initial energy can be arbitrarily large.

Remark on local boundary regularity condition of suitable weak solutions to the 3D MHD equations

Remark on local boundary regularity condition of suitable weak solutions to the 3D MHD equations

Varieties of scaling regimes in hydromagnetic turbulence

We revisit the scaling properties of the energy spectra in fully developed incompressible homogeneous turbulence in forced magnetofluids (MHD) in three dimensions (3D), which are believed to be characterised by {\em universal scaling exponents} in the inertial range. Enumerating these universal scaling exponents that characterise the energy spectra remains a theoretical challenge. To study this, we set up a scaling analysis of the 3D MHD equations, driven by large-scale external forces and with or without a mean magnetic field. We use scaling arguments to bring out various scaling regimes for the energy spectra. We obtain a variety of scaling in the inertial range, ranging from the well-known Kolmogorov spectra in the isotropic 3D ordinary MHD to more complex scaling in the anisotropic cases that depend on the magnitude of the mean magnetic field. We further dwell on the possibility that the energy spectra scales as $k^{-2}$ in the inertial range, where $k$ is a wavevector belonging to the inertial range, and also speculate on unequal scaling by the kinetic and magnetic energy spectra in the inertial range of isotropic 3D ordinary MHD. We predict the possibilities of {\em scale-dependent anisotropy} and intriguing {\em weak dynamic scaling} in the Hall MHD and electron MHD regimes of anisotropic MHD turbulence. Our results can be tested in large scale simulations and relevant laboratory-based and solar wind experiments.

Weighted Regularity Criteria of Weak Solutions to the Incompressible 3D MHD Equations

In this paper, we investigate regularity conditions of the weighted type for weak solutions to the incompressible 3D MHD equations.

Backwards Asymptotically Autonomous Dynamics for 2D MHD Equations

We consider the backwards topological property of pullback attractors for the nonautonomous MHD equations. Under some backwards assumptions of the nonautonomous force, it is shown that the theoretical existence result for such an attractor is derived from an increasing, bounded pullback absorbing and the backwards pullback flattening property. Meanwhile, some abstract results on the convergence of nonautonomous pullback attractors in asymptotically autonomous problems are established and applied to MHD equations.

A new regularity criterion of the 2D MHD equations

In this paper, we establish a new regularity criterion for the two-dimensional incompressible generalized magnetohydrodynamics equations.

Global strong solutions to the 3D incompressible MHD equations with density-dependent viscosity

This paper is concerned with the 3D incompressible MHD equations with density-dependent viscosity in smooth bounded domains. Through time-weighted a priori estimates, the global existence of strong solutions is established under the assumption that the initial energy is suitably small. This generalizes previous results for the 3D Navier–Stokes equations in Huang and Wang (2015) and Zhang (2015), which need ‖∇u0‖L2 to be small. The initial vacuum is allowed.

Turbulent Heating between 0.2 and 1 au: A Numerical Study

Abstract The heating of the solar wind is key to understanding its dynamics and acceleration process. The observed radial decrease of the proton temperature in the solar wind is slow compared to the adiabatic prediction, and it is thought to be caused by turbulent dissipation. To generate the observed 1/R decrease, the dissipation rate has to reach a specific level that varies in turn with temperature, wind speed, and heliocentric distance. We want to prove that MHD turbulent simulations can lead to the 1/R profile. We consider here the slow solar wind, characterized by a quasi-2D spectral anisotropy. We use the expanding box model equations, which incorporate into 3D MHD equations the expansion due to the mean radial wind, allowing us to follow the plasma evolution between 0.2 and 1 au. We vary the initial parameters: Mach number, expansion parameter, plasma β, and properties of the energy spectrum as the spectral range and slope. Assuming turbulence starts at 0.2 au with a Mach number equal to unity, with a 3D spectrum mainly perpendicular to the mean field, we find radial temperature profiles close to 1/R on average. This is done at the price of limiting the initial spectral extent, corresponding to the small number of modes in the inertial range available, due to the modest Reynolds number reachable with high Mach numbers.