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{30α}{7α-45}}\left( 0,T;L^{α,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{45}{7}% \leq α\leq \infty , \label{eq01} \end{equation}% and \begin{equation} j_{3}\in L^{\frac{2β}{2β-3}}\left( 0,T;L^{β,\infty }\left( \mathbb{R}^{3}\right) \right) \text{ \ \ \ with \ \ }\frac{3}{2}\leq β\leq \infty , \label{eq02} \end{equation}% which generalize some known results.
Local regularity results are obtained for the MHD equations using as global framework the setting of parabolic Morrey spaces. Indeed, by assuming some local boundedness assumptions (in the sense of parabolic Morrey spaces) for weak solutions of the MHD equations it is possible to obtain a gain of regularity for such solutions in the general setting of the Serrin regularity theory. This is the first step of a wider program that aims to study both local and partial regularity theories for the MHD equations.
In this paper, similar to the incompressible Euler equation, we prove the propagation of the Gevrey regularity of solutions to the three-dimensional incompressible ideal magnetohydrodynamics (MHD) equations. We also obtain an uniform estimate of Gevery radius for the solution of MHD equation.
We generalize here the celebrated Partial Regularity Theory of Caffarelli, Kohn and Nirenberg to the MHD equations in the framework of parabolic Morrey spaces. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory for the classical Navier-Stokes equations as well as for the MHD equations.
Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.
This work establishes two regularity criteria for the 3D incompressible MHD equations. The first one is in terms of the derivative of the velocity field in one-direction while the second one requires suitable boundedness of the derivative of the pressure in one-direction.
In this paper, we consider the global well-posedness of the incompressible Hall-MHD equations in $\mathbb{R}^3$. We prove that the solution of this system is globally regular if the initial data is axisymmetric and the swirl components of the velocity and magnetic vorticity are trivial. It should be pointed out that the initial data without any smallness and in low regularity spaces. This improves a previous result established in \cite{Fan2013}.
Whether or not the solution to 2D resistive MHD equations is globally smooth remains open. This paper establishes the global regularity of solutions to the 2D almost resistive MHD equations, which require the dissipative operators $\mathcal{L}$ weaker than any power of the fractional Laplacian. The result is an improvement of the one of Fan et al. (Global Cauchy problem of 2D generalized MHD equations, Monatsh. Math., 175 (2014), pp. 127-131) which ask for $α>0, β=1$.
In this paper we consider three-dimensional incompressible magnetohydrodynamics equations. By using interpolation inequalities in anisotropic Lebesgue space, we provide regularity criteria involving the velocity or alternatively involving the fractional derivative of velocity in one direction, which generalize some known results.
We study the regularity criteria of the three dimensional generalized MHD and Navier-Stokes systems. In particular, we show that the regularity criteria of the generalized MHD system may be reduced to depend only on two diagonal entries of the Jacobian matrix of the velocity vector field or one vorticity component and one entry of the Jacobian matrix of the velocity vector field.
Under the condition of small external forces, we obtain existence of a weak solution of the steady Hall-MHD system with Hölder continuous magnetic field. We also established regularity of weak solutions provided that magnetic fields are bounded. For sufficiently small external forces, uniqueness result is also established.
Let $(u,b)$ be a smooth enough solution of 3-D incompressible MHD system. We prove that if $(u,b)$ blows up at a finite time $T^*$, then for any $p\in]4,\infty[$, there holds $\int_0^{T^*}\bigl(\|u^3(t')\|^p_{\dH^{\frac 12+\frac 2p}}+\|b(t')\|^p_{\dH^{\frac 12+\frac 2p}}\bigr)dt'=\infty$. We remark that all these quantities are in the critical regularity of the MHD system.
The purpose of this paper is to study the incompressible non-resistive MHD equations in $\mathbb{R}^3$. We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the velocity and magnetic vorticity vanish. In particular, the special axially symmetric initial data can be arbitrarily large and satisfy low regularity assumptions.
We prove the global regularity of the solution pair to the N-dimensional logarithmically supercritical magnetohydrodynamics system with zero diffusivity. This is the endpoint case omitted in the work of [24]; it also improves some previous results logarithmically.
This paper is concerned with the global regularity of the 2D (two-dimensional) generalized magnetohydrodynamic equations with only magnetic diffusion $Λ^{2β} b$. It is proved that when $β>1 $ there exists a unique global regular solution for this equations. The obtained result improves the previous known one which requires that $β>\frac32$.
In this paper, we investigate the global regularity of 2D generalized MHD equations, in which the dissipation term and magnetic diffusion term are $ν(-Δ)^αu$ and $η(-Δ)^βb$ respectively. Let $(u_{0}, b_{0})\in H^{s}$ with $s\geq2$, it is showed that the smooth solution $(u(x,t),b(x,t))$ is globally regular for the case $ 0\leqα\leq\{1}{2}, α+β> \{3}{2}$.
We prove that a solution to the 3D Navier-Stokes or MHD equations does not blow up at $t=T$ provided $\displaystyle \limsup_{q \to \infty} \int_{\mathcal{T}_q}^T \|Δ_q(\nabla \times u)\|_\infty \, dt$ is small enough, where $u$ is the velocity, $Δ_q$ is the Littlewood-Paley projection, and $\mathcal T_q$ is a certain sequence such that $\mathcal T_q \to T$ as $q \to \infty$. This improves many existing regularity criteria.
We study here a new generalization of Caffarelli, Kohn and Nirenberg's partial regularity theory for weak solutions of the MHD equations. Indeed, in this framework some hypotheses on the pressure P are usually asked (for example P $\in$ L q t L 1 x with q > 1) and then local H{ö}lder regularity, in time and space variables, for weak solutions can be obtained over small neighborhoods. By introducing the notion of dissipative solutions, we weaken the hypothesis on the pressure (we will only assume that P $\in$ D) and we will obtain H{ö}lder regularity in the space variable for weak solutions.
In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the magnetohydrodynamics equations with the minimal dissipation. Consequently, our results significantly improve the previous works.
Whether or not the classical solutions of the two-dimensional (2D) incompressible magnetohydrodynamics (MHD) equations with only Laplacian magnetic diffusion (without velocity dissipation) are globally well-posed is a difficult problem and remains completely open. In this paper, we establish the global regularity of solutions to the 2D incompressible MHD equations with almost Laplacian magnetic diffusion in the whole space. This result can be regarded as a further improvement and generalization of the previous works. Consequently, our result is more closer to the resolution of the global regul…
In this paper, we consider regularity criteria of a class of 3D axially symmetric MHD-Boussinesq systems without magnetic resistivity or thermal diffusivity. Under some Prodi-Serrin type critical assumptions on the horizontal angular component of the velocity, we will prove that strong solutions of the axially symmetric MHD-Boussinesq system can be smoothly extended beyond the possible blow-up time $T_\ast$ if the magnetic field contains only the horizontal swirl component. No a priori assumption on the magnetic field and the temperature fluctuation is imposed.
In this paper, we will show that solutions of the three-dimensional non-resistive and non-diffusive MHD-Boussinesq system are globally regular if the initial data is axisymmetric and the swirl components of the velocity and the magnetic vorticity are zero. Our main result extends previous ones on the three-dimensional non-resistive MHD system and non-diffusive Boussinesq system, and the method used here can also be applied to the magnetic Rayleigh-Bénard convection system.
This paper establishes the global regularity of classical solution to the 2D MHD system with only horizontal dissipation and horizontal magnetic diffusion in a strip domain $\mathbb{T}\times\mathbb{R}$ when the initial data is suitable small. To prove this, we combine the Littlewood-Paley decomposition with anisotropic inequalities to establish a crucial commutator estimate. We also analysis the asymptotic behavior of the solution. In addition, the global existence and uniqueness of classical solution is obtained for the 2D simplified tropical climate model with only horizontal dissipations.
We prove a Prodi-Serrin-type global regularity condition for the three-dimensional Magnetohydrodynamic-Boussinesq system (3D MHD-Boussinesq) without thermal diffusion, in terms of only two velocity and two magnetic components. This is the first Prodi-Serrin-type criterion for a hydrodynamic system which is not fully dissipative, and indicates that such an approach may be successful on other systems. In addition, we provide a constructive proof of the local well-posedness of solutions to the fully dissipative 3D MHD-Boussinesq system, and also the fully inviscid, irresistive, non-diffusive MHD-…
It is shown in this paper that suitable weak solutions to the 6D steady incompressible Navier-Stokes and MHD equations are Hölder continuous near boundary provided that either $r^{-3}\int_{B_r^+}|u(x)|^3dx$ or $r^{-2}\int_{B_r^+}|\nabla u(x)|^2dx$ is sufficiently small, which implies that the 2D Hausdorff measure of the set of singular points near the boundary is zero. This generalizes recent interior regularity results by Dong-Strain \cite{DS}.
In this paper, we establish new $L^p$ gradient estimates of the solutions in order to discuss Cauchy problem for the full compressible magnetohydrodynamic(MHD) systems in $\mathrm{R}^3$. We use the "$\rm{div}-\rm{curl}$" decomposition technique (see \cite{{HJR},{MR}}) and new modified effective viscous flux and vorticity to calculate "$\Vert\nabla \mathbf{u}\Vert_{L^3}$" and "$\Vert\nabla \mathbb{H}\Vert_{L^3}$".As a result, we obtain global well-posedness for the solution with the initial data being in a class of space with lower regularity, while the energy of which should be suitably small.
We obtain a regularity criteria of the solution to the three-dimensional magnetohydrodynamics system to remain smooth for all time involving only one velocity and one vorticity component. Moreover, the norm in space and time with which we impose our criteria for the vorticity component is at the scaling invariant level. The proof requires a new decomposition of the four non-linear terms making use of a new identity due to the divergence-free conditions of the velocity and the magnetic vector fields.
We consider the electron magnetohydrodynamics (MHD) with static background ion flow. A special situation of $B(x,y,t)=\nabla\times (a\vec e_z)+b \vec e_z$ with scalar-valued functions $a(x,y,t)$ and $b(x,y,t)$ was studied numerically in the physics paper [7]. The authors concluded from numerical simulations that there is no evidence of dissipation cutoff for the electron MHD. In this paper we show the existence of determining wavenumber for the electron MHD, and establish a regularity condition only on the low modes of the solution. Our results suggest that the conclusion of the physics paper …
We are concerned with the uniform regularity estimates of solutions to the two dimensional compressible non-resistive magnetohydrodynamics (MHD) equations with the no-slip boundary condition on velocity in the half plane. Under the assumption that the initial magnetic field is transverse to the boundary, the uniform conormal energy estimates are established for the solutions to compressible MHD equations with respect to small viscosity coefficients. As a direct consequence, we proved the inviscid limit of solutions from viscous MHD systems to the ideal MHD systems in $L^\infty$ sense. It shows…
We prove that if $0<T_0<T\leq\infty$, $(\mathbf{u},\mathbf{b},p)$ is a suitable weak solution of the MHD equations in $\mathbb{R}^3\times(0,T)$ and either $\mathcal{F}_γ(p_-)\in L^{\infty}(0,T_0;\, L^{3/2}(\mathbb{R}^3))$ or $\mathcal{F}_γ((|\mathbf{u}|^2+ |\mathbf{b}|^2+2p)_+)\in L^{\infty}(0,T_0;\, L^{3/2}(\mathbb{R}^3))$ for some $γ>0$, where $\mathcal{F}_γ(s)=s\, [\ln{}(1+ s)]^{1+γ}$ and the subscripts "$-$" and "$+$" denote the negative and the nonnegative part, respectively, then the solution $(\mathbf{u},\mathbf{b},p)$ has no singular points in $\mathbb{R}^3\times(0,T_0]$.
The global regularity for the incompressible magnetohydrodynamic equations (MHD) in three dimensions is a long standing open problem of fluid dynamics and PDE theory. The Navier-Stokes equations can be viewed as a special case of MHD with a constant magnetic field, whose global regularity problem is known as a Clay Millennium Prize Problem. In this article, we prove the global regularity of axially symmetric solutions to the ideal MHD in three dimensions for a family of non-trivial magnetic fields. The proofs are based on the special structures of MHD and can of course also applied to the resi…
In this paper, we consider the Cauchy problem to the 3D MHD equations. We show that the Serrin--type conditions imposed on one component of the velocity $u_{3}$ and one component of magnetic fields $b_{3}$ with $$ u_{3} \in L^{p_{0},1}(-1,0;L^{q_{0}}(B(2))),\ b_{3} \in L^{p_{1},1}(-1,0;L^{q_{1}}(B(2))), $$ $\frac{2}{p_{0}}+\frac{3}{q_{0}}=\frac{2}{p_{1}}+\frac{3}{q_{1}}=1$ and $3<q_{0},q_{1}<+\infty$ imply that the suitable weak solution is regular at $(0,0)$. The proof is based on the new local energy estimates introduced by Chae-Wolf (Arch. Ration. Mech. Anal. 2021) and Wang-Wu-Zhang (arXiv:…
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 …
In this paper, we study the large time behavior of solutions to the compressible magnetohydrodynamic equations in the $L^p$-type critical Besov spaces. Precisely, we show that if the initial data in the low frequencies additionally belong to some Besov space $\dot{B}_{2,\infty}^{-σ_1}$ with $σ_1\in (1-N/2, 2N/p-N/2]$, then the $\dot{B}_{p,1}^0$ norm of the critical global solutions presents the optimal decay $t^{-\frac{N}{2}(\frac{1}{2}-\frac{1}{p})-\frac{σ_1}{2}}$ for $t\rightarrow+\infty$. The pure energy argument without the spectral analysis is performed, which allows us to remove the usua…
We prove several Liouville type results for the stationary MHD and Hall-MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces or satisfying a priori decay assumption, must be zero.
This paper establishes the global existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyperresistivity. More precisely, the equation of $b_1$ (the horizontal component of the magnetic field) involves only vertical hyperdiffusion (given by $Λ_2^{2β} b_1$) while the equation of $b_2$ (the vertical component) has only horizontal hyperdiffusion (given by $Λ_1^{2β} b_2$), where $Λ_1$ and $Λ_2$ are directional Fourier multiplier operators with the symbols being $|ξ_1|$ and $|ξ_2|$, respectively. We prove that, for $β>1$, this sy…
In this paper, we are concerned with the two-dimensional (2D) incompressible magnetohydrodynamic (MHD) equations with velocity dissipation given by $(-Δ)^α$ and magnetic diffusion given by reducing about logarithmic diffusion from standard Laplacian diffusion. More precisely, we establish the global regularity of solutions to the system as long as the power $α$ is a positive constant. In addition, we prove several global \emph{a priori} bounds for the case $α=0$. In particular, our results significantly improve previous works and take us one step closer to a complete resolution of the global r…
We consider the three-dimensional incompressible magnetohydrodynamics (MHD) equations in a bounded domain with small volume and free moving surface boundary. We establish a priori estimate for solutions with minimal regularity assumptions on the initial data in Lagrangian coordinates. In particular, due to the lack of the Cauchy invariance for MHD equations, the smallness assumption on the fluid domain is required to compensate a loss of control of the flow map. Moreover, we show that the magnetic field has certain regularizing effect which allows us to control the vorticity of the fluid and t…
We prove several Liouville type results for the stationary Hall-MHD and MHD equations. In particular, we show that the velocity and magnetic field, belonging to some Lorentz spaces, must be zero.
The magnetohydrodynamics (MHD) problem is most often studied in a framework where Dirichlet type boundary conditions on the velocity field is imposed. In this Note, we study the (MHD) system with pressure boundary condition, together with zero tangential trace for the velocity and the magnetic field. In a three-dimensional bounded possibly multiply connected domain, we first prove the existence of weak solutions in the Hilbert case, and later, the regularity in $W^{1,p}(Ω)$ for $p\geq 2$ and in $W^{2,p}(Ω)$ for $p\geq 6/5$ using the regularity results for some Stokes and elliptic problems with…
Global uniform regularity and large time behavior of solutions to three dimensional compressible MHD equations with vanishing vertical magnetic resistivity in half space
Shanghai Jiao Tong University（上海交通大学）；University of Notre Dame（圣母大学）
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 investigate the solvability, regularity and the vanishing dissipation limit of solutions to the three-dimensional viscous magneto-hydrodynamic (MHD) equations in bounded domains. On the boundary, the velocity field fulfills a Navier-slip condition, while the magnetic field satisfies the insulating condition. It is shown that the initial-boundary problem has a global weak solution for a general smooth domain. More importantly, for a flat domain, we establish the uniform local well-posedness of the strong solution with higher order uniform regularity and the asymptotic converge…
We prove some sufficient conditions of local regularity of the siutable weak solutions to the system of magnetohydrodynamics near the plane part of the boundary.
This paper studies the regularity problem for the 3D incompress- ible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi-Serrin type c…
In this paper, we are concerned with regularity of suitable weak solutions of the 3D Navier-Stokes equations in Lorentz spaces. We obtain $\varepsilon$-regularity criteria in terms of either the velocity, the gradient of the velocity, the vorticity, or deformation tensor in Lorentz spaces. As an application, this allows us to extend the result involving Leray's blow up rate in time, and to show that the number of singular points of weak solutions belonging to $ L^{p,\infty}(-1,0;L^{q,l}(\mathbb{R}^{3})) $ and $ {2}/{p}+{3}/{q}=1$ with $3<q<\infty$ and $q\leq l <\infty$ is finite.
In this paper, we prove the energy and cross-helicity conservation of weak solutions to the three-dimensional ideal MHD equations in bounded domain under the interior Besov regularity conditions which are exactly same as the three-dimensional periodic domain case in \cite{Caflisch 97}, and the boundedness and the Besov-type continuity for both the velocity and magnetic fields near the boundary, which seem crucial for the bounded domain case due to the boundary effect. Note that the Besov-type continuity condition near the boundary is consistent with the interior Besov regularity, which is a ne…
