Viktor Vyalov
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.
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Eventual regularity and asymptotic behavior of Leray-Hopf weak solutions for the Hall-MHD system
研究不可压缩、粘性、电阻性Hall-MHD系统，证明二维Leray-Hopf弱解在有限时间后变光滑，三维中构造出磁涡度场最终获得额外正则性的弱解，并在初始数据低频频谱假设下结合广义Fourier分裂方法导出速度场和磁场解耦的代数衰减率。
Existence of local-in-time unique solution and loss of smoothness of full Magnet-Hydro-Dynamics system (MHD) is considered for periodic initial data. The result is proven using Fujita-Kato's method in $\ell^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwells's equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.
We study the weak solutions to the electron-MHD system and obtain a conditional uniqueness result. In addition, we prove conservation of helicity for weak solutions to the electron-MHD system under a geometric condition.
We prove the partial regularity of the boundary suitable weak solutions to the MHD system near the plane part of the boundary.
$L^2$ decay estimates of weak solutions to 3D fractional MHD equations in exterior domains
本文研究了在Dirichlet边界条件下外部区域中三维分数阶MHD方程弱解的渐近行为，并获得了弱解的$L^2$衰减估计。
We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor's conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.
We prove the existence of a weak solution to a generalized quantum MHD equation in a 2-dimensional periodic box for large initial data. The existence of a global weak solution is established through a three-level approximation, energy estimates, and weak convergence for the adiabatic exponent γ>1.
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.
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 investigate the initial-boundary value problem for one-dimensional compressible, heat-conductive, non-resistive MHD equations of viscous, ideal polytropic fluids in the Lagrangian coordinates. The existence and Lipschitz continuous dependence on the initial data of global weak solutions are established. Uniqueness of weak solutions follows as a direct consequence of stability.
Non-uniqueness of weak solutions to the 3-D stationary MHD equations in Besov space with negative regularity index
通过凸积分方法证明三维环面上分数阶稳态MHD方程存在无穷多个非平凡奇异解，从而在次临界Besov空间中建立弱解的非唯一性。
We consider the Cauchy problem for the electron magnetohydrodynamics model in the supercritical regime. For rough initial data in $\mathcal H^{-s}(\mathbb T^n)$ with $s>0$, we obtain global in time weak solutions almost surely via an appropriate randomization of the initial data.
This paper deals with the existence of global weak solutions for 3D MHD equations when the initial data belong to the weighted spaces $L^2_{w_γ}$, with $w_γ(x)=(1+| x|)^{-γ}$ and $0 \leq γ\leq 2$. Moreover, we prove the existence of discretely self-similar solutions for 3D MHD equations for discretely self-similar initial data which are locally square integrable. Our methods are inspired of a recent work of P. Fernández-Dalgo and P.G. Lemarié-Riseusset for the 3D Navier-Stokes equations.
Energy equality of the weak solutions to the fractional Navier-Stokes / MHD equations
本文研究 3D 不可压缩分数阶 Navier-Stokes / MHD 方程弱解的能量等式，借助对称化和插值方法获得包含 Sobolev 乘子空间的新充分条件，以保障能量等式有效性，并与解的唯一性相关。
Non-unique weak solutions in Leray-Hopf class are constructed for the three dimensional magneto-hydrodynamics with Hall effect. We adapt the widely appreciated convex integration framework developed in a recent work of Buckmaster and Vicol for the Navier-Stokes equation, and with deep roots in a sequence of breakthrough papers for the Euler equation.
We establish the existence of axially symmetric weak solutions to steady incompressible magnetohydrodynamics with non-homogeneous boundary conditions. The key issue is the Bernoulli's law for the total head pressure $Φ=\f 12(|{\bf u}|^2+|{\bf h}|^2)+p$ to a special class of solutions to the inviscid, non-resistive MHD system, where the magnetic field only contains the swirl component.
We consider here the magneto-hydrodynamics (MHD) equations on the whole space. For the 3D case, in the setting of the weighted $L^2$ spaces we obtain a weak-strong uniqueness criterion provided that the velocity field and the magnetic field belong to a fairly general multipliers space. On the other hand, we study the local and global existence of weak suitable solutions for intermittent initial data, which is characterized through a local Morrey space. This large initial data space was also exhibit in a contemporary work [4] in the context of 3D Navier-Stokes equations. Finally, we make a disc…
Non-uniqueness of weak solutions to the 3D Hall-MHD equations on the plane
通过凸积分格式和新间歇流构造，证明了平面上 3D Hall-MHD 方程具有非平凡磁场的弱解非唯一性，并发现其不守恒磁螺旋性，且理想解为强零粘与零电阻极限。
Non-uniqueness of Leray weak solutions of the forced MHD equations
本研究针对受迫MHD方程，通过构造不稳定背景解及对应不稳定流形上的轨道，证明了其Leray弱解的非唯一性，且解处于已知适定性理论边界。
Global existence and uniqueness of weak solutions for the MHD equations with large $L^3$-initial values
本文通过勒雷近似技术和扰动理论，研究了大$ L^3 $初值下MHD方程弱解的全局存在性和唯一性，同时提供了不可压缩纳维-斯托克斯系统弱$ L^3 $解理论的简洁证明。
We consider a circulation system arising in turbulence modelling in fluid dynamics with unbounded eddy viscosities. Various notions of weak solutions are considered and compared. We establish existence and regularity results. In particular we study the boundedness of weak solutions. We also establish an existence result for a classical solution
Weak solutions to the Hall-MHD equations whose singular sets in time have Hausdorff dimension strictly less than 1
针对三维环面上的超黏性阻性 Hall-MHD 方程，论文利用凸积分构造非 Leray-Hopf 弱解，证明存在动能有界且时间奇异集 Hausdorff 维数严格小于 1 的非唯一弱解。
We show that $t^{s/2} \Vert (\mbox{\boldmath $u$},\mbox{\boldmath $b$})(.,t)\Vert_{\dot{H}(\mathbb{R}^{n})} \rightarrow 0,$ as $t\rightarrow \infty$ for Leray solutions $(\mbox{\boldmath $u$}, \mbox{\boldmath $b$})(.,t)$ of the incompressible MHD equations, where $2 \leq n \leq 4$ and $s \geq 0.$ As a corollary of main result described previously we have also that $\lim_{t\rightarrow\infty} t^{\frac{n}{2} - \frac{n}{2q}} \Vert(\mbox{\boldmath $u$},\mbox{\boldmath $b$})(.,t)\Vert_{L^{q}(\mathbb{R}^{n})} = 0, 2\leq q\leq \infty.$
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.
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 $\mathbb{R}^3\times(0,T_0]$.
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…
On the weak solutions for the MHD systems with controllable total energy and cross helicity
本文通过构造六种箱式流的凸积分扰动，证明了粘性电阻MHD系统在给定时间区间内总能量和交叉螺旋度可控时的弱解非唯一性，将理想MHD的结果推广至耗散情形。
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 …
We study the initial-boundary value problem for 1D compressible MHD equations of viscous non-resistive fluids in the Lagrangian mass coordinates. Based on the estimates of upper and lower bounds of the density, weak solutions are constructed by approximation of global regular solutions, the existence of which has recently been obtained by Jiang and Zhang in [17]. Uniqueness of weak solutions is also proved as a consequence of Lipschitz continuous dependence on the initial data. Furthermore, long time behavior for global solutions is investigated. Specifically, based on the uniform-in-time boun…
In this paper, we consider a two-dimensional non-resistive magnetohydrodynamic model, taking the fluctuation of absolute temperature into account. Combining the method of weak convergence developed by Lions [20], Feireisl et al. [7, 8] from compressible Navier-Stokes(- Fourier) system and the new technique of variable reduction proposed by Vasseur et al. [26] and refined by Novotny et al. [22] from compressible two-fluid models, weak solutions are shown to exist globally in time with finite energy initial data. The result is the first one on global solvability to full compressible, viscous, no…
Via convex integration, we construct weak solutions to the non-resistive electron-MHD system which do not conserve energy and magnetic helicity.
Non-unique solutions for electron MHD
本文在三维环面上构造电子磁流体力学方程的弱解，证明其非唯一性，并给出无电阻率情形下弱解的存在性。
In this paper we analyze a method of to approximation for the weak solutions of the incompressible magnetohydrodynamic equations (MHD) in unbounded domains. In particular we describe an hyperbolic version of the so called artificial compressibility method adapted to the MHD system. By exploiting the wave equation structure of the approximating system we achieve the convergence of the approximating sequences by means of dispersive estimate of Strichartz type. We prove that the soleinoidal component of the approximating velocity and magnetic fields is relatively compact and converges strongly to…
We study a general convergence theory for the analysis of numerical solutions to the magnetohydrodynamic system describing the time evolution of compressible, viscous, electrically conducting fluids in space dimension d (= 2; 3). First, we introduce the concept of dissipative weak solutions and prove the weak-strong uniqueness property for dissipative weak solutions, meaning a dissipative weak solution coincides with a classical solution emanating from the same initial data on the lifespan of the latter. Next, we introduce the concept of consistent approximations and prove the convergence of c…
Weak solutions to a compressible viscous non-resistive MHD equations with general boundary data
本文在一般流入-流出边界条件下，证明了可压缩粘性非电阻型MHD方程全局弱解的存在性，并建立了弱-强唯一性原理，扩展了先前工作。
We show that ideal 2D MHD does not possess weak solutions (or even subsolutions) with compact support in time and non-trivial magnetic field. We also show that the $Λ$-convex hull of ideal MHD has empty interior in both 2D and 3D; this is seen by finding suitable $Λ$-convex functions. As a consequence we show that mean-square magnetic potential is conserved in 2D by subsolutions and weak limits of solutions in the physically natural energy space $L^\infty_t L^2_x$, and in 3D we show the conservation of magnetic helicity by $L^3$-integrable subsolutions and weak limits of solutions. However, in…
Let us consider a semilinear boundary value problem $ - Δu= f(x,u),$ in $Ω,$ with Dirichlet boundary conditions, where $ Ω\subset \mathbb{R}^N $, $N> 2,$ is a bounded smooth domain. We provide sufficient conditions guarantying that semi-stable weak positive solutions to subcritical semilinear elliptic equations are smooth in any dimension, and as a consequence, classical solutions. By a subcritical nonlinearity we mean $f(x,s)/s^\frac{N+2}{N-2} \to 0$ as $s\to\infty$, including non-power nonlinearities, and enlarging the class of subcritical nonlinearities, which is usually reserved for power …
We consider a two-dimensional MHD model describing the evolution of viscous, compressible and electrically conducting fluids under the action of vertical magnetic field without resistivity. Existence of global weak solutions is established for any adiabatic exponent γ>1. Inspired by the approximate scheme proposed in [15], we consider a two-level approximate system with artificial diffusion and pressure term. At the first level, we prove global well-posedness of the regularized system and establish uniform-in-εestimates to the regular solutions. At the second level, we show global existence of…
Some remarks on the regularity of weak solutions for the stationary Ericksen-Leslie and MHD systems
本文研究了三维空间中受外力作用的定常 Ericksen-Leslie (SEL) 和 MHD 两个椭圆耦合系统，为 SEL 系统提出了基于齐次 Morrey 空间的弱解正则性新准则，并利用 Gevrey 类证明了 MHD 系统有限能量弱解的解析性。
In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.
Non-uniqueness for the hypo-dissipative compressible 3D magnetohydrodynamic equations
针对一般压力律下的可压缩三维磁流体力学方程组，证明了在次粘性和次电阻率条件下弱解的非唯一性，并建立了理想MHD光滑解作为次耗散MHD弱解粘性消失极限的结论。
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.
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 $α>0, β=1$.
We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the $L^3$ integrability condition for magnetic helicity conservation and provide turbulent bounde…
We characterize when radial weak solutions to Monge-Ampere equations are smooth. This paper extends previous partial results and also covers Generalized Monge-Ampere equations and infinitely vanishing right hand side.
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.
Weak and mild solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces
本文研究三维 MHD 方程与带阻尼粘弹性 Navier–Stokes 方程，在 Wiener amalgam 空间中证明温和解存在性并构造全局局部能量弱解，同时推广局部能量解的正则性与小–大唯一性结论。
We prove existence of global weak solutions to the chemotaxis system $ u_t=Δu - \nabla\cdot (u\nabla v) +κu -μu^2 $ $ v_t=Δv-v+u $ under homogeneous Neumann boundary conditions in a smooth bounded convex domain $Ω\subset R^n$, for arbitrarily small values of $μ>0$. Additionally, we show that in the three-dimensional setting, after some time, these solutions become classical solutions, provided that $κ$ is not too large. In this case, we also consider their large-time behaviour: We prove decay if $κ\leq 0$ and the existence of an absorbing set if $κ>0$ is sufficiently small. Keywords: chemotaxi…
Asymptotic stability of Landau solutions to the MHD system and energy decay
研究磁流体动力学系统中弱解的渐进行为，证明在强能量不等式下，解在Landau解附近L^2渐近稳定，并在初始扰动满足积分性假设时获得速度和磁扰动的显式代数衰减率。
We prove the existence and uniqueness of weak solutions of the three dimensional compressible magnetohydrodynamics (MHD) equations. We first obtain the existence of weak solutions with small $L^2$-norm which may display codimension-one discontinuities in density, pressure, magnetic field and velocity gradient. The weak solutions we consider here exhibit just enough regularity and structure which allow us to develop uniqueness and continuous dependence theory for the compressible MHD equations. Our results generalise and extend those for the intermediate weak solutions of compressible Navier-St…
Three dimensional stationary solutions of the Electron MHD equations
本文利用凸积分方法，在低正则性 Sobolev 空间中构造了三维电子磁流体力学方程的非平凡稳态弱解，展示了该方法在经典三维 Navier-Stokes 方程中尚无法实现的可行性。
We prove the non-uniqueness of weak solutions to 3D hyper viscous and resistive MHD in the class $L^γ_tW^{s,p}_x$, where the exponents $(s,γ,p)$ lie in two supercritical regimes. The result reveals that the scaling-invariant Ladyženskaja-Prodi-Serrin (LPS) condition is the right criterion to detect non-uniqueness, even in the highly viscous and resistive regime beyond the Lions exponent. In particular, for the classical viscous and resistive MHD, the non-uniqueness is sharp near the endpoint $(0,2,\infty)$ of the LPS condition. Moreover, the constructed weak solutions admit the partial regular…
Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations
本文通过尺度分析、线性化近似和谐函数单调性构造迭代序列，将纳维-斯托克斯技术推广到含外力的MHD方程，证明其恰当弱解奇点的一维抛物型豪斯多夫测度为零，建立了正则性理论。
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.
On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations
本文提出了一种新的弱形式，用于研究3D欧拉和无粘磁流体力学方程弱解中旋度的守恒问题，并给出了局部旋度平衡方程和磁旋度守恒的充分条件。
On Type I blowup and $\varepsilon$-regularity criteria of suitable weak solutions to the 3D incompressible MHD equations
本文研究了3D不可压缩MHD方程合适弱解的I型爆破和ε-正则化准则，扩展了Seregin的Navier-Stokes方程I型准则，并提供了I型爆破分析的自然起点。
On physically grounded boundary conditions for the compressible MHD system
该研究针对非均匀介质中的一般可压缩MHD系统，通过输运系数惩罚项执行奇异极限，严格证明了有界域上的物理基础边界条件，得到弱解存在性结果，并为复杂边界域的数值实验提供理论平台。
This article is devoted to a regularity criteria for solutions of the Navier-Stokes equations in terms of regularity along the stream lines. More precisely, we prove that a suitable weak solution for the Navier-Stokes equations is regular under some constraint on the second derivative of |u| along the stream lines.
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $ u \in L^{q}_{loc}\left(0, T ; L^{p}(Ω)\right) \text { for any } \frac{1}{q}+\frac{1}{p} \leq \frac{1}{2}, \text { with } p \geq 4,\text{ and } b \in L^{r}_{loc}\left(0, T ; L^{s}(Ω)\right) \text { for any } \frac{1}{r}+\frac{1}{s} \leq \frac{1}{2}, \text { with } s \geq 4 $. In particular, compared with the…
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…
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