Mimi Dai, Susan Friedlander
We construct non-unique Leray-Hopf solutions for some dyadic models for magnetohydrodynamics when the intermittency dimension $δ$ is less than 1. In contrast, uniqueness of Leray-Hopf solution is established in the case of $δ\geq 1$. Analogous results on uniqueness and non-uniqueness of Leray-Hopf solution are also obtained for dyadic models of MHD with fractional diffusion.
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A particular type of dyadic model for the magnetohydrodynamics (MHD) with forward energy cascade is studied. The model includes intermittency dimension $δ$ in the nonlinear scales. It is shown that when $δ$ is small, positive solution with large initial data for either the dyadic MHD model or the dyadic Hall MHD develops blow-up in finite time.
We derive dyadic models for the magnetohydrodynamics with Hall effect by including the intermittency dimension as a parameter. For such dyadic models, existence of global weak solutions is established. In addition, local strong solution is obtained; while global strong solution is obtained in the case of high intermittency dimension. Moreover, we show that positive solution with large initial data develops blow-up in finite time provided the intermittency dimension is lower than a threshold.
We study two dyadic models for incompressible ideal magnetohydrodynamics, one with a uni-directional energy cascade and the other one with both forward and backward energy cascades. Global existence of weak solutions and local well-posedness are established for both models. In addition, solutions to the model with uni-directional energy cascade associated with positive initial data are shown to develop blow-up at a finite time. Moreover, a set of fixed points is found for each model. Linear instability about some particular fixed points is proved.
The dyadic problem $\dot u_n + λ^{2n} u_n - λ^{βn} u_{n-1}^2 + λ^{β(n+1)} u_n u_{n+1} = 0$ with "smooth" initial data is considered. The uniqueness of the Leray-Hopf solution is proved.
The dyadic model $\dot u_n + λ^{2n}u_n - λ^{βn}u_{n-1}^2 + λ^{β(n+1)}u_nu_{n+1} = f_n$, $u_n(0)=0$, is considered. It is shown that in the case of non-trivial right hand side the system can have two different Leray-Hopf solutions.
Uniqueness for a stochastic ideal dyadic MHD model
研究无粘性无电阻磁流体动力学随机二进模型，针对特定随机强迫证明弱唯一性，但解耗散形式上守恒的能量。
Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model
本文在二元Obukhov壳模型中构造了临界正则性下产生两个不同Leray-Hopf解的初始数据，揭示非唯一性源于标度对称性部分破缺，并指出其非线性相互作用与凸积分一致，为完整Navier-Stokes方程的非唯一性研究提供了潜在机制。
We consider the solutions of the Cauchy problem for a dyadic model of Euler equations. We prove global existence and uniqueness of Leray-Hopf solutions in a rather large class K that implies in particular global existence and uniqueness in l^2 for all initial positive conditions in l^2.
In this paper we prove that the lack of uniqueness for solutions of the tree dyadic model of turbulence is overcome with the introduction of a suitable noise. The uniqueness is a weak probabilistic uniqueness for all $l^2$-initial conditions and is proven using a technique relying on the properties of the $q$-matrix associated to a continuous time Markov chain.
We consider the dyadic model with viscosity and additive Gaussian noise as a simplified version of the stochastic Navier-Stokes equations, with the purpose of studying uniqueness and emergence of singularities. We prove path-wise uniqueness and absence of blow-up in the intermediate intensity of the non-linearity, morally corresponding to the 3D case, and blow-up for stronger intensity. Moreover, blow-up happens with probability one for regular initial data.
For the deterministic dyadic model of turbulence, there are examples of initial conditions in $l^2$ which have more than one solution. The aim of this paper is to prove that uniqueness, for all $l^2$-initial conditions, is restored when a suitable multiplicative noise is introduced. The noise is formally energy preserving. Uniqueness is understood in the weak probabilistic sense.
We consider an inviscid stochastically forced dyadic model, where the additive noise acts only on the first component. We prove that a strong solution for this problem exists and is unique by means of uniform energy estimates. Moreover, we exploit these results to establish strong existence and uniqueness of the stationary distribution.
In this paper, we obtain the uniqueness of the 2D MHD equations, which fills the gap of recent work \cite{1} by Chemin et al.
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.
In the paper, we consider the Cauchy problem of the non-resistive MHD equations in homogeneous Besov spaces. We prove the local existence and uniqueness of the solution to the non-resistive MHD equations by using the iterative scheme and compactness arguments. Our obtained result improves considerably the recent results in \cite{C,W}.
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.
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。
Local existence and uniqueness of solutions for the 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion
针对带水平耗散和水平磁扩散的二维不可压缩MHD方程，本文通过非线性项的新估计，建立了$H^s$（$s\neq 1$）初始数据下解的局部存在性与唯一性。
In this note we consider the ideal compressible magneto-hydrodynamics (MHD) equations in a special two dimensional setting. We show that there exist particular initial data for which one obtains infinitely many entropy-conserving weak solutions by using the convex integration technique. Our result is also true for the isentropic case.
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。
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…
In this paper, we provide a much simplified proof of the main result in [Lin, Xu, Zhang, arXiv:1302.5877] concerning the global existence and uniqueness of smooth solutions to the Cauchy problem for a 2D incompressible viscous and non-resistive MHD system under the assumption that the initial data are close to some equilibrium states. Beside the classical energy method, the interpolating inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments. We combine the energy estimates with the $L^\infty$ estimates for time sli…
We study an infinite system of non-linear differential equations coupled in a tree-like structure. This system was previously introduced in the literature and it is the model from which the dyadic shell model of turbulence was derived. It mimics 3d Euler and Navier-Stokes equations in a rough approximation of a wavelet decomposition. We prove existence of finite energy solutions, anomalous dissipation in the inviscid unforced case, existence and uniqueness of stationary solutions (either conservative or not) in the forced case.
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 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。
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…
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。
Instantaneous blowup and non-uniqueness of smooth solutions of MHD
通过逆能量级联机制和凸积分方法，构造了不可压缩磁流体动力学（MHD）系统的一族解，其L∞范数以临界速率瞬时爆破，且解在爆破时间之前保持光滑。
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。
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…
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…
We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space $\mathbb{R}^{3}$ and on the torus $\mathbb{T}^{3}$. In a first step, existence of a unique strong solution are established by constructing a weak solution, proving that pathwise uniqueness holds and using the Yamada-Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.
In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in $L^p_tL^{\infty}_x$ with $1\le p<2$, which reveals the strong non-uniqueness, and the sharpness in terms of the classical Ladyzhenskaya-Prodi-Serrin criteria at endpoint $(2, \infty)$. Moreover, for any $1\le p<2$ and $ε>0$, we construct non-Leray-Hopf weak solutions in $L^p_tL^{\infty}_x\cap L^1_tC^{1-ε}$. The results of Navier-Stokes equations in \cite{1C…
We improve regolarity and uniqueness results from the literature for the inviscid dyadic model. We show that positive dyadic is globally well-posed for every rate of growth $β$ of the scaling coefficients k_n = 2^{bn}. Some regularity results are proved for positive solutions, namely \sup_n n^{-a} k_n^{1/3} X_n(t) < \infty for a.e. t and \sup_n k_n^{1/3-1/(3b)} X_n(t) \leq C t^{-1/3}$ for all $t$. Moreover it is shown that under very general hypothesis, solutions become positive after a finite time.
Magnetic norm inflation at the $\ell^1$ Besov endpoint for viscous non-resistive incompressible MHD
本文证明黏性非电阻不可压缩MHD方程在$\ell^1$ Besov端点空间中磁场范数在原点处膨胀，通过共振四波速度相互作用与磁场拉伸机制构造初值，表明解映射在端点拓扑下不连续。
Sharp non-uniqueness of weak solutions to 2D magnetohydrodynamic equations
本文证明二维粘性电阻磁流体动力学（MHD）方程在$L^2_t L^p(\mathbb{R}^2) \cap L^1_t W^{1,p}(\mathbb{R}^2)$中弱解非唯一，其中$1\le p<\infty$，展示了Ladyzhenskaya-Prodi-Serrin条件在端点$(2,\infty)$的尖锐性，且解位于Beale-Kato-Majda准则的边界上。
Nonuniqueness for high-dimensional ideal MHD equations via differential inclusion
本文通过微分包含框架，证明了高维理想磁流体方程存在无限多个弱解，揭示了解的非唯一性及能量守恒的失效。
Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data
研究在临界正则性下构造了纳维-斯托克斯方程的非唯一光滑解，证明了小数据全球适定性结果的尖锐性。
This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).
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。
This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $\mathbb{R}^d$, $d=2,3$, with initial data $B_0\in H^s(\mathbb{R}^d)$ and $u_0\in H^{s-1+\varepsilon}(\mathbb{R}^d)$ for $s>d/2$ and any $0<\varepsilon<1$. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking $\varepsilon=0$ is explained by the failure of solutions of the heat equation with initial data $u_0\in H^{s-1}$ to satisfy $u\in L^1(0,T;H^{s+1})$; we provide an explicit example of this phenome…
Existence and non-uniqueness of probabilistically strong solutions to 3D stochastic magnetohydrodynamic equations
本文针对三维随机MHD方程，在超临界LPS区域构造无穷多个概率强解，证明其非唯一性，并揭示噪声消失时解的聚点包含所有确定性MHD解。
Non-uniqueness of forced active scalar equations with even drift operators
本文通过和差凸积分方案，证明了带偶漂移算子的强迫活性标量方程在低正则性下弱解非唯一，并指出一维临界正则性为1/3。
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。
In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $\mathbb{R}^{n}$, $n=2,3$, for divergence-free initial data in certain Besov spaces, namely $\boldsymbol{u}_{0} \in B^{n/2-1}_{2,1}$ and $\boldsymbol{B}_{0} \in B^{n/2}_{2,1}$. The a priori estimates include the term $\int_{0}^{t} \| \boldsymbol{u}(s) \|_{H^{n/2}}^{2} \, \mathrm{d} s$ on the right-hand side, which thus requires an auxiliary bound in $H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate i…
This paper is concerned with the asymptotic behaviors of global strong solutions to the incompressible non-resistive viscous magnetohydrodynamic (MHD) equations with large initial perturbations in two-dimensional periodic domains in Lagrangian coordinates. First, motivated by the odevity conditions imposed in [Arch. Ration. Mech. Anal. 227 (2018), 637--662], we prove the existence and uniqueness of strong solutions under some class of large initial perturbations, where the strength of impressive magnetic fields depends increasingly on the $H^2$-norm of the initial perturbation values of both v…
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。
Ideal MHD below the classical well-posedness threshold
该研究针对理想不可压缩磁流体动力学系统，在特定正则性范围内证明了解的局部存在与唯一性，通过挖掘压力项的阿尔文零结构建立了拉格朗日理论与欧拉柯西问题的联系。
Uniqueness and Zeroth-Order Analysis of Weak Solutions to the Non-cutoff Boltzmann equation
本文研究非截断Boltzmann方程大解的唯一性，证明在有限能量条件下，弱解唯一，并通过相空间中的稀释二进制分解方法，将Boltzmann碰撞算子的分数导数结构降为零阶。
In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence,~uniqueness and continuous dependence) with initial data $(u_0,b_0)$ in critical Besov spaces $ {B}^{\frac{d}{p}+1}_{p,1}\times{B}^{\frac{d}{p}}_{p,1}$ with $1\leq p\leq\infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove the global existence in critical Besov spaces. In particular, the results of global existence also hold in Sobole…
In this paper, we deal with the $2\frac{1}{2}$ dimensional Hall MHD by taking the velocity field $u$ and the magnetic field $B$ of the form $u(t,x,y)=\left(\nabla^{\perp}ϕ(t,x,y), W(t,x,y)\right)$ and $B(t,x,y)=\left(\nabla^{\perp}ψ(t,x,y), Z(t,x,y)\right)$. We begin with the Hall equations (without the effect of the fluid part). We first show the long time behavior of weak solutions and weak-strong uniqueness. We then proceed to prove the existence of unique strong solutions locally in time and to derive a blow-up criterion. We also demonstrate that the strong solution exists globally in time…
On Onsager-type conjecture for the Elsässer energies of the ideal MHD equations
本文针对理想MHD方程的Elsässer能量Onsager型猜想，在三维构造了具有总能量耗散和交叉螺旋度守恒失效的弱解，在二维证明了Hölder连续弱解的非唯一性，并给出了Euler解可由MHD弱解逼近的结果。
We prove existence, uniqueness, and higher-order global regularity of strong solutions to a particular Voigt-regularization of the three-dimensional inviscid resistive Magnetohydrodynamic (MHD) equations. Specifically, the coupling of a resistive magnetic field to the Euler-Voigt model is introduced to form an inviscid regularization of the inviscid resistive MHD system. The results hold in both the whole space $\nR^3$ and in the context of periodic boundary conditions. Weak solutions for this regularized model are also considered, and proven to exist globally in time, but the question of uniq…
The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. Recently it has become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$α$) system, which differs from the original MHD system by including an additional non-linear terms (indexed by $α$), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form $-|ξ|^γ/ g(|ξ|)$. In a paper by Penn…
In this article, we study bounded solutions of Euler-type equations on $\mathbb{R}^d$ which have no integrability at $|x| \rightarrow +\infty$. As has been previously noted, such solutions fail to achieve uniqueness in an initial value problem, even under strong smoothness conditions. This contrasts with well-posedness results that have been obtained by using the Leray projection operator in these equations. This apparent paradox is solved by noting that using the Leray projector requires an extra condition the solutions must fulfill at $|x| \rightarrow + \infty$. Our goal is to find one such …
This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Δ)^αu$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $α=1$. This paper discovers that there are new phenomena with the case $α<1$. The approach for $α=1$ can not be directly extended to $α<1$. We establish that, for $α<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^σ_{2,\infty}(\mathbb R^d)$ with $σ> 1+\frac{d}{2}-α$ leads to a unique local solution. For the case …
Global existence and decay estimates of strong solutions for compressible non-isentropic magnetohydrodynamic flows with vacuum
本文研究了含真空状态的压缩性非等熵MHD方程的强解全球存在性及衰减估计，改进了之前关于粘性系数的条件并首次探讨了多维情况下的时间衰减率。
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