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1. Stability and Large-Time Behavior on 3D Incompressible MHD Equations with Partial Dissipation Near a Background Magnetic Field

   https://doi.org/10.1007/s00205-025-02100-4  

   Lin, Hongxia; Wu, Jiahong; Zhu, Yi (April 2025, Archive for Rational Mechanics and Analysis)

   Abstract Physical experiments and numerical simulations have observed a remarkable stabilizing phenomenon: a background magnetic field stabilizes and dampens electrically conducting fluids. This paper intends to establish this phenomenon as a mathematically rigorous fact on a magnetohydrodynamic (MHD) system with anisotropic dissipation in$$\mathbb R^3$$ $R^3$. The velocity equation in this system is the 3D Navier–Stokes equation with dissipation only in the$$x_1$$ $x_1$-direction, while the magnetic field obeys the induction equation with magnetic diffusion in two horizontal directions. We establish that any perturbation near the background magnetic field (0, 1, 0) is globally stable in the Sobolev setting$$H^3({\mathbb {R}}^3)$$ $H^3\left(R^3\right)$. In addition, explicit decay rates in$$H^2({\mathbb {R}}^3)$$ $H^2\left(R^3\right)$are also obtained. For when there is no presence of a magnetic field, the 3D anisotropic Navier–Stokes equation is not well understood and the small data global well-posedness in$$\mathbb R^3$$ $R^3$remains an intriguing open problem. This paper reveals the mechanism of how the magnetic field generates enhanced dissipation and helps to stabilize the fluid. 

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2. Stability and optimal decay for a system of 3D anisotropic Boussinesq equations

   https://doi.org/10.1088/1361-6544/ac08e9  

   Wu, Jiahong; Zhang, Qian (July 2021, Nonlinearity)

   Abstract This paper focuses on a system of three-dimensional (3D) Boussinesq equations modeling anisotropic buoyancy-driven fluids. The goal here is to solve the stability and large-time behavior problem on perturbations near the hydrostatic balance, a prominent equilibrium in fluid dynamics, atmospherics and astrophysics. Due to the lack of the vertical kinematic dissipation and the horizontal thermal diffusion, this stability problem is difficult. When the spatial domain is Ω = R 2 × T with T = [ − 1 / 2 , 1 / 2 ] being a 1D periodic box, this paper establishes the desired stability for fluids with certain symmetries. The approach here is to distinguish the vertical averages of the velocity and temperature from their corresponding oscillation parts. In addition, the oscillation parts are shown to decay exponentially to zero in time. 

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3. Stability and exponential decay for magnetohydrodynamic equations

   https://doi.org/10.1017/prm.2022.23  

   Feng, Wen; Hafeez, Farzana; Wu, Jiahong; Regmi, Dipendra (April 2022, Proceedings of the Royal Society of Edinburgh: Section A Mathematics)

   This paper focuses on a 2D magnetohydrodynamic system with only horizontal dissipation in the domain$$\Omega = \mathbb {T}\times \mathbb {R}$$with$$\mathbb {T}=[0,\,1]$$being a periodic box. The goal here is to understand the stability problem on perturbations near the background magnetic field$$(1,\,0)$$. Due to the lack of vertical dissipation, this stability problem is difficult. This paper solves the desired stability problem by simultaneously exploiting two smoothing and stabilizing mechanisms: the enhanced dissipation due to the coupling between the velocity and the magnetic fields, and the strong Poincaré type inequalities for the oscillation part of the solution, namely the difference between the solution and its horizontal average. In addition, the oscillation part of the solution is shown to converge exponentially to zero in$$H^{1}$$as$$t\to \infty$$. As a consequence, the solution converges to its horizontal average asymptotically. 

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4. Mild Ill-Posedness in L ∞ for 2D Resistive MHD Equations Near a Background Magnetic Field

   https://doi.org/10.1093/imrn/rnac007  

   Wu, Jiahong; Zhao, Jiefeng (February 2022, International Mathematics Research Notices)

   Abstract The global well-posedness on the 2D resistive MHD equations without kinematic dissipation remains an outstanding open problem. This is a critical problem. Any $L^p$-norm of the vorticity $$\omega $$ with $$1\le p<\infty $$ has been shown to be bounded globally (in time), but whether the $$L^\infty $$-norm of $$\omega $$ is globally bounded remains elusive. The global boundedness of $$\|\omega \|_{L^\infty }$$ yields the resolution of the aforementioned open problem. This paper examines the $$L^\infty $$-norm of $$\omega $$ from a different perspective. We construct a sequence of initial data near a special steady state to show that the $$L^\infty $$-norm of $$\omega $$ is actually mildly ill-posed. 

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5. On Singular Vortex Patches, I: Well-posedness Issues

   https://doi.org/10.1090/memo/1400  

   Elgindi, Tarek; Jeong, In-Jee (March 2023, Memoirs of the American Mathematical Society)

   The purpose of this work is to discuss the well-posedness theory of singular vortex patches. Our main results are of two types: well-posedness and ill-posedness. On the well-posedness side, we show that globally m m -fold symmetric vortex patches with corners emanating from the origin are globally well-posed in natural regularity classes as long as m ≥ 3. m\geq 3. In this case, all of the angles involved solve a closed ODE system which dictates the global-in-time dynamics of the corners and only depends on the initial locations and sizes of the corners. Along the way we obtain a global well-posedness result for a class of symmetric patches with boundary singular at the origin, which includes logarithmic spirals. On the ill-posedness side, we show that any other type of corner singularity in a vortex patch cannot evolve continuously in time except possibly when all corners involved have precisely the angle π 2 \frac {\pi }{2} for all time. Even in the case of vortex patches with corners of angle π 2 \frac {\pi }{2} or with corners which are only locally m m -fold symmetric, we prove that they are generically ill-posed. We expect that in these cases of ill-posedness, the vortex patches actually cusp immediately in a self-similar way and we derive some asymptotic models which may be useful in giving a more precise description of the dynamics. In a companion work from 2020 on singular vortex patches, we discuss the long-time behavior of symmetric vortex patches with corners and use them to construct patches on R 2 \mathbb {R}^2 with interesting dynamical behavior such as cusping and spiral formation in infinite time. 

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