arXiv subjects

Xiaoping Zhai

Publications and source records attributed to Xiaoping Zhai.

At least 19 recordsLinked to original sources

arXiv2026

Linear Growth and Nonlinear Stability of Two-Dimensional MHD Couette Flow with Vertical Dissipation

We study the two-dimensional incompressible magnetohydrodynamic system near the Couette equilibrium $\bigl((y,0)^{\rm T},(\beta,0)^{\rm T}\bigr)$ on $\mathbb T\times\mathbb R$, in the anisotropic regime where both viscosity and magnetic diffusivity act only in the vertical direction. For the inviscid linearized problem with $|\beta|>1/2$, we prove sharp linear-in-time growth of the vorticity and current density at the level of the time rate. In contrast, the horizontal components of the velocity and magnetic perturbations remain uniformly bounded, while the vertical components exhibit quantitative inviscid damping at the rate $\langle t\rangle^{-1}$. For the nonlinear problem, we introduce shear-adapted Fourier multipliers that simultaneously capture enhanced dissipation, critical-time effects, and echo-type resonant interactions. Under a suitable horizontal background magnetic field and a quantitative compatibility condition between the magnetic field strength, viscosity, and magnetic diffusivity, we establish global nonlinear stability for divergence-free perturbations satisfying $ \left\| \bigl( \mathbf v_{\rm in}-(y,0)^{\rm T}, \mathbf H_{\rm in}-(\beta,0)^{\rm T} \bigr) \right\|_{H^N} \leq \varepsilon_0\min\{\mu,\nu\}^{1/2}, N\geq4$. Moreover, the nonzero horizontal Fourier modes decay in a shear-adapted $H^N$ norm at the enhanced-dissipation rate $e^{-c\min\{\mu,\nu\}^{1/3}\,t},$ and the vertical velocity and magnetic components gain an additional inviscid-damping factor $\langle t\rangle^{-1}$.

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arXiv2026

Stabilization by a background magnetic field: global well-posedness of the compressible isentropic ideal MHD equations with velocity damping

We study the Cauchy problem for the three-dimensional isentropic compressible ideal (inviscid and non-resistive) magnetohydrodynamic equations with velocity damping on the periodic torus $\mathbb{T}^3$. The system admits a steady equilibrium consisting of a constant density $\bar{\rho}$ and a uniform background magnetic field $\omega\in\mathbb{R}^3$. We prove that this equilibrium is nonlinearly stable. More precisely, we show that if the initial data are a sufficiently small perturbation of $(\bar{\rho},\mathbf{0},\omega)$ in the Sobolev space $H^N(\mathbb{T}^3)$ with $N\geq 6r+4$, and if $\omega$ satisfies a Diophantine condition, then the system admits a unique global smooth solution. Moreover, the perturbations decay algebraically in time. To the best of our knowledge, this is the first global well-posedness result for the multi-dimensional isentropic compressible ideal MHD system. The proof reveals a hidden dissipation mechanism: although neither the density equation nor the magnetic field equation contains explicit diffusion or damping, the coupling between the velocity and the magnetic field through the background field $\omega$, combined with a Diophantine--Poincar\'{e} inequality, generates effective dissipation for both the density perturbation and the magnetic field perturbation, which together with the velocity damping yields global regularity and time decay.

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arXiv2025

Stability threshold of the 2D Boussinesq system near Couette flow in an infinite channel

In this paper, we study the stability threshold of the two-dimensional Boussinesq equations around the Couette flow in an infinite channel $\mathbb{R} \times [-1, 1]$ under no-slip boundary conditions. We prove that the Couette flow is asymptotically stable under initial perturbations satisfying $\| \mathbf{v}^{\mathrm{in}} -(y,0)\|_{H^2} \le \varepsilon_0 \nu^{\frac12}$, and $\| \rho^{\mathrm{in}}-1 \|_{H^1} + \big\| |\partial_x|^{\frac13} \rho^{\mathrm{in}} \big\|_{H^1} \le \varepsilon_1 \nu^{\frac56}$. Compared with the work of Masmoudi, Zhai, and Zhao [J. Funct. Anal., 284 (2023), 109736], where the asymptotic stability of the 2D Navier-Stokes-Boussinesq system around Couette flow in a finite channel $\mathbb{T} \times [-1, 1]$ was established, our result improves the stability threshold for the temperature from $\nu^{\frac{11}{12}}$ to $\nu^{\frac56}$.

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arXiv2025

Improved stability threshold for 2D Navier-Stokes Couette flow in an infinite channel

We study the nonlinear stability of the two-dimensional Navier-Stokes equations around the Couette shear flow in the channel domain $\mathbb{R}\times[-1,1]$ subject to Navier slip boundary conditions. We establish a quantitative stability threshold for perturbations of the initial vorticity $\omega_{in}$, showing that stability holds for perturbations of order $\nu^{1/2}$ measured in an anisotropic Sobolev space. This sharpens the recent work of Arbon and Bedrossian [Comm. Math. Phys., 406 (2025), Paper No. 129] who proved stability under the threshold $\nu^{1/2}(1+\ln(1/\nu))^{-1/2}$. Our result removes the logarithmic loss and identifies the natural scaling $\nu^{1/2}$ as the critical size of perturbations for nonlinear stability in this setting.

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arXiv2025

Magnetic Stabilization of Compressible Flows: Global Existence in 3D Inviscid Non-Isentropic MHD Equations

Solutions to the compressible Euler equations in all dimensions have been shown to develop finite-time singularities from smooth initial data such as shocks and cusps. There is an extraordinary list of results on this subject. When the inviscid compressible flow is coupled with the magnetic field in the 3D inviscid non-isentropic compressible magnetohydrodynamic (MHD) equations in $\mathbb{T}^3$, this paper rules out finite-time blowup and establishes the global existence of smooth and stable solutions near a suitable background magnetic field. This result rigorously confirms the stabilizing phenomenon observed in physical experiments involving electrically conducting fluids.

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arXiv2025

Global solutions to 3D compressible MHD equations with partial magnetic diffusion

The global existence of strong solutions to the compressible viscous magnetohydrodynamic (MHD) equations in $\mathbb{R}^3$ remains a significant open problem. When there is no magnetic diffusion, even small data global well-posedness is unknown. This study investigates the Cauchy problem in $\mathbb{R}^3$ for the compressible viscous MHD equations with horizontal magnetic diffusion. Using various anisotropic Sobolev inequalities and sharp estimates, we establish the existence of global solutions under small initial data within the Sobolev space framework.

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arXiv2025

Large global solutions to the Oldroyd-B model with dissipation

In the first part of this work, we investigate the Cauchy problem for the $d$-dimensional incompressible Oldroyd-B model with dissipation in the stress tensor equation. By developing a weighted Chemin-Lerner framework combined with a refined energy argument, we prove the existence and uniqueness of global solutions for the system under a mild constraint on the initial velocity field, while allowing a broad class of large initial data for the stress tensor. Notably, our analysis accommodates general divergence-free initial stress tensors ( $\mathrm{div}\tau_0=0$) and significantly relaxes the requirements on initial velocities compared to classical fluid models. This stands in sharp contrast to the finite-time singularity formation observed in the incompressible Euler equations, even for small initial data, thereby highlighting the intrinsic stabilizing role of the stress tensor in polymeric fluid dynamics. The second part of this paper focuses on the small-data regime. Through a systematic exploitation of the perturbative structure of the system, we establish global well-posedness and quantify the long-time behavior of solutions in Sobolev spaces $H^3(\mathbb{T}^d)$. Specifically, we derive exponential decay rates for perturbations, demonstrating how the dissipative mechanisms inherent to the Oldroyd-B model govern the asymptotic stability of the system.

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arXiv2025

Stability threshold for two-dimensional Boussinesq systems near-Couette shear flow in a finite channel

In this paper, we investigate the stability threshold problem of the two-dimensional Navier-Stokes Boussinesq(NSB) equations in a finite channel $ \T \times [-1,1]$, focusing on the stability around the near Couette shear flow $ (U(y), 0)$, assuming the Navier slip boundary conditions are satisfied. In particular, when the initial data for the vorticity resides in an anisotropic Sobolev space of size $ O(\min \{ \mu^{\frac{1}{2}}, \nu^{\frac{1}{2}}\})$, and the initial perturbation of the temperature resides in an anisotropic Sobolev space of size $ O(\min \{ \mu, \nu\})$, we derive the nonlinear enhanced dissipation effect and the inviscid damping effect for the NSB system.

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arXiv2024

Global well-posedness and large time behavior of solutions to the compressible Oldroyd-B model without stress diffusion

We consider the Cauchy problem ($\mathbb{R}^d, d=2,3$) and the initial boundary values problem ($\mathbb{T}^d, d=2,3$)associated to the compressible Oldroyd-B model which is first derived by Barrett, Lu and Süli [Existence of large-data finite-energy global weak solutions to a compressible Oldroyd-B model, Commun. Math. Sci., 15 (2017), 1265--1323] through micro-macro-analysis of the compressible Navier-Stokes-Fokker-Planck system.Due to lack of stress diffusion, the problems considered here are very difficult. Exploiting tools from harmonic analysis,notably the Littlewood Paley theory,we first establish the global well-posedness and time-decay rates for solutions of the model with small initial data in Besov spaces with critical regularity.Then, through deeply exploring and fully utilizing the structure of the perturbation system,we obtain the global well-posedness and exponential decay rates for solutions of the model with small initial data in the Soboles spaces $H^3(\mathbb{T}^d)$.Our obtained results improve considerably the recent results by Lu, Pokorný [Anal. Theory Appl., 36 (2020), 348--372],Wang, Wen [Math. Models Methods Appl. Sci., 30 (2020), 139--179],and Liu, Lu, Wen [SIAM J. Math. Anal., 53 (2021), 6216--6242].

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arXiv2024

Global well-posedness and large-time behavior for a special $2\frac{1}{2}$D full compressible viscous non-resistive MHD system

In this paper, we consider the full compressible, viscous, non-resistive MHD system under the assumption that the fluids move on a plane while the magnetic field is oriented vertically. Within the framework of Besov spaces, by introducing several new unknown quantities and exploiting the intrinsic structure of the system, we prove the global well-posedness of strong solutions for initial data close to a constant equilibrium state. Furthermore, under some suitable additional conditions involving only the low-frequency part of the initial perturbation, we develop a Lyapunov-type energy argument, which yields the optimal time-decay rates of the global solution. To the best of our knowledge, our result is the first one on global solvability to the full compressible, viscous, non-resistive MHD system in multi-dimensional whole space.

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arXiv2024

Stability for the $2\frac12$-D compressible viscous non-resistive and heat-conducting magnetohydrodynamic flow

In this paper, we are concerned with the initial boundary values problem associated to the compressible viscous non-resistive and heat-conducting magnetohydrodynamic flow, where the magnetic field is vertical. More precisely, by exploiting the intrinsic structure of the system and introducing several new unknown quantities, we overcome the difficulty stemming from the lack of dissipation for density and magnetic field, and prove the global well-posedness of strong solutions in the framework of Soboles spaces $H^3$. In addition, we also get the exponential decay for this non-resistive system. Different from the known results [23], [24], [42], we donot need the assumption that the background magnetic field is positive here.

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arXiv2024

Stability for the 2D incompressible MHD equations with only magnetic diffusion

This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion on the periodic domain. The stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit large-time decay rate of the solutions.

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arXiv2023

Global small solutions to the 3D compressible viscous non-resistive MHD system

Whether or not smooth solutions to the 3D compressible magnetohydrodynamic (MHD) equations without magnetic diffusion are always global in time remains an extremely challenging open problem. No global well-posedness or stability result is currently available for this 3D MHD system in the whole space $\mathbb R^3$ or the periodic box $\mathbb T^3$ even when the initial data is small or near a steady-state solution. This paper presents a global existence and stability result for smooth solutions to this 3D MHD system near any background magnetic field satisfying a Diophantine condition.

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arXiv2023

Global strong solutions to the compressible Navier-Stokes system with potential temperature transport

We study the global strong solutions to the compressible Navier-Stokes system with potential temperature transport in $\mathbb{R}^n.$ Different from the Navier-Stokes-Fourier system, the pressure is a nonlinear function of the density and the potential temperature, we can not exploit the special quasi-diagonalization structure of this system to capture any dissipation of the density. Some new idea and delicate analysis involved in high or low frequency decomposition in the Besov spaces have to be made to close the energy estimates.

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arXiv2022

Global small solutions to a special $2\frac12$-D compressible viscous non-resistive MHD system

This paper solves the global well-posedness and stability problem on a special $2\frac12$-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in $L^p$-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion, and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced dissipation and stabilizing effect resulting from the background magnetic field, a phenomenon observed in physical experiments. In addition, we obtain the solution's optimal decay rate when the initial data is further assumed to be in a Besov space of negative index.

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arXiv2021

Linear stability of the Couette flow for the non-isentropic compressible fluid

We are concerned with the linear stability of the Couette flow for the non-isentropic compressible Navier-Stokes equations with vanished shear viscosity in a domain $\mathbb{T}\times \mathbb{R}$. For a general initial data settled in Sobolev spaces, we obtain a Lyapunov type instability of the density, the temperature, the compressible part of the velocity field, and also obtain an inviscid damping for the incompressible part of the velocity field. Moreover, if the initial density, the initial temperature and the incompressible part of the initial velocity field satisfy some quality relation, we can prove the enhanced dissipation phenomenon for the velocity field.

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arXiv2021

Global well-posedness and inviscid limits of the generalized Oldroyd type models

We obtain the global small solutions to the generalized Oldroyd-B model without damping on the stress tensor in $\mathbb{R}^n$. Our result give positive answers partially to the question proposed by Elgindi and Liu (Remark 2 in Elgindi and Liu [J Differ Equ 259:1958--1966, 2015)]. The proof relies heavily on the trick of transferring dissipation from $u$ to $τ$, and a new commutator estimate which may be of interest for future works. Moreover, we prove a global result of inviscid limit of two dimensional Oldroyd type models in the Sobolev spaces. The convergence rate is also obtained simultaneously.

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arXiv2021

Global small solutions to the inviscid Hall-MHD system

The local existence of smooth solutions to the inviscid Hall-MHD system has been obtained since Chae, Degond and Liu [Ann. Inst. H. Poincaré Anal. Non Linéaire, {31} (2014), 555--565]. However, as we known, how to construct the global small solutions to the inviscid Hall-MHD system is still an open problem. In the present paper, we give a positive answer in $ \mathbb{T}^3$ when the initial magnetic field is close to a background magnetic field satisfying the Diophantine condition.

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