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Abstract ：
We revisit the global existence of solutions with some large perturbations to the incompressible, viscous, and non-resistive MHD system in a three-dimensional periodic domain, where the impressed magnetic field satisfies the Diophantine condition, and the intensity of the impressed magnetic field, denoted by m, is large compared to the perturbations. It was proved by Jiang-Jiang that the highest-order derivatives of the velocity increase with m and the convergence rate of the nonlinear system towards a linearized problem is of m-1/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m<^>{-1/2}$$\end{document} in [F. Jiang and S. Jiang, Arch. Ration. Mech. Anal., 247 (2023), 96]. In this paper, we adopt a different approach by leveraging vorticity estimates to establish the highest-order energy inequality. This strategy prevents the appearance of terms that grow with m, and thus the increasing behavior of the highest-order derivatives of the velocity with respect to m does not appear. Additionally, we use the vorticity estimates to demonstrate the convergence rate of the nonlinear system towards a linearized problem as time or m approaches infinity. Notably, our analysis reveals that the convergence rate in m is faster compared to the finding of Jiang-Jiang. Finally, a key contribution of our work is identifying an integrable time-decay of the lower-order dissipation. This finding can replace the time-decay of lower-order energy in closing the highest-order energy inequality, significantly relaxing the regularity requirements for the initial perturbations.
Keyword ：
algebraic time-decay algebraic time-decay convergence rate in term of the field intensity. convergence rate in term of the field intensity. global well-posedness global well-posedness incompressible fluids incompressible fluids large initial perturbation large initial perturbation MHD fluids MHD fluids vorticity estimate vorticity estimate
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| GB/T 7714 | Zhao, Youyi . Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations [J]. | JOURNAL OF MATHEMATICAL FLUID MECHANICS , 2025 , 27 (3) . |
| MLA | Zhao, Youyi . "Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations" . | JOURNAL OF MATHEMATICAL FLUID MECHANICS 27 . 3 (2025) . |
| APA | Zhao, Youyi . Global Solutions and Asymptotic Behavior for the Three-dimensional Viscous Non-resistive MHD System with Some Large Perturbations . | JOURNAL OF MATHEMATICAL FLUID MECHANICS , 2025 , 27 (3) . |
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Abstract ：
The inhibition phenomenon of Rayleigh-Taylor instability by surface tension in stratified incompressible viscous fluids driven by gravity has been established in [Y.J. Wang, I. Tice and C. Kim, Arch. Ration. Mech. Anal., 212 (2014), pp. 1-92] via a special flattening coordinate transformation. However, it remains an open problem whether this inhibition phenomenon can be rigorously verified in Lagrangian coordinates due to the delicate nonlinear part of the surface tension term. In this paper, we provide a new mathematical approach, together with some key observations, to prove that the Rayleigh-Taylor problem in Lagrangian coordinates admits a unique global -intime solution under the sharp stability condition nu > nu(C), where nu and nu(C) are the surface tension coefficient and the threshold of the surface tension coefficient, respectively. Furthermore, the solution decays exponentially in time to the equilibrium. Our result provides a rigorous proof of the inhibition phenomenon of Rayleigh-Taylor instability by surface tension under Lagrangian coordinates.
Keyword ：
incompressible viscous fluids incompressible viscous fluids inhibiting effect inhibiting effect Lagrangian coordinates Lagrangian coordinates Rayleigh-Taylor instability Rayleigh-Taylor instability stratified fluids stratified fluids surface tension surface tension
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| GB/T 7714 | Zhao, Youyi . ON THE INHIBITION OF RAYLEIGH-TAYLOR INSTABILITY BY SURFACE TENSION IN STRATIFIED INCOMPRESSIBLE VISCOUS FLUIDS UNDER LAGRANGIAN COORDINATES [J]. | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS , 2024 , 44 (9) : 2815-2845 . |
| MLA | Zhao, Youyi . "ON THE INHIBITION OF RAYLEIGH-TAYLOR INSTABILITY BY SURFACE TENSION IN STRATIFIED INCOMPRESSIBLE VISCOUS FLUIDS UNDER LAGRANGIAN COORDINATES" . | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS 44 . 9 (2024) : 2815-2845 . |
| APA | Zhao, Youyi . ON THE INHIBITION OF RAYLEIGH-TAYLOR INSTABILITY BY SURFACE TENSION IN STRATIFIED INCOMPRESSIBLE VISCOUS FLUIDS UNDER LAGRANGIAN COORDINATES . | DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS , 2024 , 44 (9) , 2815-2845 . |
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Abstract ：
We study the existence of unstable classical solutions of the Rayleigh-Taylor instability problem (abbr. RT problem) of an inhomogeneous incompressible viscous fluid in a bounded domain. We find that, by using an existence theory of (steady) Stokes problem and an iterative technique, the initial data of classical solutions of the linearized RT problem can be modified to new initial data, which can generate local-in-time classical solutions of the RT problem, and are close to the original initial data. Thus, we can use a classical bootstrap instability method to further obtain classical solutions of (nonlinear) RT instability based on the ones of linear RT instability.
Keyword ：
35B10 35B10 35M33 35M33 35Q35 35Q35 76D05 76D05 76E25 76E25
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| GB/T 7714 | Jiang, Fei , Zhao, Youyi . Classical Solutions of Rayleigh-Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains [J]. | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS , 2024 , 63 (4) . |
| MLA | Jiang, Fei 等. "Classical Solutions of Rayleigh-Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains" . | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS 63 . 4 (2024) . |
| APA | Jiang, Fei , Zhao, Youyi . Classical Solutions of Rayleigh-Taylor instability for inhomogeneous incompressible viscous fluids in bounded domains . | CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS , 2024 , 63 (4) . |
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