scholarly journals Global well-posedness for the three-dimensional incompressible viscous non-resistive MHD equations in an infinite slab

2021 ◽  
Author(s):  
Youyi Zhao
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Mhd Equations ◽  
Well Posedness ◽  
Lagrangian Coordinates ◽  
Mathematical Approach ◽  
Compatibility Conditions ◽  
Finite Height ◽  
Infinite Slab ◽  
Mhd System

In this paper, we investigate the global well-posedness of the system of incompressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no nonlinear compatibility conditions on the initial data.

scholarly journals Instability of the abstract Rayleigh–Taylor problem and applications

2020 ◽  
Vol 30 (12) ◽  
pp. 2299-2388 ◽  
Author(s):  
Fei Jiang ◽  
Song Jiang ◽  
Weicheng Zhan
Keyword(s):  
Initial Data ◽  
Stokes Problem ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Existence Theory ◽  
Lagrangian Coordinates ◽  
Compatibility Conditions ◽  
Unstable Solutions ◽  
Taylor Problem ◽  
Proper Values

Based on a bootstrap instability method, we prove the existence of unstable strong solutions in the sense of [Formula: see text]-norm to an abstract Rayleigh–Taylor (RT) problem arising from stratified viscous fluids in Lagrangian coordinates. In the proof we develop a method to modify the initial data of the linearized abstract RT problem by exploiting the existence theory of a unique solution to the stratified (steady) Stokes problem and an iterative technique, such that the obtained modified initial data satisfy the necessary compatibility conditions on boundary of the original (nonlinear) abstract RT problem. Applying an inverse transform of Lagrangian coordinates to the obtained unstable solutions and taking then proper values of the parameters, we can further obtain unstable solutions of the RT problem in viscoelastic, magnetohydrodynamics (MHD) flows with zero resistivity and pure viscous flows (with/without interface intension) in Eulerian coordinates.

scholarly journals Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

2020 ◽  
Vol 22 (3) ◽  
Author(s):  
Xin Liu ◽  
Edriss S. Titi
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Viscous Flows ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Physical Vacuum ◽  
Well Posedness ◽  
Small Initial Data ◽  
Anelastic Equations

Abstract We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

scholarly journals Global Well-posedness and Asymptotics of Full Compressible Non-resistive MHD System with Large External Potential Forces

2021 ◽  
Author(s):  
Wanrong Yang ◽  
Xiaokui Zhao
Keyword(s):  
Stationary Solution ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Unique Existence ◽  
Background Magnetic Field ◽  
External Forces ◽  
Mhd System ◽  
Initial Boundary Value

We consider the global well-posedness and asymptotic behavior of compressible viscous, heat-conductive, and non-resistive magnetohydrodynamics (MHD) fluid in a field of external forces over three-dimensional periodic thin domain $\Omega=\mathbb{T}^2\times(0,\delta)$. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and periodic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. Moreover, if the external forces are small or disappeared in an appropriate Sobolev space, then $\delta$ can be a general constant. Our proof relies on the two-tier energy method for the reformulated system in Lagrangian coordinates and the background magnetic field which is perpendicular to the flat layer. Compared to the work of Tan and Wang (SIAM J. Math. Anal. 50:1432–1470, 2018), we not only overcome the difficulties caused by temperature, but also consider the big external forces.

scholarly journals Local Well-Posedness of Strong Solutions to the Three-Dimensional Compressible Primitive Equations

2021 ◽  
Author(s):  
Xin Liu ◽  
Edriss S. Titi
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Atmospheric Dynamics ◽  
Primitive Equations ◽  
Well Posedness ◽  
Short Time

AbstractThis work is devoted to establishing the local-in-time well-posedness of strong solutions to the three-dimensional compressible primitive equations of atmospheric dynamics. It is shown that strong solutions exist, are unique, and depend continuously on the initial data, for a short time in two cases: with gravity but without vacuum, and with vacuum but without gravity.

scholarly journals Long time decay of 3D-NSE in Lei-Lin-Gevrey spaces

2020 ◽  
Vol 70 (4) ◽  
pp. 877-892
Author(s):  
Jamel Benameur ◽  
Lotfi Jlali
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Well Posedness ◽  
Time Decay ◽  
Navier Stokes Equation ◽  
Gevrey Spaces ◽  
Gevrey Space ◽  
Long Time ◽  
Incompressible Navier Stokes Equation

AbstractIn this paper, we prove a global well-posedness of the three-dimensional incompressible Navier-Stokes equation under initial data, which belongs to the Lei-Lin-Gevrey space $\begin{array}{} Z^{-1}_{a,\sigma} \end{array}$(ℝ3) and if the norm of the initial data in the Lei-Lin space 𝓧−1 is controlled by the viscosity. Moreover, we will show that the norm of this global solution in the Lei-Lin-Gevrey space decays to zero as time approaches to infinity.

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