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 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.

ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1089-1121 ◽  
Author(s):  
A. BABIN ◽  
A. MAHALOV ◽  
B. NICOLAENKO
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Boussinesq Equations ◽  
Stable Stratification ◽  
Primitive Equations ◽  
Time Interval ◽  
Regular Solutions ◽  
Time Intervals ◽  
Long Time

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in Hα, α≥ 3/4. Existence on a long-time interval T*of regular solutions to the 3-D inviscid equations is proven for initial data in Hα, α > 5/2 (T*→∞ as the frequency of gravity waves →∞).

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 The Cauchy Problem for a Dissipative Periodic 2-Component Degasperis-Procesi System

2014 ◽  
Vol 2014 ◽  
pp. 1-17
Author(s):  
Sen Ming ◽  
Han Yang ◽  
Ls Yong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Transport Equation ◽  
Wave Breaking ◽  
A Priori Estimates ◽  
A Priori ◽  
Strong Solutions ◽  
Well Posedness ◽  
Priori Estimates ◽  
The Cauchy Problem

The dissipative periodic 2-component Degasperis-Procesi system is investigated. A local well-posedness for the system in Besov space is established by using the Littlewood-Paley theory and a priori estimates for the solutions of transport equation. The wave-breaking criterions for strong solutions to the system with certain initial data are derived.

Global existence of the 3D rotating second grade fluid system

2020 ◽  
pp. 1-32
Author(s):  
Basma Jaffal-Mourtada
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Second Grade ◽  
Second Grade Fluid ◽  
Fluid System ◽  
Forcing Term ◽  
First Case ◽  
Grade Fluid

We consider the equations of a rotating incompressible non-Newtonian fluid flow of grade two in a three dimensional torus. We prove two different results of global existence of strong solutions. In the first case, we consider that the elasticity coefficient α is arbitrary and we suppose that the third components of the vertical average of the initial data and of the forcing term are small compared to the horizontal components. In the second case, we consider a forcing term and initial data of arbitrary size but we restrict the size of α. In both cases, we show that the limit system is composed of a linear system and a second grade fluid system with two variables and three components.

Global well-posedness and decay estimates of strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

2020 ◽  
pp. 1-25
Author(s):  
Xin Zhong
Keyword(s):  
Initial Data ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Decay Estimates ◽  
Two Dimensional ◽  
Global Strong Solution ◽  
Initial Boundary Value

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

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