On the Cauchy problems for certain Boussinesq-α equations

2010 ◽  
Vol 140 (2) ◽  
pp. 319-327 ◽  
Author(s):  
Yong Zhou ◽  
Jishan Fan
Keyword(s):  
Cauchy Problem ◽  
Smooth Solution ◽  
Boussinesq Equations ◽  
Regularity Criterion ◽  
Boussinesq System ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Boussinesq Systems

We study the Cauchy problem of certain Boussinesq-α equations in n dimensions with n = 2 or 3. We establish regularity for the solution under ▽u ∈ L1 (0, T; Ḃ0∞,∞(ℝn)). As a corollary, the smooth solution of the Leray-α–Boussinesq system exists globally, when n = 2. For the Lagrangian averaged Boussinesq equations, a regularity criterion ▽θ ∈ L1(0, T;L∞(ℝ2)) is established. Other Boussinesq systems with partial viscosity are also discussed in the paper.

scholarly journals The Cauchy problem and decay rates for strong solutions of a Boussinesq system

2005 ◽  
Vol 2005 (7) ◽  
pp. 757-766 ◽  
Author(s):  
Francisco Guillén González ◽  
Márcio Santos da Rocha ◽  
Marko Rojas Medar
Keyword(s):  
Heat Transfer ◽  
Cauchy Problem ◽  
Viscous Fluid ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Decay Rates ◽  
Boussinesq System ◽  
Three Dimension ◽  
The Cauchy Problem ◽  
Derivatives Of

The Boussinesq equations describe the motion of an incompressible viscous fluid subject to convective heat transfer. Decay rates of derivatives of solutions of the three-dimension-al Cauchy problem for a Boussinesq system are studied in this work.

Fundamental Serrin type regularity criteria for 3D MHD fluid passing through the porous medium

Filomat ◽  
2017 ◽  
Vol 31 (5) ◽  
pp. 1287-1293 ◽  
Author(s):  
Zujin Zhang ◽  
Dingxing Zhong ◽  
Shujing Gao ◽  
Shulin Qiu
Keyword(s):  
Cauchy Problem ◽  
Porous Medium ◽  
Regularity Criterion ◽  
Regularity Criteria ◽  
The Cauchy Problem ◽  
Appl Math

In this paper, we consider the Cauchy problem for the 3D MHD fluid passing through the porous medium, and provide some fundamental Serrin type regularity criteria involving the velocity or its gradient, the pressure or its gradient. This extends and improves [S. Rahman, Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure, J. Comput. Appl. Math., 270 (2014), 88-99].

scholarly journals Positivity of solutions to the Cauchy problem for linear and semilinear biharmonic heat equations

2020 ◽  
Vol 10 (1) ◽  
pp. 353-370 ◽  
Author(s):  
Hans-Christoph Grunau ◽  
Nobuhito Miyake ◽  
Shinya Okabe
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Fourth Order ◽  
Parabolic Equations ◽  
Sufficient Conditions ◽  
Nonlinear Parabolic Equations ◽  
Heat Equations ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Positivity Of Solutions

Abstract This paper is concerned with the positivity of solutions to the Cauchy problem for linear and nonlinear parabolic equations with the biharmonic operator as fourth order elliptic principal part. Generally, Cauchy problems for parabolic equations of fourth order have no positivity preserving property due to the change of sign of the fundamental solution. One has eventual local positivity for positive initial data, but on short time scales, one will in general have also regions of negativity. The first goal of this paper is to find sufficient conditions on initial data which ensure the existence of solutions to the Cauchy problem for the linear biharmonic heat equation which are positive for all times and in the whole space. The second goal is to apply these results to show existence of globally positive solutions to the Cauchy problem for a semilinear biharmonic parabolic equation.

Smoothing effect on the damping mechanism for an inviscid two-phase gas—liquid model

2014 ◽  
Vol 144 (2) ◽  
pp. 351-362 ◽  
Author(s):  
Lizhi Ruan
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
External Force ◽  
Smooth Solution ◽  
Smoothing Effect ◽  
Two Phase ◽  
Global Smooth Solution ◽  
Liquid Model ◽  
The Cauchy Problem

In this paper, we consider the Cauchy problem for an inviscid two-phase gas—liquid model with external force, in order to demonstrate the smoothing effect on the damping mechanism. It is shown that the Cauchy problem admits a unique global smooth solution provided that the initial data are smooth and the C0-norm of the derivative of the initial data are small.

scholarly journals The behavior of solutions of non-linear differential equations in Hilbert space. I

1988 ◽  
Vol 11 (1) ◽  
pp. 143-165 ◽  
Author(s):  
Vladimir Schuchman
Keyword(s):  
Hilbert Space ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Differential Equations ◽  
Linear Case ◽  
Cauchy Problems ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Degenerate Linear

This paper deals with the behavior of solutions of ordinary differential equations in a Hilbert Space. Under certain conditions, we obtain lower estimates or upper estimates (or both) for the norm of solutions of two kinds of equations. We also obtain results about the uniqueness and the quasi-uniqueness of the Cauchy problems of these equations. A method similar to that of Agmon-Nirenberg is used to study the uniqueness of the Cauchy problem for the non-degenerate linear case.

scholarly journals On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jiafa Xu ◽  
Lishan Liu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Life Span ◽  
Boussinesq Equations ◽  
Classical Solutions ◽  
Buoyancy Frequency ◽  
General Initial Data ◽  
The Cauchy Problem ◽  
Long Time

In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with N being the buoyancy frequency. It is proved that for general initial data u 0 ∈ H s with s > 3 , the life span of the classical solutions satisfies T > C ln     N 3 / 4 .

scholarly journals Estimates for Unimodular Multipliers on Modulation Hardy Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-16 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan ◽  
Lijing Sun ◽  
Chunjie Zhang
Keyword(s):  
Cauchy Problem ◽  
Hardy Spaces ◽  
Asymptotic Estimate ◽  
Nonlinear Partial Differential Equations ◽  
Modulation Spaces ◽  
Mixed Norm ◽  
Well Posedness ◽  
Cauchy Problems ◽  
Norm Estimates ◽  
The Cauchy Problem

It is known that the unimodular Fourier multiplierseit|Δ|α/2,α>0,are bounded on all modulation spacesMp,qsfor1≤p,q≤∞. We extend such boundedness to the case of all0<p,q≤∞and obtain its asymptotic estimate astgoes to infinity. As applications, we give the grow-up rate of the solution for the Cauchy problems for the free Schrödinger equation with the initial data in a modulation space, as well as some mixed norm estimates. We also study theMp1,qs→Mp2,qsboundedness for the operatoreit|Δ|α/2, for the case0<α≤2andα≠1.Finally, we investigate the boundedness of the operatoreit|Δ|α/2forα>0and obtain the local well-posedness for the Cauchy problem of some nonlinear partial differential equations with fundamental semigroupeit|Δ|α/2.

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