Local well-posedness of Boussinesq equations for MHD convection with fractional thermal diffusion in sobolev space Hs(Rn)×Hs+1−ϵ(Rn)×Hs+α−ϵ(Rn)

2021 ◽  
Vol 62 ◽  
pp. 103355
Author(s):  
Mohammad Ghani
Keyword(s):  
Sobolev Space ◽  
Thermal Diffusion ◽  
Boussinesq Equations ◽  
Well Posedness

scholarly journals Global Well-Posedness of the Ocean Primitive Equations with Nonlinear Thermodynamics

2021 ◽  
Vol 23 (3) ◽  
Author(s):  
Peter Korn
Keyword(s):  
Equation Of State ◽  
Equations Of State ◽  
Boussinesq Equations ◽  
Climate Science ◽  
Rational Functions ◽  
Ocean Dynamics ◽  
Global Ocean ◽  
Primitive Equations ◽  
Well Posedness ◽  
Nonlinear Thermodynamics

AbstractWe consider the hydrostatic Boussinesq equations of global ocean dynamics, also known as the “primitive equations”, coupled to advection–diffusion equations for temperature and salt. The system of equations is closed by an equation of state that expresses density as a function of temperature, salinity and pressure. The equation of state TEOS-10, the official description of seawater and ice properties in marine science of the Intergovernmental Oceanographic Commission, is the most accurate equations of state with respect to ocean observation and rests on the firm theoretical foundation of the Gibbs formalism of thermodynamics. We study several specifications of the TEOS-10 equation of state that comply with the assumption underlying the primitive equations. These equations of state take the form of high-order polynomials or rational functions of temperature, salinity and pressure. The ocean primitive equations with a nonlinear equation of state describe richer dynamical phenomena than the system with a linear equation of state. We prove well-posedness for the ocean primitive equations with nonlinear thermodynamics in the Sobolev space $${{\mathcal {H}}^{1}}$$ H 1 . The proof rests upon the fundamental work of Cao and Titi (Ann. Math. 166:245–267, 2007) and also on the results of Kukavica and Ziane (Nonlinearity 20:2739–2753, 2007). Alternative and older nonlinear equations of state are also considered. Our results narrow the gap between the mathematical analysis of the ocean primitive equations and the equations underlying numerical ocean models used in ocean and climate science.

Global Well-Posedness of Cauchy Problem for 1D Generalized Boussinesq Equations at Both Low Initial Energy Level and Critical Initial Energy Level

2011 ◽  
Author(s):  
Jiang Xiaoli ◽  
Ding Yunhua ◽  
Liu Yacheng ◽  
Xu Runzhang ◽  
Theodore E. Simos ◽  
...  
Keyword(s):  
Cauchy Problem ◽  
Energy Level ◽  
Boussinesq Equations ◽  
Initial Energy ◽  
Well Posedness

Local well-posedness for the quantum Zakharov system in one spatial dimension

2017 ◽  
Vol 14 (01) ◽  
pp. 157-192 ◽  
Author(s):  
Yung-Fu Fang ◽  
Hsi-Wei Shih ◽  
Kuan-Hsiang Wang
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Electric Field ◽  
Classical Result ◽  
Spatial Dimension ◽  
Well Posedness ◽  
Ion Density ◽  
Zakharov System ◽  
Quantum Parameter ◽  
Quantum Zakharov System

We consider the quantum Zakharov system in one spatial dimension and establish a local well-posedness theory when the initial data of the electric field and the deviation of the ion density lie in a Sobolev space with suitable regularity. As the quantum parameter approaches zero, we formally recover a classical result by Ginibre, Tsutsumi, and Velo. We also improve their result concerning the Zakharov system and a result by Jiang, Lin, and Shao concerning the quantum Zakharov system.

Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations

2012 ◽  
Vol 1 (1) ◽  
pp. 57-80 ◽  
Author(s):  
Igor Chueshov ◽  
Alexey Shcherbina
Keyword(s):  
Boussinesq Equations ◽  
Well Posedness

scholarly journals Improved global well-posedness for defocusing sixth-order Boussinesq equations

2020 ◽  
Vol 191 ◽  
pp. 111632 ◽  
Author(s):  
Dan-Andrei Geba ◽  
Evan Witz
Keyword(s):  
Boussinesq Equations ◽  
Sixth Order ◽  
Well Posedness

scholarly journals On the local well-posedness of a Benjamin-Ono-Boussinesq system

2005 ◽  
Vol 2005 (22) ◽  
pp. 3609-3630
Author(s):  
Ruying Xue
Keyword(s):  
Sobolev Space ◽  
Boussinesq System ◽  
Well Posedness ◽  
Well Posed

Consider a Benjamin-Ono-Boussinesq systemηt+ux+auxxx+(uη)x=0,ut+ηx+uux+cηxxx−duxxt=0, wherea,c, anddare constants satisfyinga=c>0,d>0ora<0,c<0,d>0. We prove that this system is locally well posed in Sobolev spaceHs(ℝ)×Hs+1(ℝ), withs>1/4.

scholarly journals Global well-posedness of the 2D Boussinesq equations with fractional Laplacian dissipation

2016 ◽  
Vol 260 (8) ◽  
pp. 6716-6744 ◽  
Author(s):  
Zhuan Ye ◽  
Xiaojing Xu
Keyword(s):  
Fractional Laplacian ◽  
Boussinesq Equations ◽  
Well Posedness ◽  
2D Boussinesq Equations
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