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).

scholarly journals Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Menglong Su
Keyword(s):  
Strong Solution ◽  
Blow Up ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Density Dependent ◽  
Global Strong Solution ◽  
Mhd System ◽  
Initial Boundary Value ◽  
Dependent Viscosity

AbstractIn this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if $\|\nabla \mu (\rho )\|_{L^{\infty }(0, T; L^{p})}$ ∥ ∇ μ ( ρ ) ∥ L ∞ ( 0 , T ; L p ) is bounded. Second, we prove the strong solution exists globally (in time) only if $\|\nabla \mu (\rho _{0})\|_{L^{p}}$ ∥ ∇ μ ( ρ 0 ) ∥ L p is suitably small, even the presence of vacuum is permitted.

scholarly journals A note on construction of nonnegative initial data inducing unbounded solutions to some two-dimensional Keller–Segel systems

2021 ◽  
Vol 4 (6) ◽  
pp. 1-12
Author(s):  
Kentaro Fujie ◽  
◽  
Jie Jiang ◽  
Keyword(s):  
Initial Data ◽  
Boundary Value Problem ◽  
Total Mass ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Negative Energy ◽  
Two Dimensional ◽  
Unbounded Solutions ◽  
Initial Boundary Value ◽  
Symmetric Initial Data

<abstract><p>It was shown that unbounded solutions of the Neumann initial-boundary value problem to the two-dimensional Keller–Segel system can be induced by initial data having large negative energy if the total mass $ \Lambda \in (4\pi, \infty)\setminus 4\pi \cdot \mathbb{N} $ and an example of such an initial datum was given for some transformed system and its associated energy in Horstmann–Wang (2001). In this work, we provide an alternative construction of nonnegative nonradially symmetric initial data enforcing unbounded solutions to the original Keller–Segel model.</p></abstract>

scholarly journals Global existence of classical solutions to the two-dimensional compressible Boussinesq equations in a square domain

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xucheng Huang ◽  
Zhaoyang Shang ◽  
Na Zhang
Keyword(s):  
A Priori Estimates ◽  
A Priori ◽  
Boussinesq Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Classical Solutions ◽  
Two Dimensional ◽  
Priori Estimates ◽  
Constant Viscosity ◽  
Initial Boundary Value

Abstract In this paper, we consider the initial boundary value problem of two-dimensional isentropic compressible Boussinesq equations with constant viscosity and thermal diffusivity in a square domain. Based on the time-independent lower-order and time-dependent higher-order a priori estimates, we prove that the classical solution exists globally in time provided the initial mass $\|\rho _{0}\|_{L^{1}}$ ∥ ρ 0 ∥ L 1 of the fluid is small. Here, we have no small requirements for the initial velocity and temperature.

scholarly journals Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yong Zhou ◽  
Jishan Fan ◽  
Gen Nakamura
Keyword(s):  
Strong Solution ◽  
Initial Density ◽  
Regularity Criterion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Density Dependent ◽  
Global Strong Solution ◽  
Dependent Flow ◽  
Density Dependent Flow ◽  
Initial Boundary Value

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocityu0and small∇d0. We also give a regularity criterion∇d∈Lp(0,T;Lq(Ω))  (2/q)+(2/p)=1, 2<q≤∞of the problem with the Dirichlet boundary conditionu=0,d=d0on∂Ω.

scholarly journals Strong Solutions of the Incompressible Navier–Stokes–Voigt Model

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 181
Author(s):  
Evgenii S. Baranovskii
Keyword(s):  
Three Dimensional ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Evolutionary Equation ◽  
Long Time ◽  
Special Basis ◽  
External Forces ◽  
Initial Boundary Value

This paper deals with an initial-boundary value problem for the Navier–Stokes–Voigt equations describing unsteady flows of an incompressible non-Newtonian fluid. We give the strong formulation of this problem as a nonlinear evolutionary equation in Sobolev spaces. Using the Faedo–Galerkin method with a special basis of eigenfunctions of the Stokes operator, we construct a global-in-time strong solution, which is unique in both two-dimensional and three-dimensional domains. We also study the long-time asymptotic behavior of the velocity field under the assumption that the external forces field is conservative.

scholarly journals Well-posedness of time-fractional advection-diffusion-reaction equations

2019 ◽  
Vol 22 (4) ◽  
pp. 918-944 ◽  
Author(s):  
William McLean ◽  
Kassem Mustapha ◽  
Raed Ali ◽  
Omar Knio
Keyword(s):  
Linear Time ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Diffusion Reaction ◽  
Fractional Integrals ◽  
Well Posedness ◽  
Advection Diffusion ◽  
Initial Boundary Value ◽  
Reaction Equations ◽  
Advection Diffusion Reaction Equations

Abstract We establish the well-posedness of an initial-boundary value problem for a general class of linear time-fractional, advection-diffusion-reaction equations, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our analysis relies on novel energy methods in combination with a fractional Gronwall inequality and properties of fractional integrals.

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