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Global existence and convergence rates for the smooth solutions to the compressible magnetohydrodynamic equations in the half space

Qing Chen, Zhong Tan

2014
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Environmental Systems Research
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With the characteristics of low pollution and low energy consumption, the magnetohydrodynamics has made widely attention. This paper provides the standard energy method to solve the magnetohydrodynamic equations (MHD) in the half space R 3 + . It proves the global existence for the compressible (MHD) by combining the careful a priori estimates and the local existence result. This study also considers the large time behaviors of the solutions. Results: The interactions between the viscous,
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... ssible fluid motion and the magnetic field are modeled by the magnetohydrodynamic system which describes the coupling between the compressible Navier-Stokes equations and the magnetic equations. This study has applied the analytical method to obtain the solutions to (MHD) in R 3 + . It proves that under the assumption that the initial data are close to the constant state, the global existence of smooth solutions can be established. Moreover, the various decay rates of such solutions in L p -norm with 2 ≤ p ≤ +∞ and their derivatives in L 2 -norm can also be derived from combining the decay estimates of the linearized system and the energy method. Conclusions: This study demonstrates that the global existence and the decay rates for the compressible (MHD) can be established under the similar initial assumptions as for the compressible Navier-Stokes equations. Especially, the results suggest that if the initial velocity is small, the velocity decays at a certain rate. This implies that only under the initial assumption that the data are large, it may reach the requirements of (MHD) power generation, which can be used to achieve the value of industrial application and environmental protection. energy and mitigate pollution in order to protect the environment. In virtue of the industrial importance and theoretical challenges, the study on (MHD) has attracted many scientists. In the present paper, we are interested in the well-posedness theory of (MHD). Many results concerning the existence and uniqueness of (weak, strong or smooth) solutions in one dimension can be found in Wang 2002, 2003; Kawashima and Okada 1982) and the references cited therein. In multidimensional case the global existence of weak solutions for the bounded domains has been established recently in (Ducomet and Feireisl 2006; Tan and Wang 2009 ). The local unique strong solution has been obtained in Proposition 3.6. Let s ≥ 4. Under the assumptions of Proposition 3.5, if there exists a positive constants δ 2 such that | 0 , v 0 , H 0 | s + | 0 , v 0 , H 0 | L 1 ≤ δ 2 , then we have |∂ x ( , v, H)| 2 ≤ Cδ 2 (1 + t) − 5 4 −ε 2 , ∀0 < ε 2 < ε for all t ≥ 0, where ε ≤ ε is some positive number.

doi:10.1186/s40068-014-0020-z
fatcat:746rv6tc6rfe5ppywbn5l76bi4