Optimal decay rate of classical solutions to the compressible magnetohydrodynamic equations

Abstract

The three-dimensional compressible magnetohydrodynamic equation in the whole space are studied in this paper. The global classical solution is established when the initial data are small perturbations of some given constant state. Moreover, the optimal decay rate of the solution is also obtained.

Global existence and temporal decay for the 3D compressible Hall-magnetohydrodynamic system

“…For incompressible MHD equations, many problems have been investigated including the blow-up criterion, the uniqueness of weak solutions and the well-posedness of the smooth solutions, and for more details, see [7,8,9,10,15,16,21,39,40] and the references therein. On the other hand, there are also many results regarding the global existence of the solutions and the decay of the smooth solutions to the compressible MHD equations, see [23,24,28,35]. It is well known that the study for optimal decay rates of the solutions to the fluid dynamics equations is interesting in mathematical analysis.…”

“…On the study of decay rates for compressible MHD equations, Umeda, Kawashima and Shizuta [28] studied the global existence and time decay rates of smooth solutions to the linearized 3D compressible MHD equations. Recently, the global existence and optimal decay estimates of smooth solutions to the 3D compressible MHD system were obtained in [24] when the initial data are close to an equilibrium state and belong to…”

“…In these known results mentioned above (See, [13,24,28]), L 1 (R 3 x ) integrability plays an important role in the proof of the optimal decay rates based on the spectral analysis of the semigroup. In general, for evolution equations in which L 2 (R 3 x ) based norms can be propagated by the solution, it is common to make a bounded assumption on the L 1 (R 3 x ) norm of the initial data and combine this with L 2 (R 3 x ) type estimates in order to obtain large time decay estimates.…”

Decay of the compressible magnetohydrodynamic equations

“…For three-dimensional compressible MHD equations, Li and Yu [20] obtained the optimal decay rate of classical solutions, while Chen and Tan [2] studied the large time behavior of smooth solutions when the initial data move toward a constant state. The readers may also refer to [1,13,16,29] and references therein.…”

Long time behavior of solutions to the compressible MHD system in multi-dimensions

“…The compressible MHD system is divided into the isentropic MHD system and the nonisentropic MHD system. For the initial value problem for three-dimensional isentropic MHD system (1.1), lots of interesting results have been established; we refer to [3,15,20,26]. Li and Yu [15] obtained the global existence and the L 2 decay rate of classical solutions under the suitable conditions on small initial value.…”