Strong solutions to 3D compressible magnetohydrodynamic equations with Navier‐slip condition

Abstract

We consider the short time strong solutions to the compressible magnetohydrodynamic equations with initial vacuum, in which the velocity field satisfies the Navier-slip condition. The Navier-slip condition differs in many aspects from no-slip conditions, and it has attracted considerable attention in nanoscale and microscale flows research. Inspired by Kato and Lax's idea, we use the Lax-Milgram theorem and contraction mapping argument to prove local existence. Moreover, under the Navier-slip condition, we establish a criterion for possible breakdown of such solutions at finite time in terms of the temporal integral of L 1 norm of the deformation tensor D.u/.

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in a 3D Exterior Domain

“…Finally, we have the following local existence of classical solution of (1.1)-(1.5), which can be proven in a similar manner as that in [12,29].…”

“…Lv-Huang [25] obtained the local existence of classical solutions in R 2 with vacuum as far field density. Tang-Gao [29] obtained the local strong solutions to the compressible MHD equations in a 3D bounded domain with the Navier-slip condition. For global existence, Kawashima [18] first established the global smooth solutions to the general electro-magneto-fluid equations in two dimensions with non-vacuum.…”

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in 3D Bounded Domains

“…To this end, we recall the following local existence theorem of classical solution of (1.1)-(1.4), which can be proved in a similar manner as that in [36,40], base on the standard contraction mapping principle. Lemma 2.5 Assume that the initial date (ρ 0 , u 0 , H 0 ) satisfy the conditions (1.9), (1.10) and (1.12).…”

“…It is rather complicated to investigate the well-posedness and dynamical behaviors of the compressible MHD system with slip boundary condition due to the compatibility issues of the nonlinear terms with the slip boundary conditions. Tang and Gao [36] consider the local strong solutions to the compressible MHD equations with initial vacuum, in which the velocity field satisfies the Navier-slip condition. Considering the full compressible MHD system, Xi and Hao [40] proved the local existence of the classical solutions to the initial-boundary value problem with slip boundary condition for the full compressible MHD system without thermal conductivity, where the initial data contains vacuum and satisfies some initial layer compatibility condition.…”

Global Strong and Weak Solutions to the Initial-boundary-value Problem of 2D Compressible MHD System with Large Initial Data and Vacuum

“…For the initial-boundary-value problem with non-slip boundary condition for the velocity, Hu-Wang [19] proved the global existence of renormalized solutions for general large initial data, also see [11,17] for the non-isentropic compressible MHD equations. As far as the slip boundary is concerned, Tang-Gao [41] obtained the local strong solutions to the compressible MHD equations in a 3D bounded domain with the Navier-slip condition. Dou et al [7] prove the global existence and uniqueness of smooth solutions around a rest state in a 2D bounded domain with slip boundary condition.…”