Time periodic solutions to the three--dimensional equations of compressible magnetohydrodynamic flows

Abstract

In this paper, the compressible magnetohydrodynamic system with some smallness and symmetry assumptions on the time periodic external force is considered in R 3. Based on the uniform estimates and the topological degree theory, we prove the existence of a time periodic solution in a bounded domain. Then by a limiting process, the result in the whole space R 3 is obtained.

Time‐periodic solutions for the full quantum Euler equation

“…Notice that the assumption holds in general due to the conservation of mass. Different from the previous works, 16,17,21,24,26,29 the symmetry of the external force is unnecessary since we can obtain the L 2 estimates for the velocity and the temperature by the structure of (1.3) directly. Moreover, inspired by the idea in Tan et al, 24 we linearize the term u · ∇ u in the case of the quantum Euler system in order to guarantee the closed estimate in Lemma 2.2.…”

“…In the study of the Navier–Stokes equation, one can refer to Jin and Yang 21 and Tan et al 24 for the case of periodic domain and other studies 22,23,25,26 for the whole space. See also Cai et al 17 and Hong et al 27 for the Navier–Stokes–Korteweg system and other studies 16,19,28,29 for the magnetohydrodynamic equations. In addition to these results, one can see other studies 18,20,30 and references therein for the initial boundary value problem.…”

Global well‐posedness of classical solutions to 3D compressible magnetohydrodynamic equations with large potential force

“…Tan and Wang studied the existence, uniqueness, and time asymptotic stability of time periodic solutions in the whole space . Cai and Tan obtained the existence and uniqueness of a time periodic solution under some smallness and symmetry assumptions on the external force; later, Cai and Tan considered the existence of a periodic solution under some smallness and structure conditions on the time periodic external force in the whole space .…”

Stability of stationary solution for the compressible viscous magnetohydrodynamic equations with large potential force in bounded domain

“…Hao [13] discussed the well-posedness to the compressible viscous MHD system in Besov space. Cai and Tan [2,3] obtained time periodic solutions to the three-dimensional equations of compressible MHD flows. Xu and Zhang [36] gave a blow-up criterion for 3D compressible magnetohydrodynamic equations with vacuum.…”

“…Assume that (σ 0 , u 0 , H 0 )(x) ∈ H 3 (Ω)×H 3 (Ω)×H 3 (Ω) satisfying the compatible conditions (1.9), then there exists a positive constant T such that there exist a local solution of the initial boundary value problem (4.1)-(4.3) (σ, u, H)(t, x) ∈ X(0, T ), which satisfies (σ, u, H)(•, t) ∇u, ∇H)(•, s)2 3 ds ≤ C.The a-priori estimate of the solution (σ, u, H) to the initial boundary value problem (4.1)-(4.3) is stated as follows.…”