Smooth global solutions for the one-dimensional equations in magnetohydrodynamics

Global Existence and Large-Time Behavior of Solutions to the Three-Dimensional Equations of Compressible Magnetohydrodynamic Flows

“…In particular, the one-dimensional problem has been studied in many papers, for examples, [3,4,7,15,17,22,25] and so on. However, many fundamental problems for MHD are still open.…”

Global Solutions of Nonlinear Magnetohydrodynamics with Large Initial Data

“…We are interested in the well-posedness and regularity of global solutions of the initial-boundary value problem. For perfect gases with small smooth initial data, the existence of global solutions was proved in [15], and the largetime behavior was studied in [25]. For large initial data, these problems have additional difficulties because of the presence of the magnetic field and its interaction with the hydrodynamic motion of the flow of large oscillation.…”

Compressible Navier–Stokes Equations with Zero Heat Conductivity

“…To close the analysis, however, we need the help of the energy estimate for (1.1). The following result is due to Kawashima and Okada [4]. They established the energy estimate for more general case when magnetic field is present, including (1.1) as a special case.…”

“…The asymptotic shape of s (1.4), is analyzed using the third equation in (1.1)$. In the study of the dissipation of u and p we combine the energy method [4] and the pointwise estimates based on the Green's function. Such an approach was introduced in [6] for dissipative systems.…”