On the Cauchy Problem for Composite Systems of Nonlinear Differential Equations

L∞ continuation principle to the non‐baratropic non‐resistive magnetohydrodynamic equations without heat conductivity

“…In MHDs, the displacement current can be neglected [1]. As a consequence, equation (1.1) 4 is called the induction equation, and the electric field can be written in terms of the magnetic field H and the velocity u, e D r H u H.…”

“…The equations (1.3) will be studied with initial condition: 4) and one of the following boundary conditions:…”

Decay of the compressible magnetohydrodynamic equations

“…However, many physically important and mathematically fundamental problems are still open due to the lack of smoothing mechanism and the strong nonlinearity. For example, although the local strong solutions to the compressible MHD system (1.1) with large initial data were, respectively, obtained by Fan and Yu [11] and Volpert and Khudiaev [30] in the cases that the initial density is strictly positive and that the density is allowed to vanish initially, whether the unique local strong solution can exist globally is an outstanding challenging open problem.…”

Regularity of large solutions for the compressible magnetohydrodynamic equations

“…The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20]. Chen and Wang [15] investigated a free boundary problem with general large initial data with exponents r [0, 1], q ≥ 2r + 2.…”

“…For the non-radiative and non self-gravitation magnetohydrodynamic flows, there have been a number of studies under various conditions by several authors (see, e.g., [2,[15][16][17][18][19][20][21][22]). The existence and uniqueness of local smooth solutions were first obtained in [21]; moreover, the existence of global smooth solutions with small smooth initial data was shown in [20].…”