A note on global existence of weak solutions to the compressible magnetohydrodynamic equations with Coulomb force

Abstract

In this note, for the case of (9 + √ 33 )/12 < γ 4/3, we prove the existence of globalin-time finite energy weak solution of the equations of a two-dimensional magnetohydrodynamics with Coulomb force, where γ denotes the adiabatic exponent. The value (9 + √ 33 )/12 is the optimal lower bound of γ to establish global-in-time finite energy weak solution under current frame.

Global Weak Solutions to the Equations of Compressible Flow of Nematic Liquid Crystals in Two Dimensions

“…for any constant θ ∈ (0, γ − 1) (the uniform estimate on ρ δ L γ+θ (Q T ) can be found in [18] for the two-dimensional case). Here G is a positive constant independent of δ.…”

Weak-Strong Uniqueness for Compressible Magnetohydrodynamic Equations with Coulomb Force

“…(2) When 9 + ffiffiffiffiffi 33 p /12 < γ ≤ 4/3, Jiang proved the existence of finite-energy weak solution ðρ, u, Φ, MÞ of the system (1)-( 5) in [25].…”

“…In [24], Tan and Wang consider global existence and large time behavior of weak solution to the system (1)-(4). The existence of the finite-energy weak solution to the problem (1)-( 5) is established by Feireisl et al in [19] with γ ≥ 4/3 and Jiang et al in [25] with ð9 + ffiffiffiffiffi 33 p Þ/12 < γ ≤ 4/3. Although the existence of weak solution has been established, the uniqueness problem is still an open problem.…”

“…Many researchers are interested in studying MHD system because of its physical importance, mathematical complexity and extensive applications(see [19][20][21][22][23] and references cited therein). There are also many references about the system (1)-(4)(see [19,[24][25][26]). In [24], Tan and Wang consider global existence and large time behavior of weak solution to the system (1)-(4).…”

Global weak solutions to the Cauchy problem of compressible Navier-Stokes-Vlasov-Fokker-Planck equations

“…To prove Theorem , we adopt a classical approximation scheme that consists of the Faedo–Galerkin approximation, artificial viscosity, artificial pressure, and the celebrated weak continuity of the effective viscous flux to overcome the difficulty of possible large oscillations of the density, and establish the existence of weak solutions. These techniques were developed by Lion, Feireisl, et al for the compressible Navier‐Stokes equations in , and have also been successfully used to establish the existence of weak solutions to other models from fluid dynamics . We refer to for more details.…”