On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

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

“…Then, motivated by the work in [6], we show that an improvement on the integrability of density can ensure the effectiveness and convergence of our approximation scheme. More specifically, we show that the uniform bound of ρ γ ln(1 + ρ) in L 1 , rather than the uniform bound of ρ γ+θ in L 1 for some θ > 0 as used in [10,9,21], ensures the vanishing of artificial pressure and the strong convergence of the density. To overcome the difficulty arising from the possible large oscillations of the density ρ, we adopt the method in Lions [21] and Feireisl [9] which is based on the celebrated weak continuity of the effective viscous flux p − (λ + 2µ)divu (see also Hoff [13]).…”

“…In this section, we establish the existence of solutions to (2.10) following the approach in [10] with the extra efforts to overcome the difficulty arising from the magnetic field. Let…”

“…The density ρ n = ρ[u n ] is determined uniquely as the solution of the Neumann initial-boundary value problem (cf. Lemma 2.2 in [10]):…”

“…In Section 4, following the method in [9] and the result obtained in Section 3, we show the existence of solutions to the approximation system. In Section 5, we follow the technique in [10] with some modifications to get the strong convergence of ρ in L 1 ((0, T ) × Ω). In Section 6, motivated by [11] we study the large-time behavior of global weak solutions to (1.1)-(1.2).…”

Continuity of Drag and Domain Stability in the Low Mach Number Limits

“…Note that existence of global-in-time weak solution, under hypothesis (1.25), can be proved by the methods developed by Lions [11] and [7].…”