Global Classical Solutions to the Compressible Navier-Stokes Equations with Slip Boundary Conditions in 3D Exterior Domains

Abstract

We are concerned with the global existence of classical solutions to the barotropic compressible Navier-Stokes equations with slip boundary condition in a threedimensional (3D) exterior domain. We demonstrate that the classical solutions exists globally in time provided that the initial total energy is suitably small. It is worth noting that the initial density is allowed to have large oscillations and contain vacuum states. For our purpose, some new techniques and methods are adopted to obtain necessary a priori estimates, especially the estimates on the boundary. Moveover, we also give the large-time behavior of the classical solutions what we have gotten.

Global Strong Solutions to the Compressible Magnetohydrodynamic Equations with Slip Boundary Conditions in a 3D Exterior Domain

“…By Lemmas 2.7 and 3.9-3.12, it indicates that (ρ, u, H) is in fact the unique classical solution defined on Ω × (0, T ] for any 0 < T < T * = ∞. Finally, with (2.7), (2.9), (3.14), (2.11), (3.15) and (3.26) at hand, (1.13) can be obtained in similar arguments as used in [3], and we omit the details. The proof of Theorem 1.1 is finished.…”

“…In this subsection, we derive the time-dependent higher order estimates, which are necessary for the global existence of classical solutions. The procedure is similar as that in [3,19,20], and we sketch it here for completeness. From now on, assume that the initial energy C 0 ≤ ε 6 , and the positive constant C may depend on T, µ, λ, ν, a, γ, ρ ∞ , ρ, Ω, M 1 , M 2 , ∇u 0 H 1 , ∇H 0 H 1 , ρ 0 − ρ ∞ W 2,q , P (ρ 0 ) − P ∞ W 2,q , g L 2 for q ∈ (3, 6) where g ∈ L 2 (Ω) is given by compatibility condition (1.10).…”

“…In this section, we are prepared to proof the main results of this paper. Based on the estimates in Section 3, we follows the procedure in [3,20] to give the sketch of the proof.…”

“…In this paper, we are concerned with the global existence of classical solutions of (1.1) in the exterior domain of bounded region with slip boundary condition in R 3 , which can be regarded as a continuation of our work in [7]. Throughout this paper, let D be a simply connected bounded domain in R 3 with smooth boundary and contained in the ball B R {x ∈ R 3 | |x| < R} for the fixed R > 0.…”

“…In addition, the difficulties caused by the slip boundary still exist and cannot be deal with by the methods in [7] directly. To deal with this difficulty, we adapt the idea of [3] to obtain the boundary estimates (see Lemma 2.10), which are used frequently to control the boundary terms in this paper. Furthermore, in order to estimate the derivatives of the solutions, we recall the similar Beale-Kato-Majda-type inequality in the exterior domain (see Lemma 2.6) to prove the important estimates on the gradients of the density and velocity.…”