Abstract
This paper is concerned with the dynamics for the compressible Navier–Stokes equations with density-dependent viscosity in bounded annular domains in R^{3}. In the paper, we shall analyze the spherical symmetric model and establish the regularity in H^{2} and H^{4} under certain assumptions imposed on the initial data.
Highlights
Assume that the initial data satisfies (1.19) and (ρ0, u0) ∈ H4( ) × H4( ), there exists a unique generalized global solution (ρ(t), u(t)) ∈ (H4( ))2 to the problem (1.15)–(1.18) verifying that, for any T > 0, ρ ∈ L∞ [0, T], H4( ) ∩ L2 [0, T], H4( ) , u ∈ L∞ [0, T], H4( ) ∩ L2 [0, T], H4( ) , ut ∈ L∞ [0, T], H2( ) ∩ L2 [0, T], H3( ) , utt ∈ L∞ [0, T], L2( ) ∩ L2 [0, T], H1( )
1 Introduction It is well known that the compressible isentropic Navier–Stokes equations which describe the motion of compressible fluids can be written in Eulerian coordinates as ρt + div(ρU) = 0, (1.1)
We show the regularity in H2 and H4 under certain assumptions imposed on the initial data
Summary
Assume that the initial data satisfies (1.19) and (ρ0, u0) ∈ H4( ) × H4( ), there exists a unique generalized global solution (ρ(t), u(t)) ∈ (H4( ))2 to the problem (1.15)–(1.18) verifying that, for any T > 0, ρ ∈ L∞ [0, T], H4( ) ∩ L2 [0, T], H4( ) , u ∈ L∞ [0, T], H4( ) ∩ L2 [0, T], H4( ) , ut ∈ L∞ [0, T], H2( ) ∩ L2 [0, T], H3( ) , utt ∈ L∞ [0, T], L2( ) ∩ L2 [0, T], H1( ) . Lemma 3.1 The following estimate holds for any T > 0: t t utt(t) 2 + uttx(s) 2 ds ≤ C4 + C2 utxx(s) 2 ds, t ∈ [0, T].
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