Global classical large solutions to 1D compressible Navier–Stokes equations with density-dependent viscosity and vacuum

Abstract

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique) for any T > 0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ, u) ∈ C 1 ([0, T ]; H 3 (R 1 )) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ 2 u tt , such as Lemmas 3.2-3.6. This leads to further regularities of (ρ, u)T ]; H 3 ([0, 1])). It is still open whether the regularity of u could be improved to C 1 ([0, T ]; H 3 ([0, 1])) with the appearance of vacuum, since it is not obvious that the solutions in C 1 ([0, T ]; H 3 ([0, 1])) to the initial boundary value problem must blow up in finite time.Crown

Global Solutions to the Three-Dimensional Full Compressible Navier--Stokes Equations with Vacuum at Infinity in Some Classes of Large Data

“…In this case, the initial mass (i.e., C 0 ) could be large if µ are sufficiently large. In fact, for isentropic flow (no temperature equation), there have been some works on the global large regular solutions with vacuum for the initial-boundary-value problem in one dimension and in high dimensions (symmetric initial data), Cauchy problem and periodic problem in two dimensions [7,8,23,24]. For the full system, please refer to our previous works [33,34] for the initial-boundaryvalue problem in one dimension and in high dimensions (symmetric initial data).…”

Global classical solution to 1D compressible Navier–Stokes equations with no vacuum at infinity

“…Comparing with Ding et al . and Jiu et al . , the index β ≥0 is more general in this paper, and our result holds true for the constant viscosity while β = 0.…”

“…For one‐dimensional case, Ding et al . obtain a unique classical solution to the initial boundary value problem for isentropic compressible Navier–Stokes equations, where the viscosity coefficient is required to satisfy μ ( ρ )∈ C 2 [0,+ ∞ ),0 < C 1 ≤ μ ( ρ )≤ C 2 (1 + ρ γ ), f o r a n y ρ ≥0. Recently, Jiu et al .…”

“…When the viscosity . / depends on the density, the one-dimensional case has been widely studied (see [1,2,[7][8][9][10][11][12][13][14][15][16] and references therein). However, for the multi-dimensional case, the progress is very limited, even the local existence of classical solution or the global existence of the general weak solutions to the compressible Navier-Stokes equations is still open.…”

Blow‐up of the smooth solutions to the compressible Navier–Stokes equations

“…There have been extensive studies on the CNS equations. The global existence and uniqueness of the strong (classical) solution in the one‐dimensional case have been well‐understood .On the classical solutions, refer to for the local existence in three space dimension, and refer to for global existence with small initial energy in 3D. On the other hand, there are main researchers who paid attention to the well‐posedness of CNS equations in the multi‐dimensional case.…”