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 global classical solution ( ρ , u ) where ρ ∈ C 1 ( [ 0 , T ] ; H 1 ( [ 0 , 1 ] ) ) and u ∈ H 1 ( [ 0 , T ] ; H 2 ( [ 0 , 1 ] ) ) 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 t t , such as Lemmas 3.2–3.6. This leads to further regularities of ( ρ , u ) where ρ ∈ C 1 ( [ 0 , T ] ; H 3 ( [ 0 , 1 ] ) ) , u ∈ H 1 ( [ 0 , 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.