Abstract:

This paper mainly studies the large-time asymptotic behavior of the global solution of the density dependent one-dimensional isentropic compressible Navier-Stokes equations Cauchy problem. The main purpose of this paper is to improve the result of [7] to $\gamma>1, \kappa \geq 0 $. It is noted that when $\gamma=2,\kappa=1 $, the one-dimensional isentropic compressible Navier-Stokes equations correspond to the Saint-Venant shallow water wave equations, which describe the law of surface shallow water movement and have important applications in physics and oceanography ^[1,4,6]. Note that in [7], the method^[19] of Kanel is used to derive the uniform upper and lower bound estimation of specific volume. When obtaining the upper bound of specific volume, this method requires $\kappa<\frac{1}{2}$. For the problem studied in this paper, we need to use Kanel's method^[19] to derive the uniform upper and lower bound estimation of specific volume. In order to expand the value range of $\kappa$, it is also necessary to make a more precise energy estimation of the upper and lower bounds of the specific volume. After obtaining the uniform upper and lower bound estimation of specific volume, the local solution of Navier-Stokes equations can be extended into the global solution step by step through carefully designed continuity techniques, and the corresponding large-time asymptotic behavior can be obtained.

Key words: One dimensional isentropic compressible Navier-Stokes equations, Viscous shock waves, Large time asymptotic behavior, Nonlinear stability, Density-dependent viscosity, Large initial perturbation.