Local Weak Solution of the Isentropic Compressible Navier–Stokes Equations with Variable Viscosity

Abstract

In this paper, we consider the 3-D compressible isentropic Navier–Stokes equations with constant shear viscosity \(\mu \) and the bulk one \(\lambda =b\rho ^\beta \), here b is a positive constant, \(\beta \ge 0\). This model was first introduced and well studied by Vaigant and Kazhikhov (Sib Math J 36(6):1283–1316, 1995) in 2D domain. In this paper, under the assumption that \(\gamma >1\), the local existence of weak solutions with higher regularity for the 3D periodic domain is established in the presence of vacuum without any smallness on the initial data. This generalize the previous paper (Desjardins in Commun Partial Differ Equ 22(5):977–1008, 1997; Huang and Yan in J Math Phys 62(11):111504, 2021) to variable viscosity coefficients. Also this is the first result concerning the local weak solution with high regularity for the Kazhikhov model in 3D case.