Local weak solution of the isentropic compressible Navier–Stokes equations

Abstract

Whether the three dimensional isentropic compressible Navier–Stokes equations admit weak solutions for arbitrary initial data with adiabatic exponent γ > 1 remains a challenging problem. The only available results under γ > 1 were achieved by either assuming the initial data with small energy due to Hoff [J. Differ. Equations 120(1), 215–254 (1995)] or under the spherically symmetric condition by Jiang and Zhang [Commun. Math. Phys. 215, 559–581 (2001)] and Huang [J. Differ. Equations 262, 1341–1358 (2017)]. In this paper, we establish the existence of weak solutions with higher regularity of the three-dimensional periodic compressible isentropic Navier–Stokes equations in small time for the adiabatic exponent γ > 1 in the presence of vacuum. It can be viewed as a local version of Hoff’s work and also extends the result of Desjardins [Commun. Partial Differ. Equations 22(5–6), 977–1008 (1997)] by removing the assumption of γ > 3.