Communications in Mathematical Sciences

Volume 13 (2015)

Number 2

Classical solutions to the Cauchy problem for 2D viscous polytropic fluids with vacuum and zero heat-conduction

Pages: 327 – 345



Zhilei Liang (School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China)

Xiaoding Shi (Department of Mathematics, Graduate School of Science, Beijing University of Chemical Technology, Beijing, China)


This paper is concerned with viscous polytropic fluids in two-dimensional (2D) space with vacuum as far field density. By means of weighted initial density, we obtain the local existence of classical solutions to the Cauchy problem, in the case that the initial data satisfy a natural compatibility condition and the heat-conduction coefficient is zero. Recalling the blowup result of Xin [Z. Xin, Comm. Pure Appl. Math., 51, 229–240, 1998], one should not expect a global smooth solution because the compactly supported initial density is included in our case.


compressible Navier-Stokes, vacuum, classical solution, 2D Cauchy problem, zero heat-conduction

2010 Mathematics Subject Classification

35B45, 35M10, 76N10

Published 3 December 2014