Blow-up of viscous heat-conducting compressible flows

AbstractWe show the blow-up of strong solution of viscous heat-conducting flow when the initial density is compactly supported. This is an extension of Z. Xin's result [Z. Xin, Blow up of smooth solutions to the compressible Navier–Stokes equations with compact density, Comm. Pure Appl. Math. 51 (1998) 229–240] to the case of positive heat conduction coefficient but we do not need any information for the time decay of total pressure nor the lower bound of the entropy. We control the lower bound of second moment by total energy and obtain the exact relationship between the size of support of initial density and the existence time. We also provide a sufficient condition for the blow-up in case that the initial density is positive but has a decay at infinity