On the non-existence of global solutions of the Navier-Stokes equations for a compressible viscous heat-conducting fluid

The problem of global existence for the solutions of the multidimensional Navier-Stokes equations for the compressible viscous fluids has generated great interest among mathematicians. Starting from the pioneering papers of D. Gaffi [1] and J. Serrin [2] the basic initial boundary value problems were formulated and the uniqueness of the classic solutions was established. The first existence result was obtained by J. Nash [3] for the classic solution of the Cauchy problem on a sufficiently small time interval. For mixed problems local solvability in time was proved by V.A. Solonnikov [4] in the case of a baratropic fluid and by A. Tani [5] in the case of heat-conducting fluid.

We study the question of global solvability of an initial boundary value problem for the Navier-Stokes equations of a compressible viscous heat-conducting fluid in function spaces in which there is local solvability. Examples are given for the solution destructs in finite time: the solution loses smoothness.

1. D.Graffi, J.Rat.Mech.Anal.2 (1953), 99-106
2. J. Serrin, Arch.Rat.Mech.Anal. 3 (1959), 271-288
3. J. Nash, Bull.Soc.Math. France 90 (1962), 487-497
4. V. A. Solonnikov, Engl. transl. in J.Soviet.Math. 14 (1980), no. 2
5. A. Tani, Publ.Res.Inst.Math.Sci. 13 (1977/78), 193-253