On the ill-posedness of the compressible Navier-Stokes equations

Details On the ill-posedness of the compressible Navier-Stokes equations On the ill-posedness of the compressible Navier-Stokes equations

2011-09-28

arxiv.org/abs/1109.6092v1

Mathematics

Analysis of PDEs

17pages

Scientific paper

We prove the ill-posedness of three dimensional compressible viscous heat-conductive flows for the initial data belonging to the critical Besov space $(\dot{B}^{\frac3p}_{p,1}+\bar{\rho},\,\dot{B}^{\frac3p-1}_{p,1},\,\dot{B}^{\frac3p-2}_{p,1})$ for $p>3$, here $\bar{\rho}$ is a positive constant. Especially, this result means that it seems impossible to construct a global solution for the highly oscillating initial velocity for the viscous heat-conductive flows. We also prove that the baratropic Navier-Stokes equation is ill-posed for the initial data belonging to the critical Besov space $(\dot{B}^{\frac3p}_{p,1}+\bar{\rho},\,\dot{B}^{\frac3p-1}_{p,1})$ for $p>6$.

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