On the nonexistence of time dependent global weak solutions to the compressible Navier-Stokes equations

Abstract

In this paper we prove the nonexistence of global weak solutions to the compressible Navier-Stokes equations for the isentropic gas in ${\mathbb{R}}^N,N\ge 3,$ where the pressure law given by $p(\rho )=a{\rho }^{\gamma },$ $a>0,1<\gamma \le \frac{N}{4}+\frac{1}{2}$. In this case if the initial data satisfies ${\int }_{{\mathbb{R}}^N}{\rho }_0(x)v_0(x)\cdot xdx>0$, then there exists no finite energy global weak solution which satisfies the integrability conditions $\rho |x{|}^2\in L_{\mathrm{loc}}^1(0,\infty ;L^1({\mathbb{R}}^N))$ and $v\in L_{\mathrm{loc}}^1(0,\infty ;L^{\frac{N}{N-1}}({\mathbb{R}}^N))$.

8 pages(revised and substantially reduced version of the previous one)

References in corpus (1)

• On the nonexistence of stationary weak solutions to the compressible fluid equations