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 $\Bbb R^N, N\geq 3,$ where the pressure law given by $p(\rho)=a\rho^{\gamma}, $ $a>0, 1<\gamma \leq \frac{N}{4}+\frac12$. In this case if the initial data satisfies $\int_{\Bbb R^N} \rho_0 (x)v_0 (x)\cdot x dx >0$, then there exists no finite energy global weak solution which satisfies the integrability conditions $ \rho |x|^2 \in L^1_{\mathrm{loc}} (0, \infty; L^1 (\Bbb R^N))$ and $ v\in L^1_{\mathrm{loc}} (0, \infty; L^{\frac{N}{N-1}} (\Bbb R^N))$.