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

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

2009-01-21

arxiv.org/abs/0901.3278v4

Mathematics

Analysis of PDEs

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

Scientific paper

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))$.

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