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

TL;DR

This paper proves that under certain conditions on initial data and the pressure law, no global weak solutions exist for the compressible Navier-Stokes equations in dimensions three and higher.

Contribution

It establishes the nonexistence of time-dependent global weak solutions for the compressible Navier-Stokes equations with specific pressure laws and initial data constraints.

Findings

No finite energy global weak solutions exist under the given conditions.

Nonexistence holds for initial data with positive momentum integral.

Results apply to isentropic gases in dimensions N ≥ 3.

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))$.