## On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities.(English)Zbl 1091.35056

Summary: We study the Navier-Stokes equations for compressible barotropic fluids in a bounded or unbounded domain $$\Omega$$ of $$\mathbb R^3$$. We first prove the local existence of solutions $$(\rho,u)$$ in $C([0,T_*]; (\rho^\infty + H^3(\Omega))\times (D_0^1)\cap D^3)(\Omega),$ $$D_0^1= \{v\in L^6(\Omega)$$, $$|\nabla v|_{L^2}<\infty$$, $$v=0$$ on $$\partial\Omega\}$$, $$D^3= \{v\in L_{\text{loc}}^1(\Omega)$$, $$|\nabla^3v|_{L^2}<\infty\}$$, under the assumption that the data satisfies a natural compatibility condition. Then deriving the smoothing effect of the velocity $$u$$ in $$t >0$$, we conclude that $$(\rho, u)$$ is a classical solution in $$(0, T_{**})\times \Omega$$ for some $$T_{**} \in (0, T_{*}]$$. For these results, the initial density needs not be bounded below away from zero and may vanish in an open subset (vacuum) of $$\Omega$$.

### MSC:

 35Q30 Navier-Stokes equations 76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics 35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
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### References:

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