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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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