Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum

Details Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum Global Classical Large Solutions to Navier-Stokes Equations for Viscous Compressible and Heat Conducting Fluids with Vacuum

2011-03-08

arxiv.org/abs/1103.1421v1

Mathematics

Analysis of PDEs

Scientific paper

In this paper, we consider the 1D Navier-Stokes equations for viscous compressible and heat conducting fluids (i.e., the full Navier-Stokes equations). We get a unique global classical solution to the equations with large initial data and vacuum. Because of the strong nonlinearity and degeneration of the equations brought by the temperature equation and by vanishing of density (i.e., appearance of vacuum) respectively, to our best knowledge, there are only two results until now about global existence of solutions to the full Navier-Stokes equations with special pressure, viscosity and heat conductivity when vacuum appears (see \cite{Feireisl-book} where the viscosity $ \mu=$const and the so-called {\em variational} solutions were obtained, and see \cite{Bresch-Desjardins} where the viscosity $ \mu=\mu(\rho)$ degenerated when the density vanishes and the global weak solutions were got). It is open whether the global strong or classical solutions exist. By applying our ideas which were used in our former paper \cite{Ding-Wen-Zhu} to get $H^3-$estimates of $u$ and $\theta$ (see Lemma \ref{non-le:3.14}, Lemma \ref{non-le:3.15}, Lemma \ref{non-rle:3.12} and the corresponding corollaries), we get the existence and uniqueness of the global classical solutions (see Theorem \ref{non-rth:1.1}).

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