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# On the compressible Navier-Stokes equations in the whole space: From non-isentropic flow to isentropic flow

• The present work aims at the mathematical derivation of the equations for the isentropic flow from those for the non-isentropic flow for perfect gases in the whole space. Suppose that the following things hold for the entropy equation: (1). both conduction of heat and its generation by dissipation of mechanical energy are sufficiently weak(with the order of $\varepsilon$); (2). initially the entropy $S^{N}_ \varepsilon$ is around a constant $c_S$, that is, $S^{N}_ \varepsilon |_{t = 0} = c_S+O( \varepsilon )$. Then the non-isentropic compressible Navier-Stokes equations admit a unique and global solution $( \rho^{N}_ \varepsilon , \, \boldsymbol{u}^{N}_ \varepsilon , \, S^{N}_ \varepsilon )$ with the initial data $( \rho_0, \, \boldsymbol{u}_0, \, c_S+ \varepsilon S_0)$, which is a perturbation of the equilibrium $(1, \boldsymbol{0}, c_S)$. Moreover, $( \rho^{N}_ \varepsilon , u^{N}_ \varepsilon )$ can be approximated by $( \rho^{I}, \, u^{I})$, the solution to the associated isentropic compressible Navier-Stokes equations equipped with the initial data $( \rho_0, \, \boldsymbol{u}_0)$, in the sense that

$\begin{eqnarray*} ( \rho^{N}_ \varepsilon (t), \, \boldsymbol{u}^{N}_ \varepsilon (t)) = ( \rho^{I}(t), \, \boldsymbol{u}^{I}(t))+O(\epsilon), \end{eqnarray*}$

which holds globally in the so-called critical Besov spaces for the compressible Navier-Stokes equations.

Mathematics Subject Classification: 35Q35, 35B40, 76N10.

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