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.