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 ε); (2). initially the entropy S_ε^N is around a constant cS, that is, S_ε^N |_t=0 = c_S + O(ε). Then the non-isentropic compressible Navier-Stokes equations admit a unique and global solution (ρ^N_ε , u^N_ε , S_ε^N) with the initial data (ρ0, u0, c_S + εS₀), which is a perturbation of the equilibrium (1, 0, c_S). Moreover, (ρ^N_ε , u^N_ε ) can be approximated by (ρ^I, u^I), the solution to the associated isentropic compressible Navier-Stokes equations equipped with the initial data (ρ0, u0), in the sense that (ρ^N_ε (t), u^N_ε (t)) = (ρ^I(t), u^I(t)) + O(ε), which holds globally in the so-called critical Besov spaces for the compressible Navier-Stokes equations.