Global well-posedness and time-decay estimates for compressible Navier-Stokes equations with reaction diffusion

Abstract

We consider the full compressible Navier-Stokes equations with reaction diffusion. A global existence and uniqueness result of the strong solution is established for the Cauchy problem when the initial data is in a neighborhood of a trivially stationary solution. The appearance of the difference between energy gained and energy lost due to the reaction is a new feature for the flow and brings new difficulties. To handle these, we construct a new linearized system in terms of a combination of the solutions. Moreover, some optimal time-decay estimates of the solutions are derived when the initial perturbation is additionally bounded in L1. It is worth noticing that there is no decay loss for the highest-order spatial derivatives of the solution so that the long time behavior for the hyperbolic-parabolic system is exactly the same as that for the heat equation. As a byproduct, the above time-decay estimate at the highest order is also valid for compressible Navier-Stokes equations. The proof is accomplished by virtue of Fourier theory and a new observation for cancellation of a low-medium-frequency quantity.