Optimal time decay of the compressible Navier–Stokes equations for a reacting mixture

Abstract

Abstract In this paper, we are concerned with the large-time behavior of solutions to the Cauchy problem on the compressible Navier–Stokes equations for ideal reacting gases. The asymptotic stability of the constant equilibrium state with strictly positive constant density, temperature and the vanishing velocity, mass fraction of the reactant is established under suitable small initial perturbation in H N ( R 3 ) ( N ⩾ 3 ) . Precisely, we show the convergence of the density, velocity and temperature towards the corresponding equilibrium state with the optimal rate ( 1 + t ) − 3 4 in L 2-norm as well as the convergence of the mass fraction to the equilibrium state with the optimal rate e − ϕ ( 1 ) t ( 1 + t ) − 3 4 in L 2-norm. Furthermore, the optimal decay rates for the spatial-derivatives of the solution are also obtained. The proof is based on the time-weighted energy estimate and continuation argument.