The Investigation On The Large Time Behavior Of The Solutions To The Viscous And Heat-conductive Equations For Reactive Flows

This thesis is concerned a model for the viscous,heat-conductive,reactive flows,which describing the motion of the fluid micelle of the reactants and products during the chemical reaction.It has a wide range of applications in aerospace,material science,physical and chemical engineering.The investigation on the mathematical theory of the viscous heat-conductive equations for reactive flows is significant in the field of pure mathematics.There are still many open problems that have not been solved.In this thesis,we would like to prove the optimal decay rates as well as the decay estimates of the classical solutions to the three dimensional viscous and heat-conductive equations for reactive flows and the stability of rarefaction waves for Cauchy problem,the nonlinear stability of composite waves for one-dimensional viscous and heat-conductive equations for reactive flows.Precisely,In Chapter 2,we are concerned with the large-time behavior of solutions to the Cauchy problem on the viscous and heat-conductive equations for reactive flows. The asymptotic stability of the constant equilibrium state with strictly positive constant density,temperature and the vanishing velocity,mass fraction of the re-actant is established under suit-able small initial perturbation H^N(R³)(?3).Precisely,we show the convergence of the density,velocity and temperature toward-s the corresponding equilibrium state with the optimal rate(1+T)^-3/4L²-norm as well as the convergence of the mass fraction to the equilibrium state with the optimal rate e⁽-?(?)(1+t)^-3/4 norm.Furthermore,the optimal decay rates forthe spatial-derivatives of the solution are also obtained.The proof is based on thetime-weighted energy estimate and continuation argument.In Chapter 3,we investigate the stability of rarefaction wave to the Cauchy problem of the one-dimensional viscous and heat-conductive equations for reactive flows with discontinuous reaction rate function.We prove that under the background ofZ_-=Z₊=0,when the initial disturbance is small,the large-time behavior of the solution to the viscous heat-conductive equations for reactive flows is consistent with the large-time behavior of the solutions to the non-isentropic Navier–Stokes equations,converging to the corresponding 1-rarefaction wave solution of the Eulerequations.In Chapter 4,we investigate the stability of composite waves for the one-dimensional viscous and heat-conductive equations for reactive flows on half line.When the initial data satisfies some smallness assumption,we give the asymptotic stability of not only stationary solution for the impermeability problem but also the composite waves consisting of the subsonic BL-solution,the contact wave,and the rarefaction wave for the inflow problem of the viscous and heat-conductive equations for reactive flows.