The Research On The Large-Time Behavior Of Solution To The Non-Isentropic Navier-Stokes/Allen-Cahn Equations

This thesis is concerned with the asymptotic stability and large-time behavior of solutions to the Cauchy problem on the non-isentropic Navier-Stokes/Allen-Cahn equa-tions.The thesis consists of three chapters:In chapter one,we mainly formulate a model,that is,the non-isentropic Navier-Stokes/Allen-Cahn equations is derived;In chapter two,the asymptotic stability and optimal convergence of solutions have been obtained near a constant equilibrium state provided that the initial perturbation is sufficiently small;In chapter three,we prove the stability of the composite wave.Specifically,the main research content of this thesis is as follows:Firstly,we choose a special Helmholtz energy F,such that the pressure P=Rp?and internal energy E=?-1—R?thus we derive a non-isentropic Navier-Stokes/Allen-Cahn equations which are matched with the classical non-isentropic Navier-Stokes equations.In Chapter 2,we are concerned with the large-time behavior of solutions to the Cauchy problem on the non-isentropic Navier-Stokes/Allen-Cahn system when initial da-ta are around a constant equilibrium state.Moreover the convergence rates are obtained by combining the Lp—Lq estimates for the linearized equations with time-weighted es-timate.Under some conditions on initial data,we show the convergence of the density,velocity and temperature tend time-asymptotically to the corresponding equilibrium s-tate with rate(1+t)-4-3 in L2 and the phase variable also tends to the equilibrium state with the faster rate e-Bt(1+t)-4-3(B>0)in L2 norm.Futhermore,the optimal L2-decay estimates are obtained for all derivatives up to N-1 order.In Chapter 3,we are concerned with the study of the nonlinear stability of the com-posite wave consisting of two rarefactions and a viscous contact discontinuity of the non-isentropic Navier-Stokes/Allen-Cahn system.It is expected that the large-time asymp-totic profiles of solutions of Navier-Stokes/Allen-Cahn system behaves as the same as that of Navier-Stokes system in the case of x+=x-1.We prove that the compos-ite wave is time-asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier-Stokes/Allen-Cahn equations.The analysis is based on the techniques developed in[26]and an elementary L2 energy method.