Asymptotics Of Global Solutions To The Compressible Navier-Stokes-Korteweg Equations

The motion of the compressible viscous fluids with internal capillarity can be de-scribed by the compressible Navier-Stokes-Korteweg equations, which originates from the pioneering works by J. D. Van der Waals [1] and D. J. Korteweg [2] between the end of the19th century and the beginning of the20th century and was derived rigorously by J. E. Dunn and J. Serrin [3] in the1980’s by using the second gradient theory. Due to the complexity of the phenomena described by the compressible Navier-Stokes-Korteweg equations and the high nonlinearities of the equations, the corresponding mathemati-cal theory on the well-posedness of the compressible Navier-Stokes-Korteweg equations provides many challenging mathematical problems and has attracted the attentions of many mathematicians. In fact it is one of the hottest topics in the study of nonlinear partial differential equations and up to now. there have been a lot of results on global solvability of the compressible Navier-Stokes-Korteweg equations which focus on the case when the derived global solutions are small perturbations of some constants which are trivial profiles of the compressible Navier-Stokes-Korteweg equations. For the cor-responding mathematical results on the nonlinear stability of some nontrivial profiles such as rarefaction waves, traveling waves, contact discontinuity, stationary solution-s, time-periodic solutions, etc. decsribed by the compressible Navier-Stokes-Korteweg equations, to the best of our knowledge, fewer results have been obtained.This thesis is concerned with nonlinear stability of some nontrivial profiles such as rarefaction waves, traveling waves, contact discontinuity, stationary solutions, time-periodic solutions, etc. decsribed by the compressible Navier-Stokes-Korteweg equation-s. The main results of this thesis contain the following three parts.The first part, which is the main content of Chapter2, is concerned with the nonlin-ear stability of basic wave patterns for the one dimensional compressible Navier-Stokes-Korteweg equations. Namely, we consider the nonlinear stability of traveling waves and rarefaction waves to the following Cauchy problem of the isentropic compressible Navier-Stokes-Korteweg equations in the Lagrange coordinate and the nonlinear stability of viscous contact waves to the Cauchy problem of the non-isentropic compressible Navier-Stokes-Korteweg equations: where v>0,υ, P, e,θ>0denote the specific volume, the velocity, the pressure, the internal energy and the absolute temperature of the fluids, respectively. The constantss μ, κ and α are the viscosity coefficient, the capillary coefficient and heat conductive coefficient, respectively. v±>0, u±and θ±>0are given constants. Firstly, we show the existence of traveling wave solutions to system (0.0.5)1,2by using the phase plane analysis. Secondly, by the L2energy method, the nonlinear stability of weak traveling waves and strong rarefaction waves to the Cauchy problem (0.0.5) and the nonlinear stability of weak viscous contact waves to the Cauchy problem (0.0.6) are obtained under some small initial perturbations. Here weak (strong) traveling waves (rarefaction waves, viscous contact waves) means the strengths of the traveling waves (rarefaction waves, viscous contact waves) are small (large).The second part, which is the main content of Chapter3, is devoted to the ex-istence and nonlinear stability of stationary solutions to the following non-isentropic compressible Navier-Stokes-Korteweg equations Here (t,x)∈R+×R3, ρ>0denotes the density of the fluid, G(x), F(x) and H(x) are the given mass source, the general external force and the energy source, respectively. The viscous stress tensor S and the Cauchy stress tensor T are given by where μ,μ’ are the viscosity coefficients,I denotes the n×n unite matrix,(▽u)T denotes the transports of the matrix▽u. W=-κρ▽·υ▽ρ+κ▽·(G(x)▽ρ) is the interstitial working flux. The physical meaning of the other variables in (0.0.7) are the same as those of (0.0.6). Here, the coefficients μ,μ’,a and κ are assumed to be positive constants. Based on the weighted L2method and the contraction mapping principle, we prove the existence and uniqueness of stationary solution to system (0.0.7)1,2,3under some smallness assumptions on G(x), F(x) and H(x). The proof of the stability result is given in the113framework by an elementary energy method and relies on some intrinsic properties of the full compressible Navier-Stokes-Korteweg system.The third part, which is the main content of Chapter4, is concerned with the existence, nonlinear stability and temporal decay rate of time periodic solutions to the following isentropic compressible Navier-Stokes-Korteweg equations in n dimensions: with Here (t,x)∈R+×Rn,u=(u1,u2,...,un)∈Rn is the velocity, f(t,x)=(f1(t,x)f2(t, x),...,fn(t,x)) is a given time-periodic external force with period T>0.ρ∞>0 is a given constant. The physical meaning of the other variables in (0.0.8) are the same as those of (0.0.7). Based on the L2energy method and the temporal decay estimates of solutions to the linearized system of (0.0.8), we show the existence and uniqueness of time periodic solution to (0.0.8) when the space dimension n>5under some smallness assumptions on f. Furthermore, the nonlinear stability and the temporal decay estimates of the time periodic solution are deduced by the energy method.