Analysis Of Several Nonlinear Composite Partial Differential Equations

This doctoral thesis concerns mathematical theories of several problems for nonlinear composite partial differential equations. We are mainly interested in how different type operators, such as hyperbolic and parabolic operators, in the composite systems in?uence the propagation of highly oscillatory waves, the singular limit of boundary value problems and the long time behavior of solutions. For these problems, we shall study quasilinear hyperbolic-parabolic coupled systems with two speeds, the boundary value problem of the incompressible magneto-hydrodynamical equations, and the semilinear composite systems of elasticity and thermoelasticity with second sound respectively.In the introduction, we brie?y introduce physical background of these three kinds of problems, and recall certain known works. Meanwhile, we present three nonlinear hyperbolic and parabolic composite problems to be studied, and state the main results of this thesis.In Chapter 2, we consider the propagation of the high frequency oscillatory waves in a quasilinear hyperbolic-parabolic coupled system. To the Cauchy problem for a quasilinear hyperbolic-parabolic coupled system in several space variables with highly oscillatory initial data and small viscosity, by means of the nonlinear geometric optics,we obtain the asymptotic expansions of oscillatory waves and deduce that the leading oscillation pro?les satisfy the Burgers type quasilinear hyperbolic-parabolic coupled equations with integral terms, from which we obtain that the oscillations are propagated along the characteristics of the hyperbolic operators, and partial pro?les of oscillations satisfy hyperbolic problems while the other pro?les of oscillations are dissipated by the parabolic effect of the original coupled system. Next, we establish the existence of the solutions to the differential-integral equations of the oscillation pro?les by using an iterative scheme. Furthermore, by using the energy method in weighted Sobolev spaces we rigorously justify the asymptotic expansions and obtain the existence of the highly oscillatory waves in a time interval independent of the wavelength.In Chapter 3, we study the estimate of solutions uniformly in viscosity in the small viscosity limit for the initial-boundary value problem of the incompressible magnetohydrodynamic(MHD)equations. For the incompressible MHD equations with Navierfriction boundary condition, we are going to establish an energy estimate of solutions in conormal Soboloev spaces with viscosity as a parameter. First, by studying the commutators we obtain the conormal estimates of tangential derivatives of solutions.Secondly, we derive the estimate of the normal derivatives of solutions by studying the behavior of the vorticity. The L∞estimate of solutions is obtained by using the maximal principle. Finally, by combining the estimate of the pressure we close an uniform estimate of solutions to the incompressible MHD system. Moreover, as a consequence of this uniform estimate, we obtain that the solution of the viscous MHD system converges strongly to the ideal MHD system from a compactness argument, which shows that the boundary layers are very weak in this small viscosity limit.In Chapter 4, we study the long time behavior of solutions to a nonlinear boundary value problem for elasticity and thermoelasticity composite systems. We study this problem for the one-dimensional semilinear elasticity and thermoelasticity composite systems. In this system, the vibration of middle part of the bar is sensitive for the thermal change, which is described by the thermoelasticity with second sound, and the outer parts of the bar are insensitive for the thermal change, described by the pure elasticity. First, by the energy method, we show that the initial boundary value problem for this system is locally well-posed. In the process of proof, we obtain the existence of a solution to the system by using the Fredholm Alternative Principle, i.e. via studying the uniqueness of the corresponding conjugate problem. Furthermore, we obtain that this coupled system is stable exponentially as the time goes to in?nity, under the conditions that the wave speeds in the outer parts are properly large, and the nonlinear terms satisfy certain growth constrain. Moreover, this initial boundary value problem for the composite systems is well-posed globally in time.