Analysis of liquid-vapor interfaces and periodic waves in dynamical lattice systems

The aim of this thesis is the analytical investigation of liquid-vapor interfaces and periodic traveling waves in dynamical lattice systems. This work consists of two parts. The first part is the main part of this work and deals with the analysis of the stability and the persistence of liquid-vapor interfaces in a multidimensional Van der Waals fluid. The subsonic phase boundary is considered as a sharp free boundary connecting liquid and vapor bulk phase dynamics given by the isothermal Euler equations. Surface tension and configurational forces in the form of a kinetic relation determine the evolution of the interface. The system of equations and trace conditions are linearized so that we obtain a generalized hyperbolic initial boundary value problem with higher-order boundary conditions. Linearization about a non-constant reference state results in a linear system with variable coefficients. Stability and global existence theorems for the linearized system with constant coefficients and also for the variable system are shown. By using the results of these linearized systems, an iteration scheme yields the local existence of solutions to the fully nonlinear problem which is stated as the main result of this part in Theorem 5.4. The second part of this thesis concerns the analysis of periodic traveling waves in dynamical lattice systems. Three classes of nonlinear lattice systems resulting from spatial discretization of partial differential equations are considered. We analyze the stability of periodic traveling waves in these lattice systems. First, we derive the so-called modulated equations (or Whitham equations). Then, we investigate the spectral stability in a more analytical framework. The major issue of this part is to give the link between the modulated equations to the stability at the spectral level. We receive that modulation theory is indeed related to spectral stability of periodic traveling waves. In particular, we obtain that the characteristic speeds of the modulated equations have to be real. The main result is given for the reaction-diffusion type lattice system and stated in Theorem 5.1.