The mathematical theory of splitting-merging patterns in phase transition dynamics

Abstract

For nonlinear hyperbolic systems of conservation laws in one space variable, we establish the existence of nonclassical entropy solutions exhibiting nonlinear interactions between shock waves with strong strength. The proposed theory is relevant in the theory of phase transition dynamics, and the solutions under consideration enjoy a splitting--merging pattern, consisting of (compressive) classical and (undercompressive) nonclassical waves, interacting together as well as with classical waves of smaller strength. Our analysis is based on three novel ideas. First, a generalization of Hayes--LeFloch's nonclassical Riemann solver is introduced for systems and is based on prescribing, on one hand, a kinetic relation for the propagation of nonclassical undercompressive shocks and, on the other hand, a nucleation criterion that selects between classical and nonclassical behavior. Second, we extend LeFloch-Shearer's theorem to systems and we prove that the presence of a nucleation condition implies that only a finite number of splitting and merging cycles can occur. Third, our arguments of nonlinear stability build upon recent work by the last two authors who identified a natural total variation functional for scalar conservation laws and, specifically, for systems of conservation laws we introduce here novel functionals which measure the total variation and wave interaction of nonclassical and classical waves.