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For nonlinear hyperbolic systems of conservation laws in one space variable,we establish the existence of nonclassical entropy solutions exhibitingnonlinear interactions between shock waves with strong strength. The proposedtheory is relevant in the theory of phase transition dynamics, and thesolutions under consideration enjoy a splitting--merging pattern, consisting of(compressive) classical and (undercompressive) nonclassical waves, interactingtogether as well as with classical waves of smaller strength. Our analysis isbased on three novel ideas. First, a generalization of Hayes--LeFloch'snonclassical Riemann solver is introduced for systems and is based onprescribing, on one hand, a kinetic relation for the propagation ofnonclassical undercompressive shocks and, on the other hand, a nucleationcriterion that selects between classical and nonclassical behavior. Second, weextend LeFloch-Shearer's theorem to systems and we prove that the presence of anucleation condition implies that only a finite number of splitting and mergingcycles can occur. Third, our arguments of nonlinear stability build upon recentwork by the last two authors who identified a natural total variationfunctional for scalar conservation laws and, specifically, for systems ofconservation laws we introduce here novel functionals which measure the totalvariation and wave interaction of nonclassical and classical waves.

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