Diminishing functionals for nonclassical entropy solutions selected by kinetic relations

Abstract

We consider nonclassical entropy solutions to scalar conservation laws with concaveconvex flux functions, whose set of left-and right-hand admissible states u l , ur across undercompressive shocks is selected by a kinetic function ur = ϕ ♭ (u l ). We introduce a new definition for the (generalized) strength of classical and nonclassical shocks, allowing us to propose a generalized notion of total variation functional. Relying only upon the natural assumption that the composite function ϕ ♭ • ϕ ♭ is uniformly contracting, we prove that the generalized total variation of front-tracking approximations is non-increasing in time, and we conclude with the existence of nonclassical solutions to the initial-value problem. We also propose two definitions of generalized interaction potentials which are adapted to handle nonclassical entropy solutions and we investigate their monotonicity properties. In particular, we exhibit an interaction functional which is globally non-increasing along a splitting-merging interaction pattern.

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

“…This is a generalization of an earlier result by LeFloch and Shearer in the scalar case [30]. The proof does not follow the techniques of the earlier work [30], which nonetheless inspired preliminary work of two of the authors on the total variation for scalar laws [25], and hence the current definition of wave strength. The key step of the proof presented here is an extension of Lemma 5.2 from [25] which translates the nucleation condition into a lower bound on the signed variation of the waves crossing N ↓ ± and C ↑ during a single splitting-merging cycle.…”

“…As a step towards this objective, this research has treated in detail small perturbations to a single strong isolated nonclassical shocks using LeFloch's theory of kinetic functions in order to constrain the family of admissible shocks. The main novelties are the new measure of shock strength, extending a previous study of scalar problems [25], and a new weighted interaction potential inspired by earlier works [31,23]. This work explicitly identifies the relationship between the strength of the nonclassical shocks and of the perturbation, even as the strength of the nonclassical wave vanishes.…”

“…All solutions satisfy a single entropy inequality associated with a given entropy pair, and we note that the kinetic relation is equivalent to imposing the rate of entropy dissipation across nonclassical shocks. The existence of nonclassical entropy solutions for scalar equations was recently established for the initial value problem by the last two authors [25] who, building on the earlier work [3,4,5,28,30], succeeded in identifying a "natural" total variation functional adapted to deal with nonclassical shock waves and their interactions. By natural, we mean that the construction of the total variation relies explicitly on the main property of the kinetic function, namely on the contraction property of its iterates.…”

“…Waves will be measured by a generalized strength which extends the earlier definitions for scalar equations [28,30,25].…”

Kinetic relations for undercompressive shock waves. Physical, mathematical, and numerical issues

“…Perturbations of a given nonclassical wave are analyzed in LeFloch [100], Corli and Sablé-Tougeron [47,47], Colombo and Corli [41,42,43], Hattori [69], Laforest and LeFloch [94]. For a version of Glimm's wave interaction potential adapted to nonclassical solutions, we refer to [93]. The L 1 continuous dependence of nonclassical entropy solutions is still an open problem, and it would be interesting to generalize to nonclassical shocks the techniques developed by Bressan et al [27], LeFloch et al [79,66,102], and Liu and Yang [121].…”