Dafermos' principle and Brenier's duality scheme for defocusing dispersive equations

TL;DR

This paper uncovers an abstract structure for certain nonlinear dispersive equations, introducing a dual variational formulation and proving a principle that constrains entropy dissipation, with implications for equations like Euler.

Contribution

It introduces a novel dual matrix-valued variational framework for defocusing dispersive equations and establishes Dafermos' principle within this abstract setting.

Findings

Established a dual variational formulation in anisotropic Orlicz spaces.

Proved the consistency of the duality scheme over large time intervals.

Demonstrated that subsolutions cannot dissipate entropy faster than strong solutions.

Abstract

We discover an abstract structure behind several nonlinear dispersive equations (including the NLS, NLKG and GKdV equations with generic defocusing power-law nonlinearities) that is reminiscent of hyperbolic conservation laws. The underlying abstract problem admits an "entropy" that is formally conserved. The entropy is determined by a strictly convex function that naturally generates an anisotropic Orlicz space. For such problems, we introduce the dual matrix-valued variational formulation in the spirit of [Y. Brenier. Comm. Math. Phys. (2018) 364(2) 579-605]. Employing time-adaptive weights, we are able to prove consistency of the duality scheme on large time intervals. We also prove solvability of the dual problem in the corresponding anisotropic Orlicz spaces. As an application, we show that no subsolution of the PDEs that fit into our framework is able to dissipate the total entropy earlier or faster than the strong solution on the interval of existence of the latter. This result (we call it Dafermos' principle) is new even for "isotropic" problems such as the incompressible Euler system.