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

Dmitry Vorotnikov

TL;DR

The paper develops an abstract variational framework for defocusing dispersive PDEs with a formally conserved entropy, realized in anisotropic Orlicz spaces. A dual matrix-valued formulation, inspired by Brenier, is paired with time-adaptive weights to establish large-time consistency and a Dafermos-type entropy principle, linking dual maximizers to strong primal solutions. Existence of dual solutions is shown under a strong trace condition on the operator, and strong solutions are proved unique via a Jeffreys divergence argument, yielding weak-strong uniqueness in the dual setting. The framework is applied to scalar conservation laws, GKdV, defocusing NLS, and complex NLKG, showing how these models fit the abstract structure and benefit from the dual formulation. Overall, the work extends Brenier-type duality to nonlinear dispersive systems, providing a robust tool for entropy-based selection and long-time analysis in anisotropic Orlicz spaces.

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.

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

TL;DR

The paper develops an abstract variational framework for defocusing dispersive PDEs with a formally conserved entropy, realized in anisotropic Orlicz spaces. A dual matrix-valued formulation, inspired by Brenier, is paired with time-adaptive weights to establish large-time consistency and a Dafermos-type entropy principle, linking dual maximizers to strong primal solutions. Existence of dual solutions is shown under a strong trace condition on the operator, and strong solutions are proved unique via a Jeffreys divergence argument, yielding weak-strong uniqueness in the dual setting. The framework is applied to scalar conservation laws, GKdV, defocusing NLS, and complex NLKG, showing how these models fit the abstract structure and benefit from the dual formulation. Overall, the work extends Brenier-type duality to nonlinear dispersive systems, providing a robust tool for entropy-based selection and long-time analysis in anisotropic Orlicz spaces.

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.
Paper Structure (13 sections, 12 theorems, 225 equations)

This paper contains 13 sections, 12 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Anisotropic Orlicz spaces
  3. The abstract setting
  4. Consistency and Dafermos' principle
  5. Solvability of the dual problem
  6. Uniqueness of strong solutions
  7. Applications to PDEs
  8. Scalar conservation laws
  9. GKdV
  10. Defocusing NLS
  11. Complex NLKG
  12. An anisotropic Hardy inequality
  13. Ballistic optimal transport

Key Result

Proposition 2.7

For every $f\in L_{\mathbf{K}}(Q;\mathbb{R}^n)$ and $g\in L_{\mathbf{K}^*}(Q;\mathbb{R}^n)$ one has

Theorems & Definitions (46)

  • Definition 2.1: Young function
  • Definition 2.2: $N$-function
  • Definition 2.3: $\Delta_2$-condition
  • Definition 2.5: Modular and anisotropic Orlicz class
  • Definition 2.6: Luxemburg and Orlicz norms
  • Proposition 2.7: Hölder inequality Gwiazda
  • Proposition 2.8: Duality Gwiazda
  • Proposition 2.9
  • Proposition 2.10: Density of smooth functions
  • proof
  • ...and 36 more