Entropy Structures and Long-Time Relaxation for 3-Wave Kinetic Equations

TL;DR

This paper introduces a new class of entropy estimates for 3-wave kinetic equations, enabling the construction of global solutions and demonstrating their long-term relaxation to equilibrium.

Contribution

It develops novel entropy structures based on a one-sided algebraic balance condition, not previously seen in wave turbulence literature.

Findings

Established new entropy estimates for 3-wave kinetic equations.

Constructed global weak solutions using entropy compactness.

Proved solutions relax to zero equilibrium as time approaches infinity.

Abstract

We establish a new class of entropy structures for \(3\)-wave kinetic equations with a broad family of interaction weights. Unlike the classical entropies arising from detailed balance, these estimates are generated by a one-sided algebraic balance condition encoded in the interaction weights. To the best of our knowledge, this family of entropy estimates has not previously appeared in the physical literature on wave turbulence. These estimates form the central a priori mechanism of the paper and are the key ingredient in the construction of global weak \(L^1_{\mathrm{loc}}\) solutions. We also prove a long-time rigidity result, showing that the solutions obtained by this entropy compactness method relax locally to the zero equilibrium as \(t\to\infty\).