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

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 $L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1$ 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 $t\to \mathrm{\infty }$.