Local well-posedness of wave kinetic equations associated with water waves

Abstract

We prove two conceptually different local well-posedness results for wave kinetic equations associated with water-wave systems. For pure gravity waves with 2D interfaces, we establish a classical well-posedness theory in weighted Sobolev spaces: regularity is propagated, and solutions depend continuously on the initial data in the same topology. For gravity-capillary waves with one-dimensional interfaces, the situation is fundamentally different. We prove well-posedness in weighted $L^\infty$ spaces (in the sense of existence, uniqueness, and continuity in a weaker topology), but we also show that, in general, any additional regularity of the initial data is instantly lost. Moreover, continuity in time and continuous dependence on the initial data fail in the weighted $L^\infty$ norm, even for smooth compactly supported initial data. This instability is caused by degenerate three-wave resonances, which produce singular collision rates and arbitrarily short relaxation time scales. The distinction between the two theories reflects a fundamental difference in their resonant geometry.