Resonant Geometry and Well-Posedness for the 2D Boussinesq Wave Kinetic Equation

Abstract

We study the wave kinetic equation (WKE) derived by Shavit--Bühler--Shatah from the two-dimensional Boussinesq system of internal waves. The equation is anisotropic and vector-valued, with a two-branch sign-changing dispersion relation, coupled propagation branches, and a sign-indefinite pseudo-momentum invariant. The resonant manifold has angular degeneracies that obstruct the standard analytic treatment of the collision operator. We introduce a natural angular cut-off adapted to these degeneracies, prove boundedness of the cut-off collision operator, and establish local well-posedness in weighted $L^\infty$ spaces. We also show that, even with the zero-frequency cut-off retained, removing the cut-off near $|\cosθ|=\tfrac12$ makes the collision operator unbounded on these weighted spaces. This provides, to our knowledge, the first rigorous analytic framework for an anisotropic vector-valued WKE arising from Boussinesq dynamics.