Global well-posedness and stability of the inhomogeneous kinetic wave equation near vacuum

Abstract

In this paper, we prove global in time existence, uniqueness and stability of mild solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation, for Laplacian dispersion relation in dimension $d=2,3$. We also show that for non-negative initial data, the solution remains non-negative. This is achieved by connecting the inhomogeneous kinetic wave equation to the cubic part of a quantum Boltzmann-type equation with moderately hard potential and no collisional averaging.

arXiv comment17 pages

References in corpus (10)

• The phonon Boltzmann equation, properties and link to weakly anharmonic lattice dynamics
• Weakly nonlinear Schrödinger equation with random initial data
• On the derivation of the wave kinetic equation for NLS
• Full derivation of the wave kinetic equation
• Distributional and classical solutions to the Cauchy Boltzmann problem for soft potentials with integrable angular cross section
• On the derivation of the homogeneous kinetic wave equation
• Global well-posedness of a binary-ternary Boltzmann equation
• Propagation of chaos and the higher order statistics in the wave kinetic theory
• Long time justification of wave turbulence theory
• Inhomogeneous turbulence for the Wick nonlinear Schrödinger equation

Cited by in corpus (2)

• One-dimensional wave kinetic theory
• On the optimal local well-posedness of the wave kinetic equation in $L^r$