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Inhomogeneous wave kinetic equation and its hierarchy in polynomially weighted $L^\infty$ spaces
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Inspired by ideas stemming from the analysis of the Boltzmann equation, in this paper we expand well-posedness theory of the spatially inhomogeneous 4-wave kinetic equation, and also analyze an infinite hierarchy of PDE associated with this nonlinear equation. More precisely, we show global in time well-posedness of the spatially inhomogeneous 4-wave kinetic equation for polynomially decaying initial data. For the associated infinite hierarchy, we construct global in time solutions using the solutions of the wave kinetic equation and the Hewitt-Savage theorem. Uniqueness of these solutions is proved by using a combinatorial board game argument tailored to this context, which allows us to control the factorial growth of the Dyson series.
Forward citations
Cited by 2 Pith papers
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On the ill-posedness of kinetic wave equations
For kinetic wave equations from quasilinear Schrödinger models, local well-posedness in weighted L∞ spaces holds exactly when the derivative-loss parameter β ≤ 1/4, and fails for β > 1/4.
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Inhomogeneous six-wave kinetic equation in exponentially weighted $L^\infty$ spaces
First global well-posedness and scattering for the spatially inhomogeneous six-wave kinetic equation in d=1 with small data in exponentially weighted L-infinity spaces.
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