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The weak turbulence model for electrostatic system, also known as the quasilinear theory in plasma physics, has been a cornerstone in modeling resonant particle-wave interactions in plasmas. This reduced model stems from either Vlasov-Poisson or Vlasov-Poisson/Maxwell systems under weak turbulence assumptions, incorporating the random phase approximation and ergodicity. The Vlasov-Poisson system reduction has been rigorous  justified in the work of Besse and Bardos (2021). 

      The interaction between particles and waves (plasmons) can be treated as a stochastic process, whose transition probability bridges the momentum space and the spectral space. We establish the existence of global weak,  and also  bounded solutions for well prepared data, for the system modeling electrostatic plasmas.  Our key contribution consists of associating the original integral-differential system to a degenerate inhomogeneous porous medium equation (PME) with nonlinear source terms.  This approach opens a novel pathway for analyzing weak turbulence models in plasma physics and may bring new tools for tackling related problems in the broader context of nonlinear, nonlocal PDE systems.

This is work in collaboration with Kun Huang and William Porteous.