Weak solutions for weak turbulence models in electrostatic plasmas

TL;DR

This paper proves the existence of global weak solutions for a one-dimensional electrostatic plasma turbulence model by linking it to a degenerate porous medium equation, introducing new analytical methods.

Contribution

It introduces a novel approach by connecting the quasilinear plasma model to porous medium equations, enabling the first well-posedness results for this class of turbulence models.

Findings

Established existence of global weak solutions

Linked plasma turbulence model to porous medium equations

Provided new analytical tools for nonlinear nonlocal PDEs

Abstract

The weak turbulence model, 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 the Vlasov-Poisson/Maxwell system under the weak turbulence assumption, incorporating the random phase approximation and ergodicity. 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. Therefore, the operators on the right hand side resemble collision forms, such as those in Boltzmann and Landau interacting models. For them, there have been results on well-posedness and regularity of solutions. However, as far as we know, there is no such preceding work for the quasilinear theory addressed in this manuscript. In this paper, we establish the existence of global weak solutions for the system modeling electrostatic plasmas in one dimension. Our key contribution consists of associating the original integral-differential system to a degenerate inhomogeneous porous medium equation(PME) with nonlinear source terms, and leveraging advanced techniques from the PME literature. This approach opens a novel pathway for analyzing weak turbulence models in plasma physics. Moreover, our work offers new tools for tackling related problems in the broader context of nonlinear nonlocal PDEs.