Existence and Regularizing Effects of a Nonlinear Diffusion Model for Plasma Instabilities

2025/03/18 by Porteous, William, Gamba, Irene M., Huang, Kun
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Plasma Physics (physics.plasm-ph)

paper · doi:10.48550/arxiv.2503.13922

Abstract

We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem ∂_tu - ρ_λ(x)uΔu = ρ_λ(x)g₀(x)u on the half line (0,∞). We find weak solutions from L^p (p < ∞) initial data, and by means of a Benilan-Crandall inequality, show solutions are jointly Holder continuous, and locally, spatially Lipschitz on the parabolic interior. We identify special solutions which saturate these bounds. The Benilan-Crandall inequality, derived from time-scaling arguments, is of independent interest for exposing a regularizing effect of the parabolic uΔu operator. Recently considered in [11], this problem originates in the theory of nonlinear instability damping via wave-particle interactions in plasma physics (see [8, 22]).

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