Existence and Smoothing Effects for a Degenerate Diffusion from Plasma Instability Theory

Abstract

We prove existence and positive-time smoothing of weak solutions to the parabolic Cauchy--Dirichlet problem $\partial_{t} u - ρ_λ(x) u \partial_{x}^2 u = ρ_λ(x) g(x) u$ on the half-line $(0, \infty)$. This problem arises from a system of equations known as the quasilinear theory of plasma waves. We construct weak solutions from weighted $L^p$ initial data ($p < \infty$) and bounded forcing $ρ_λ g \in L^\infty$, a substantially broader data class than previously considered. We identify a parabolic smoothing mechanism for these solutions: a Bénilan--Crandall inequality which provides a one-sided lower bound on $\partial_{t} u$. Driven by this inequality, our solutions become jointly Hölder in space and time and locally Lipschitz in space at positive times. Explicit examples show this spatial regularity is sharp. To our knowledge, this parabolic smoothing from discontinuous data has not previously been established within the family of degenerate quasilinear nondivergence equations with prototype $\partial_{t} u - u Δu = 0$. The Bénilan--Crandall inequality is also new in this setting, and its proof by time-scaling extends formally to other equations of this family.