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

William Porteous, Irene M. Gamba, Kun Huang

TL;DR

This work establishes the existence of weak energy solutions for a weighted nonlinear parabolic equation on the half-line, cast in divergence form with a gradient nonlinearity $|\nabla u|^2$ and weight $\rho_{\lambda}$. The authors construct approximate, regularized problems on finite intervals, obtain uniform a priori estimates, and employ Aubin-Lions-type compactness together with gradient-truncation techniques to pass to the limit. A central contribution is the Benilan-Crandall inequality, which yields a time-scaling regularizing effect that implies interior Hölder continuity and spatial Lipschitz regularity in the parabolic interior, with sharpness illustrated by special solutions. The results provide a rigorous foundation for a plasma-instability damping model, and they introduce a robust regularity framework for parabolic equations with a quadratic gradient term, potentially applicable to broader nonlinear diffusion problems.

Abstract

We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem $\partial_{t}u - ρ_λ(x)uΔu = ρ_λ(x)g_{0}(x)u$ on the half line $(0,\infty)$. We find weak solutions from $L^p\ (p < \infty)$ 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 u$Δ$u 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]).

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

TL;DR

This work establishes the existence of weak energy solutions for a weighted nonlinear parabolic equation on the half-line, cast in divergence form with a gradient nonlinearity and weight . The authors construct approximate, regularized problems on finite intervals, obtain uniform a priori estimates, and employ Aubin-Lions-type compactness together with gradient-truncation techniques to pass to the limit. A central contribution is the Benilan-Crandall inequality, which yields a time-scaling regularizing effect that implies interior Hölder continuity and spatial Lipschitz regularity in the parabolic interior, with sharpness illustrated by special solutions. The results provide a rigorous foundation for a plasma-instability damping model, and they introduce a robust regularity framework for parabolic equations with a quadratic gradient term, potentially applicable to broader nonlinear diffusion problems.

Abstract

We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem on the half line . We find weak solutions from 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 uu 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]).

Paper Structure

This paper contains 20 sections, 33 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results
  4. Approximate Problems
  5. Uniform Estimates
  6. Compactness
  7. Existence of Weak Solutions
  8. Gradient Tail Bounds
  9. Strong Convergence of Gradients
  10. Proof of Theorem 1
  11. Regularizing Effects
  12. A Benilan-Crandall Inequality
  13. Improved Estimates on Approximate Problems
  14. Improved Regularity
  15. Special Solutions
  16. ...and 5 more sections

Key Result

Theorem 1

When $\lambda = 0$, suppose $u_0 \in L^{\delta}((0,\infty);d\nu_{0}) \cap L^{3}((0,\infty);d\nu_{0})$ for some $0 < \delta < 1$, and when $\lambda > 0$, suppose $u_0 \in L^{1}((0,\infty);d\nu_{\lambda}) \cap L^3((0,\infty);d\nu_{\lambda})$. Suppose $g_0(x)\rho_{\lambda}(x) \in C_{0}((0,\infty))$. Th

Theorems & Definitions (72)

  • Definition 1
  • Theorem 1
  • Theorem 2: Distributional Benilan-Crandall
  • Theorem 3: Local Regularity
  • Remark : $L^{\delta}$ Requirement
  • Remark : Continuity of $\rho_{\lambda}(x)g_0(x)$
  • Definition 2: Regularized Dirichlet Problem
  • Proposition 1
  • proof
  • Lemma 1
  • ...and 62 more