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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.13922 |
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| _version_ | 1866910880385466368 |
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| author | Porteous, William Gamba, Irene M. Huang, Kun |
| author_facet | Porteous, William Gamba, Irene M. Huang, Kun |
| contents | 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]). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_13922 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence and Regularizing Effects of a Nonlinear Diffusion Model for Plasma Instabilities Porteous, William Gamba, Irene M. Huang, Kun Analysis of PDEs Mathematical Physics Plasma Physics 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]). |
| title | Existence and Regularizing Effects of a Nonlinear Diffusion Model for Plasma Instabilities |
| topic | Analysis of PDEs Mathematical Physics Plasma Physics |
| url | https://arxiv.org/abs/2503.13922 |