标题： 非线性扩散模型对于等离子体不稳定性的存在性和正则化效应 显示英文标题

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

摘要： 我们研究了非线性抛物型Dirichlet问题$\partial_{t}u - \rho_{\lambda}(x)u\Delta u = \rho_{\lambda}(x)g_{0}(x)u$在半线$(0,\infty)$上弱解的存在性和正则性。我们从$L^p\ (p < \infty)$初始数据中找到弱解，并通过Benilan-Crandall不等式证明解在抛物型内部是联合Holder连续的，并且局部空间上是Lipschitz连续的。我们识别出达到这些界值的特殊解。 从时间尺度论证推导出的Benilan-Crandall不等式，独立地揭示了抛物型算子 u$\Delta$u 的正则化效应，具有重要意义。 最近在[11]中考虑的问题起源于等离子体物理中波粒相互作用引起的非线性不稳定性阻尼理论（参见[8, 22]）。 显示英文摘要

摘要： We study existence and regularity of weak solutions to a nonlinear parabolic Dirichlet problem $\partial_{t}u - \rho_{\lambda}(x)u\Delta u = \rho_{\lambda}(x)g_{0}(x)u$ 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$\Delta$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]).