边界具有非光滑速率的Lifshitz-Slyozov方程的Initial-boundary value problem
The Initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary
摘要：
我们证明了Lifshitz-Slyozov方程（半直线上的非线性迁移方程）的初边值问题解的存在性和唯一性，重点讨论了动力学速率在原点处具有无界导数的情况。我们的理论涵盖了特别是那些情况下，与率表现为幂律在原点，预计流入行为和描述成核现象的边界条件需要施加。我们在这里介绍的方法来证明存在性是基于一个公式的特征，仔细分析的行为附近的奇异边界。作为一个副产品，我们提供了一个一般理论的半线与输运场退化的边界上的线性连续性方程。我们还解决了最大性和唯一性的流入解决方案的Lifshitz-Slyozov模型，利用相关的输运方程的单调性。
英文摘要：
We prove existence and uniqueness of solutions to the initial-boundary value problem for the Lifshitz--Slyozov equation (a nonlinear transport equation on the half-line), focusing on the case of kinetic rates with unbounded derivative at the origin. Our theory covers in particular those cases with rates behaving as power laws at the origin, for which an inflow behavior is expected and a boundary condition describing nucleation phenomena needs to be imposed. The method we introduce here to prove existence is based on a formulation in terms of characteristics, with a careful analysis on the behavior near the singular boundary. As a byproduct we provide a general theory for linear continuity equations on a half-line with transport fields that degenerate at the boundary. We also address both the maximality and the uniqueness of inflow solutions to the Lifshitz--Slyozov model, exploiting monotonicity properties of the associated transport equation.
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