The initial-boundary value problem for the Lifshitz–Slyozov equation with non-smooth rates at the boundary

We prove existence and uniqueness of solutions to the initialboundary 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 rat...

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Bibliographic Details
Published in:Nonlinearity 2021-04, Vol.34 (4), p.1975-2017
Main Authors: Calvo, Juan, Hingant, Erwan, Yvinec, Romain
Format: Article
Language:English
Subjects:
Analysis of PDEs
Mathematics
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Summary:We prove existence and uniqueness of solutions to the initialboundary 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.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/abd3f3