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

Juan Calvo 1, * Erwan Hingant 2 Romain Yvinec 3, 4
* Corresponding author
3 MUSCA - Dynamiques de populations multi-échelles pour des systèmes physiologiques
Inria Saclay - Ile de France, PRC - Physiologie de la reproduction et des comportements [Nouzilly], MaIAGE - Mathématiques et Informatique Appliquées du Génome à l'Environnement [Jouy-En-Josas]
Abstract : 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.
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https://hal.inrae.fr/hal-03109431
Contributor : Romain Yvinec <>
Submitted on : Wednesday, January 13, 2021 - 5:29:57 PM
Last modification on : Thursday, May 13, 2021 - 3:47:50 AM

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  • HAL Id : hal-03109431, version 1
  • ARXIV : 2004.01947

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Juan Calvo, Erwan Hingant, Romain Yvinec. The initial-boundary value problem for the Lifshitz-Slyozov equation with non-smooth rates at the boundary. Nonlinearity, IOP Publishing, In press. ⟨hal-03109431⟩

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