The initial-boundary value problem for the Lifshitz–Slyozov equation with non-smooth rates at the boundary
MetadataShow full item record
Institute of Physics Publishing
Nonlinear transport equationSingular initial-boundary value problemDynamic boundary conditionCharacteristics formulationOswald ripeningNucleation theoryPolymerization
Published version: Juan Calvo et al 2021 Nonlinearity 34 1975. [10.1088/1361-6544/abd3f3]
SponsorshipSpanish Government European Commission MTM2017-91054-EXP RTI2018-098850-B-IOO; Plan Propio de Investigacion, Universidad de Granada, Programa 9 - partially through FEDER (ERDF) funds; Junta de Andalucia European Commission P18-RT-2422 A-FQM-311-UGR18; Comision Nacional de Investigacion Cientifica y Tecnologica (CONICYT) CONICYT FONDECYT 11170655
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.