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

Abstract

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.

Some Remarks About the Well-Posedness of Lifshitz–Slyozov’s Equations with Nucleation Kinetics

“…A suitable adaptation of the results and proofs in [2,4] provides us the following local existence result. Proposition 2.4.…”

“…We need to consider several notions of (weak) solutions. Those are readapted from [2]. Definition 2.2 (Inflow and outflow solutions).…”

“…For more general sets of kinetic rates, we cannot presently rule out concentration behavior at zero size when ρ < Φ inf -think for instance in a(x) = x α with α ∈ (0, 1). See [3] for a more detailed investigation of this pathological phenomenon. At any rate, we will impose the condition Φ 0 < ρ in the sequel, in order to allow for inflow solutions with nucleation kinetics.…”

“…Some instances of this boundary condition have been deduced in [6,8]. The well-posedness for (1.1)-(1.4) has been treated in [2], where it is shown that only local-in-time solutions can be expected in the general case. This again is due to the fact that the transport field may change its character at x = 0 in a dynamic fashion, in such a way that a boundary condition may be needed at earlier times but not at later times.…”

“…The aim of this contribution is to give some partial results about its global wellposedness. Some of these results were announced without proof in [2].…”