Classical Solutions for Diffusion-Induced Grain-Boundary Motion

Abstract

We prove existence and uniqueness of classical solutions for the motion of hypersurfaces driven by mean curvature and diffusion of a solute along the surface. This free boundary problem involves solving a coupled system of nonlinear partial differential equations.

Self-Intersections for Willmore Flow

“…A proof for the existence of a unique trajectory {z(τ, x) ∈ R 3 ; τ ∈ (−ε, ε)} with the above properties can for instance be found in [11,Lemma 2.1]. For further use we introduce the manifold M := t∈(0,T ) {t} × Γ(t).…”

On diffusion-induced grain-boundary motion

“…In this paper we consider a sharp interface model for DIGM. The same model has been studied in [14], where existence and uniqueness of classical Hölder solutions is proved. Here we improve this result considerably.…”

“…A detailed analysis shows that (1.1) is a fully nonlinear coupled system, where the fully nonlinear character comes in through the term VH Γ u. It is shown in [14] that (1.1) admits classical solutions which are smooth in time and C 2+α in space for given initial data in C 2+β , where 0 < α < β < 1.…”

“…In order to investigate system (1.1) we represent the moving hypersurface Γ(t) as a graph over a fixed reference manifold Σ and then transform (1.1) to an evolution equation over Σ. This leads to a fully nonlinear system which is parabolic (in the sense that the linearization generates an analytic semigroup on an appropriate function space), as is shown in [14]. Since the fully nonlinear term occurs on the cross diagonal we will be able to combine maximal regularity results and bootstrapping arguments to show that solutions immediately regularize for positive times.…”

On Thermodynamically Consistent Stefan Problems with Variable Surface Energy

“…with the above properties is not completely obvious, see for instance [22] for a proof. We note that the (non-degenerate) equilibria for this problem are the same as those for (1.1): the temperature is constant, and the disperse phase Ω 1 consists of finitely many nonintersecting balls of the same radius.…”