Global stability of steady states in the classical Stefan problem for general boundary shapes

Abstract

The classical one-phase Stefan problem (without surface tension) allows for a continuum of steadystate solutions, given by an arbitrary (but sufficiently smooth) domain together with zero temperature. We prove global-in-time stability of such steady states, assuming a sufficient degree of smoothness on the initial domain, but without any a priori restriction on the convexity properties of the initial shape. This is an extension of our previous result (Hadžić & Shkoller 2014 Commun. Pure Appl. Math. 68, 689-757 (doi:10.1002) in which we studied nearly spherical shapes.

Local Well-Posedness and Global Stability of the Two-Phase Stefan Problem

“…To this end, we develop a higher-order energy with natural weights adapted to the problem and combine it with Hopf-type inequalities. This extends the previous work by Hadžić & Shkoller [31,32] on the one-phase Stefan problem to the setting of two-phase problems, and simplifies the proof significantly.…”

“…With the bounds for the matrices κĀ being analogous as those of Lemma A.1, combined with a modified Poincaré inequality detailed in equation (4.6) of [32], we obtain the desired inequality. Now, taking T κ small enough on equation (3.57) gives us m-independent bounds for q m ttt ∈ L ∞ ([0,T κ ],L 2 (Ω ± )) ∩ L 2 ([0,T κ ],H 1 (Ω ± )), and moreover, from the weak formulation (3.55), we can obtain as well that q m tttt ∈ L 2 ([0,T ],H 1 (Ω ± ) * ) with bounds independent of m. As a consequence, q m ttt ∈ C([0,T κ ];L 2 (Ω ± )) ∩ L 2 ([0,T κ ];H 1 (Ω ± )), and therefore we can use elliptic regularity in succession on the time differentiated problems to obtain the desired m-independent estimates for ∂ l t q m ∈ C([0,T κ ];H 6−2l (Ω)) ∩ L 2 ([0,T κ ];H 7−2l (Ω)) for l = 0,1,2,3.…”

“…In the absence of surface tension, the nonlinear stability of nearly spherical steady state solutions to the one-phase Stefan problem was proved in [31] and the authors generalized that result to allow for arbitrary (bounded) initial domains in [32]. Due to the infinitedimensional space of steady states, the nonlinear stability theory cannot be viewed as a perturbation of a given linear profile; thus, a novel hybrid methodology was developed in [31,32], which combined energy methods with pointwise maximum principle techniques to establish exponential-in-time lower-bounds on the Rayleigh-Taylor stability condition. Maximum principles together with energy estimates were also used in [14] for the analysis of the related Muskat problem.…”

“…We implement a bootstrap scheme, where we first assume such bounds and use them to prove important energy-norm equivalence Lemmas in Section 4.1. Just like in [32], to show that the lower bound is dynamically preserved, we make a very sharp use of the Pucci operators and comparison principles as explained in Section 4.4. Finally, using standard continuity arguments and the improvement of the bootstrap bounds, we present the proof of Theorem 2.2 in Section 4.7.…”

“…We will omit the proof as it is detailed in Section 2.6 of [32], where a comparison function is constructed using the so-called demi-eigenvalues and demi-eigenfunctions of the maximal Pucci operators detailed in [42]. 4.5.…”