Continuity for bounded solutions of multiphase Stefan problem (original) (raw)

Regularity and Asymptotic Behavior of Nonlinear Stefan Problems

Archive for Rational Mechanics and Analysis, 2014

We study the following nonlinear Stefan problem      ut − d∆u = g(u) for x ∈ Ω(t), t > 0, u = 0 and ut = µ|∇xu| 2 for x ∈ Γ(t), t > 0, u(0, x) = u0(x) for x ∈ Ω0, where Ω(t) ⊂ R n (n ≥ 2) is bounded by the free boundary Γ(t), with Ω(0) = Ω0, µ and d are given positive constants. The initial function u0 is positive in Ω0 and vanishes on ∂Ω0. The class of nonlinear functions g(u) includes the standard monostable, bistable and combustion type nonlinearities. We show that the free boundary Γ(t) is smooth outside the closed convex hull of Ω0, and as t → ∞, either Ω(t) expands to the entire R n , or it stays bounded. Moreover, in the former case, Γ(t) converges to the unit sphere when normalized, and in the latter case, u → 0 uniformly. When g(u) = au − bu 2 , we further prove that in the case Ω(t) expands to R n , u → a/b as t → ∞, and the spreading speed of the free boundary converges to a positive constant; moreover, there exists µ * ≥ 0 such that Ω(t) expands to R n exactly when µ > µ * .

Regularity results for multiphase Stefan-like equations

Modelling and Optimization of Distributed Parameter Systems Applications to engineering, 1996

In this note the local continuity of any bounded local weak solution of degenerate multiphase Stefan problem is proved. Moreover the modulus of continuity can be determined a priori only in terms of the data.

Regularity for solutions of the two-phase Stefan problem

Communications on Pure and Applied Analysis, 2008

We consider the two-phase Stefan problem u t = ∆α(u) where α(u) = u + 1 for u < −1, α(u) = 0 for −1 ≤ u ≤ 1, and α(u) = u − 1 for u ≥ 1. We show that if u is an L 2 loc distributional solution then α(u) has L 2 loc derivatives in time and space. We also show |α(u)| is subcaloric and conclude that α(u) is continuous.

Regularity of weak solutions of one-dimensional two-phase Stefan problems

Annali di Matematica Pura ed Applicata, Series 4, 1977

Sunto.-Si consldera il problems di Ste]an unidimensiona~v a duv ]asl e si dimostra ~'esistenza di soluzio~d vlasslche sotto ipotesi minimali sui dat~ (continuit~ a tratti e limitatezza). 2~elle stesse ipotesl sl dlmostra vhe tall soluzloni dipendono iv~ modo conti'~uo da~ dati, conseguendo un visultato che $ pi~ generale anvhe di quelto noto pet le sohtzioni deboli. l.-Introduction.

Singular limits for the two-phase Stefan problem

Discrete and Continuous Dynamical Systems, 2013

We prove strong convergence to singular limits for a linearized fully inhomogeneous Stefan problem subject to surface tension and kinetic undercooling effects. Different combinations of σ → σ 0 and δ → δ 0 , where σ, σ 0 ≥ 0 and δ, δ 0 ≥ 0 denote surface tension and kinetic undercooling coefficients respectively, altogether lead to five different types of singular limits. Their strong convergence is based on uniform maximal regularity estimates.

Nonincrease of mushy region in a nonhomogeneous Stefan problem

Quarterly of Applied Mathematics

For an arbitrary bounded solution of the Stefan problem the mushy region is nonincreasing in time in a sense of the theory of sets. This result takes place for the nonhomogeneous Stefan problem under some conditions on the behavior of a heat source in the mushy region.

Sharp boundedness and continuity results for the singular porous medium equation

Israel Journal of Mathematics

We consider non-homogeneous, singular (0 < m < 1) parabolic equations of porous medium type of the form ut − div A(x, t, u, Du) = µ in E T , where E T is a space time cylinder, and µ is a Radon-measure having finite total mass µ(E T). In the range (N −2) + N < m < 1 we establish sufficient conditions for the boundedness and the continuity of u in terms of a natural Riesz potential of the right-hand side measure µ.