[Paper Review] Viscosity Solutions for the two-phase Stefan Problem

Inwon C. Kim, Norbert Požár|arXiv (Cornell University)|Oct 20, 2010

Nonlinear Partial Differential Equations15 references4 citations

TL;DR

This paper introduces a viscosity solution framework for the two-phase Stefan problem that accounts for initial mushy regions—regions of zero temperature with positive measure—commonly arising from initial data. It establishes a comparison principle for viscosity solutions and proves their equivalence to weak solutions when no initial mushy region exists, ensuring uniqueness and consistency with established theories for the heat equation and free boundary problems.

ABSTRACT

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.

Motivation & Objective

• To develop a robust viscosity solution theory for the two-phase Stefan problem that accommodates initial mushy regions generated by the initial data.
• To establish a comparison principle between viscosity sub- and supersolutions, ensuring existence of maximal and minimal viscosity solutions.
• To investigate the relationship between viscosity solutions and weak solutions defined via integration by parts, particularly in the absence of initial mushy regions.
• To demonstrate that viscosity solutions coincide with weak solutions when the initial data do not contain a mushy region, thereby ensuring uniqueness.
• To extend the applicability of viscosity solution methods to the two-phase setting, where free boundary evolution is non-monotone and more complex than in the one-phase case.

Proposed method

• Define viscosity solutions for the two-phase Stefan problem using the enthalpy formulation, where the temperature $ u = \chi(h) $ is derived from the enthalpy $ h $, with $ \chi $ piecewise linear.
• Incorporate the possibility of a mushy region $ \Gamma(u) = \{ -1 \leq h \leq 0 \} $, which may persist or shrink over time, particularly when $ |\{-1 \leq h_0 \leq 0\}| > 0 $.
• Establish a comparison principle between viscosity sub- and supersolutions by constructing appropriate test functions that account for the non-monotonic evolution of the free boundary.
• Use the viscosity solution framework to show that the maximal and minimal viscosity solutions correspond to weak solutions with maximal and minimal initial enthalpy for a given temperature profile.
• Prove that in the absence of an initial mushy region, the viscosity solution coincides with the unique weak solution, leveraging continuity and regularity results from Caffarelli and Evans.
• Adapt techniques from Crandall-Lions viscosity theory to the two-phase setting, extending them beyond the one-phase case and handling the non-monotonicity of the free boundary.

Experimental results

Research questions

• RQ1 Can a viscosity solution framework be consistently defined for the two-phase Stefan problem that includes the possibility of an initial mushy region?
• RQ2 Does a comparison principle hold for viscosity solutions in the two-phase Stefan problem, even when the free boundary evolves non-monotonically?
• RQ3 Under what conditions do viscosity solutions coincide with weak solutions defined via integration by parts?
• RQ4 Is the viscosity solution unique when there is no initial mushy region, and how does it relate to existing viscosity solution theories for the heat equation?
• RQ5 How does the presence of an initial mushy region affect the regularity and propagation speed of the free boundary in the viscosity solution framework?

Key findings

• A viscosity solution exists for the two-phase Stefan problem in the domain $ Q $, and a comparison principle holds between viscosity sub- and supersolutions, ensuring the existence of maximal and minimal solutions.
• The maximal and minimal viscosity solutions correspond to weak solutions of the two-phase Stefan problem with maximal and minimal initial enthalpy, respectively, for a fixed initial temperature $ u_0 $.
• Every weak solution in the sense of Caffarelli and Evans is also a viscosity solution, establishing a direct link between the two solution concepts.
• When the initial data contain no mushy region (i.e., $ |\{-1 \leq h_0 \leq 0\}| = 0 $), the viscosity solution is unique and coincides with the unique weak solution.
• In the absence of an initial mushy region, the viscosity solution agrees with the notion of viscosity solutions defined in Crandall-Lions and related works, validating consistency with existing theory.
• The method is robust and extends to a wider class of parabolic operators beyond the heat equation, as demonstrated by the generality of the comparison principle and solution construction.