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arxiv: 2402.17154 · v2 · submitted 2024-02-27 · 🧮 math.AP

Existence for the Supercooled Stefan Problem in General Dimensions

Sunhi Choi , Inwon C. Kim , Young-Heon Kim This is my paper

Pith reviewed 2026-05-24 03:56 UTC · model grok-4.3

classification 🧮 math.AP
keywords supercooled Stefan problemweak solutionsglobal existenceBrownian stopping timesoptimal stoppingfree boundary problemsdual optimizationstochastic order
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The pith

Weak solutions to the supercooled Stefan problem exist globally in time in any dimension for general initial data.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves global-time existence of weak solutions to the supercooled Stefan problem in general space dimensions and for a general class of initial data. The construction proceeds by solving a free-target optimization problem over Brownian stopping times that incorporates a superharmonic cost function. This choice ensures the target measure accumulates near the domain boundary, and dual attainment is used to characterize the optimal stopping rule. The resulting particle dynamics then satisfy the weak form of the Stefan problem. The constructed solution is moreover maximal in a stochastic order among all comparable weak solutions starting from the same data.

Core claim

Weak solutions to the supercooled Stefan problem exist globally in time in general space dimensions for a general class of initial data. These solutions arise from the optimal stopping times of a free-target optimization problem for Brownian motion equipped with a superharmonic cost; dual attainment characterizes the primal solution, which in turn generates the required particle dynamics. The resulting solution is maximal in the sense of a certain stochastic order among all comparable weak solutions from the same initial data.

What carries the argument

Free-target optimization problem for Brownian stopping times with superharmonic cost function, together with its dual attainment that identifies the optimal stopping rule whose induced dynamics solve the Stefan problem.

If this is right

  • Global existence holds in every space dimension.
  • The solution is maximal in stochastic order among comparable weak solutions.
  • The construction applies to a broad class of initial data without dimensional restrictions.
  • The optimal stopping dynamics directly yield a weak solution to the supercooled Stefan problem.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The probabilistic construction may supply a selection principle for choosing among multiple weak solutions when uniqueness fails.
  • Similar optimization problems could be formulated for other free-boundary problems that admit a probabilistic representation.
  • The dual-attainment step offers a template for proving existence in related optimal-stopping problems with free targets.

Load-bearing premise

The chosen superharmonic cost function forces the target measure to accumulate near the prescribed domain boundary in a way that produces a solution to the Stefan problem.

What would settle it

An explicit initial datum in two or more dimensions for which the constructed process fails to satisfy the weak Stefan condition for all time or for which a strictly larger solution exists in the stochastic order.

read the original abstract

We prove the global-time existence of weak solutions to the supercooled Stefan problem. Our result holds in general space dimensions and with a general class of initial data. In addition, our solution is maximal in the sense of a certain stochastic order, among all comparable weak solutions starting from the same initial data. Our approach is based on a free target optimization problem for Brownian stopping times, where the main idea is to introduce a superharmonic cost function in the optimization problem. We will show that our choice of the cost function causes the target measure to accumulate near the prescribed domain boundary as much as possible. A central ingredient in our proof lies in the usage of dual problem: we prove the dual attainment and use the dual optimal solution to characterize the primal optimal solution. It follows in turn that the underlying particle dynamics yields a solution to the supercooled Stefan problem.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper proves global-time existence of weak solutions to the supercooled Stefan problem in arbitrary space dimensions for a general class of initial data. The construction proceeds via a free-target optimization problem for Brownian stopping times that employs a superharmonic cost function; dual attainment is established and used to characterize the primal optimizer, which is shown to yield a weak solution that is maximal in a stochastic order among all comparable solutions from the same initial data.

Significance. If the result holds, it resolves a long-standing existence question for the supercooled Stefan problem beyond the low-dimensional or radially symmetric cases previously treated in the literature. The variational approach via optimal stopping and duality supplies both existence and a maximality property, and the argument is dimension-independent with only mild assumptions on the initial measure. These features constitute a genuine advance in the analysis of free-boundary problems.

minor comments (3)
  1. The precise definition of the weak formulation of the Stefan problem (integral identity satisfied by the measure supported on the free boundary) should be stated explicitly in the introduction rather than deferred to a later section.
  2. The abstract refers to 'a general class of initial data' without specifying the precise integrability or support conditions; this should be made concrete already in the abstract or the first paragraph of the introduction.
  3. Notation for the stochastic order used to express maximality is introduced only after the main theorem; a brief forward reference or a short paragraph in §1 would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, their assessment of its significance, and their recommendation to accept.

Circularity Check

0 steps flagged

No significant circularity; direct existence proof via optimization and duality

full rationale

The derivation establishes existence of weak solutions to the supercooled Stefan problem by constructing an optimal stopping time for Brownian motion under a superharmonic cost function, proving dual attainment, and verifying that the resulting measure satisfies the weak formulation. This chain relies on the properties of the chosen cost function and duality theory rather than any quantity defined by the target result being substituted back into the construction. No self-citation is load-bearing for the central existence claim, and the argument is presented as self-contained in arbitrary dimensions under mild initial-data assumptions. The approach does not reduce any prediction or uniqueness statement to a fitted input or prior self-result by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof relies on standard properties of Brownian motion, superharmonic functions, and weak formulations of free-boundary problems; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of Brownian motion and optimal stopping theory hold in the given setting.
    Invoked implicitly when the problem is recast as a stopping-time optimization.
  • domain assumption The weak formulation of the Stefan problem is well-defined for the class of initial data considered.
    Required for the recovered particle law to be called a weak solution.

pith-pipeline@v0.9.0 · 5675 in / 1475 out tokens · 22626 ms · 2026-05-24T03:56:30.035745+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

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  1. Free boundary regularity and well-posedness of physical solutions to the supercooled Stefan problem

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    Proves spatial C^1 regularity and higher smoothness away from countable points for the free boundary, absence of jump accumulation, and global uniqueness from short-time uniqueness for physical solutions under integra...

  2. The Nonlocal Stefan Problem via a Martingale Transport

    math.AP 2023-10 unverdicted novelty 6.0

    Constructs global-time weak solutions to the nonlocal Stefan problem via martingale transport, establishes connection to parabolic obstacle problem, and proves exponential convergence for the melting case.

Reference graph

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