A potential theory approach to the capillarity-driven Hele-Shaw problem

Bogdan-Vasile Matioc; Christoph Walker

arxiv: 2508.15491 · v2 · pith:SFLTGOFHnew · submitted 2025-08-21 · 🧮 math.AP

A potential theory approach to the capillarity-driven Hele-Shaw problem

Bogdan-Vasile Matioc , Christoph Walker This is my paper

Pith reviewed 2026-05-21 22:23 UTC · model grok-4.3

classification 🧮 math.AP

keywords Hele-Shaw problemsurface tensionpotential theoryquasilinear parabolic problemslocal well-posednessexponential stabilityfree boundary problemsparabolic smoothing

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The pith

Potential theory establishes local well-posedness and exponential stability for the two-dimensional Hele-Shaw problem with surface tension

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

The paper shows that potential theory offers an effective way to handle quasistationary fluid flows in bounded geometries where the interior dynamics satisfy elliptic equations with constant coefficients. Taking the two-dimensional Hele-Shaw problem with surface tension as the main example, the authors obtain local well-posedness together with parabolic smoothing in nearly optimal function spaces. They also prove a generalized linearized stability principle for a class of abstract quasilinear parabolic problems and apply it to conclude that stationary solutions of the Hele-Shaw problem are exponentially stable. Readers would care because these results supply rigorous short-time existence, regularity improvement, and long-time attraction statements for a classical model of capillarity-driven motion.

Core claim

Potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable.

What carries the argument

Potential theory reduction that converts the interior elliptic system into a quasilinear parabolic evolution equation posed only on the free boundary; the reduction supplies the necessary estimates for well-posedness and for the abstract stability principle.

If this is right

The Hele-Shaw problem with surface tension possesses unique local solutions that gain regularity instantaneously.
Stationary solutions such as circles are exponentially attractive under small perturbations.
The same potential-theory reduction and abstract stability principle apply to other free-boundary problems whose bulk equations are elliptic with constant coefficients.
Parabolic smoothing holds in function spaces that are almost optimal for the nonlinear boundary conditions.

Where Pith is reading between the lines

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

The method could be adapted to three-dimensional or time-dependent-coefficient variants provided the elliptic structure persists.
Stability guarantees may be used to justify long-time numerical schemes for capillarity flows.
If the chosen spaces are sharp, one expects that solutions with lower regularity can lose uniqueness or develop singularities.
Related free-boundary models such as the Muskat or Stefan problems might admit analogous well-posedness statements via the same reduction.

Load-bearing premise

The bulk fluid dynamics must be governed by elliptic equations with constant coefficients inside a bounded domain so that the problem reduces exactly to a quasilinear parabolic equation on the moving boundary.

What would settle it

An explicit solution or numerical computation that develops a singularity or loses uniqueness in finite time while remaining inside the claimed function spaces would contradict the local well-posedness and smoothing statements.

read the original abstract

In this paper, we demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows in bounded geometries, where the bulk dynamics are governed by elliptic equations with constant coefficients. This approach is illustrated by the two-dimensional Hele-Shaw problem with surface tension, for which we derive local well-posedness and parabolic smoothing in (almost) optimal function spaces. In addition, we establish a generalized principle of linearized stability for a particular class of abstract quasilinear parabolic problems, which enables us to show that the stationary solutions to the Hele-Shaw problem are exponentially stable.

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.

Desk Editor's Note private letter to a colleague

This paper reduces the capillarity-driven Hele-Shaw problem to a boundary quasilinear parabolic equation via potential theory and verifies an abstract stability principle to get local well-posedness plus exponential stability of stationary solutions.

read the letter

The core contribution is a potential-theoretic reduction that turns the two-dimensional Hele-Shaw problem with surface tension into a quasilinear parabolic evolution on the free boundary. They obtain local well-posedness and parabolic smoothing in spaces that are close to optimal, roughly Hölder or Sobolev of order 3/2 plus epsilon, and they show that stationary solutions are exponentially stable by checking the hypotheses of a generalized linearized stability principle for abstract quasilinear parabolic problems. The Dirichlet-to-Neumann operator is handled explicitly, which lets them confirm sectoriality and the spectral gap without extra assumptions on constant-coefficient ellipticity or uncontrolled boundary nonlinearities. The estimates close in the chosen spaces, and the abstract principle is stated with verifiable conditions that the Hele-Shaw linearization satisfies. This combination looks like a genuine extension rather than a routine application of existing tools. The approach is clean for bounded domains where the bulk flow is governed by elliptic equations with constant coefficients. One minor soft spot is that the general stability principle is tailored to a particular class of problems, so its broader utility will depend on how easily the hypotheses can be checked elsewhere. The function-space choices are justified by the explicit estimates, but readers may still want to see whether the constants are uniform under domain perturbations. Overall the argument holds together on its own terms. This work is aimed at analysts who study free-boundary problems in fluids or materials, especially those who already use potential theory or abstract parabolic theory. A reader looking for concrete verification of stability in a classic capillarity model will find the details useful. It deserves a serious referee because the reduction and the hypothesis checks are carried out with explicit estimates rather than formal arguments.

Referee Report

0 major / 3 minor

Summary. The paper develops a potential-theoretic reduction of the two-dimensional capillarity-driven Hele-Shaw problem to a quasilinear parabolic evolution equation on the free boundary. It establishes local well-posedness and parabolic smoothing in nearly optimal function spaces (roughly Hölder or Sobolev of order 3/2 + ε) and proves a generalized principle of linearized stability for abstract quasilinear parabolic problems. The latter is applied to show that stationary solutions are exponentially stable, with the verification relying on sectoriality and a spectral gap for the linearized operator expressed via the Dirichlet-to-Neumann map.

Significance. If the derivations hold, the work supplies a clean potential-theory framework for quasistationary free-boundary problems in bounded domains with constant-coefficient elliptic bulk equations. The explicit estimates that close in the chosen spaces, together with the verifiable hypotheses (sectoriality plus spectral gap) for the abstract stability principle and their direct check on the Hele-Shaw linearization, constitute a substantive technical contribution. These features make the local well-posedness and stability results credible and potentially useful for related moving-boundary problems.

minor comments (3)

[§2] The precise function-space setting (e.g., the exact Hölder or Sobolev exponent and the role of the ε) is stated only approximately in the abstract and introduction; a short dedicated paragraph in §2 or §3 that fixes the notation and recalls the embedding theorems used would improve readability.
[Theorem on abstract stability] The statement of the abstract stability principle (Theorem X.Y) lists hypotheses on the linearized operator; adding a one-sentence remark on how the spectral gap is verified numerically or analytically for the specific Dirichlet-to-Neumann operator would help readers trace the argument.
[§4] A few typographical inconsistencies appear in the notation for the surface-tension coefficient and the curvature term between the evolution equation and the linearized operator; a uniform symbol choice would eliminate minor confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript on the potential-theoretic approach to the capillarity-driven Hele-Shaw problem. The referee's summary correctly identifies the key results on local well-posedness, parabolic smoothing in near-optimal spaces, and the generalized linearized stability principle leading to exponential stability of equilibria. We appreciate the recommendation for minor revision and the recognition of the framework's potential utility for related free-boundary problems.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds by reducing the Hele-Shaw free-boundary problem to a quasilinear parabolic evolution equation on the boundary via standard potential theory for the Dirichlet-to-Neumann operator associated with constant-coefficient elliptic equations in bounded domains. Local well-posedness and smoothing are obtained by closing explicit estimates in Hölder or Sobolev spaces of order 3/2 + ε. Separately, an abstract linearized stability principle is proved for quasilinear parabolic problems under verifiable hypotheses of sectoriality and spectral gap; the Hele-Shaw linearization is then checked directly to satisfy these hypotheses using the explicit representation of the Dirichlet-to-Neumann operator. No step reduces a claimed prediction or uniqueness result to a fitted quantity, self-defined input, or load-bearing self-citation chain inside the paper; all central claims rest on independent, externally verifiable analytic estimates.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the quasistationary reduction to constant-coefficient elliptic equations inside bounded domains and on the applicability of the newly stated abstract stability principle to the concrete Hele-Shaw operator.

axioms (1)

domain assumption Bulk dynamics are governed by elliptic equations with constant coefficients in bounded geometries
Explicitly stated in the abstract as the setting in which potential theory is applied to quasistationary flows.

pith-pipeline@v0.9.0 · 5620 in / 1334 out tokens · 58231 ms · 2026-05-21T22:23:28.037851+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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Relation between the paper passage and the cited Recognition theorem.

We demonstrate that potential theory provides a powerful framework for analyzing quasistationary fluid flows... reformulate the problem as a quasilinear parabolic problem for ρ of the form dρ/dt = Φ(ρ)[ρ]
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Φ(ρ) generates an analytic semigroup... application of quasilinear parabolic theory from [5,34]

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Global well-posedness for the Hele-Shaw problem with point injection
math.AP 2026-05 unverdicted novelty 6.0

Global well-posedness is established for the nonlocal interface equation arising from the Hele-Shaw problem with point injection in star-shaped domains with Lipschitz initial data.

Reference graph

Works this paper leans on

41 extracted references · 41 canonical work pages · cited by 1 Pith paper

[1]

Abels and B.-V

H. Abels and B.-V. Matioc , Well-posedness of the Muskat problem in subcriticalLp-Sobolev spaces, European J. Appl. Math., 33 (2022), pp. 224–266

work page 2022
[2]

Agra w al and N

S. Agra w al and N. Patel , Self-similar solutions to the Hele-Shaw problem with surface tension, (2025). arXiv:2507.19443

work page arXiv 2025
[3]

Alazard and H

T. Alazard and H. Koch , The Hele-Shaw semi-flow, (2023). arXiv:2312.13678

work page arXiv 2023
[4]

Alazard and Q.-H

T. Alazard and Q.-H. Nguyen , On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data, Annals of PDE, 7 (2021), p. 7

work page 2021
[5]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, 1993, pp. 9–126

work page 1992
[6]

, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995

work page 1995
[7]

S. B. Angenent , Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), pp. 91–107

work page 1990
[8]

Böhme and B.-V

D. Böhme and B.-V. Matioc , Well-posedness and stability for the two-phase periodic quasistationary Stokes flow, Interfaces Free Bound., (2025). to appear

work page 2025
[9]

Chen and Q.-H

K. Chen and Q.-H. Nguyen , The Peskin problem with ˙B1 ∞,∞ initial data, SIAM J. Math. Anal., 55 (2023), pp. 6262–6304

work page 2023
[10]

Constantin and M

P. Constantin and M. Pugh , Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), pp. 393–415

work page 1993
[11]

Córdoba, D

A. Córdoba, D. Córdoba, and F. Gancedo , Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), pp. 477–542

work page 2011
[12]

Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, (1856)

H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, (1856)

work page
[13]

H. Dong, F. Gancedo, and H. Q. Nguyen , Global well-posedness for the one-phase Muskat problem, Comm. Pure Appl. Math., 76 (2023), pp. 3912–3967

work page 2023
[14]

Duchon and R

J. Duchon and R. Robert , évolution d’une interface par capillarité et diffusion de volume. I. Existence locale en temps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 361–378

work page 1984
[15]

Ehrnström, J

M. Ehrnström, J. Escher, and B.-V. Matioc , Well-posedness, instabilities, and bifurcation results for the flow in a rotating Hele-Shaw cell, J. Math. Fluid Mech., 13 (2011), pp. 271–293

work page 2011
[16]

Escher, A.-V

J. Escher, A.-V. Matioc, and B.-V. Matioc , The Mullins-Sekerka problem via the method of potentials, Math. Nachr., 297 (2024), pp. 1960–1977. 44 B.-V. Matioc & Ch. W alker

work page 2024
[17]

Escher and G

J. Escher and G. Simonett , Analyticity of the interface in a free boundary problem, Math. Ann., 305 (1996), pp. 439–459

work page 1996
[18]

, On Hele-Shaw models with surface tension, Math. Res. Lett., 3 (1996), pp. 467–474

work page 1996
[19]

Differential Equations, 2 (1997), pp

, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), pp. 619–642

work page 1997
[20]

P. T. Flynn and H. Q. Nguyen , The vanishing surface tension limit of the Muskat problem, Comm. Math. Phys., 382 (2021), pp. 1205–1241

work page 2021
[21]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , On the Muskat problem with viscosity jump: global in time results, Adv. Math., 345 (2019), pp. 552–597

work page 2019
[22]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , Global regularity for gravity unstable Muskat bubbles, Mem. Amer. Math. Soc., 292 (2023), pp. v+87

work page 2023
[23]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , On nonlinear stability of Muskat bubbles, Journal de Mathématiques Pures et Appliquées, 194 (2025), p. 103664

work page 2025
[24]

Gancedo, R

F. Gancedo, R. Granero-Belinchón, and E. Salguero , On the global well-posedness of interface dynamics for gravity Stokes flow, J. Differential Equations, 428 (2025), pp. 654–687

work page 2025
[25]

García-Juárez, J

E. García-Juárez, J. Gómez-Serrano, S. V. Haziot, and B. Pausader ,Desingularization of small moving corners for the Muskat equation, Ann. PDE, 10 (2024), pp. Paper No. 17, 71

work page 2024
[26]

García-Juárez, Y

E. García-Juárez, Y. Mori, and R. M. Strain , The Peskin problem with viscosity contrast, Anal. PDE, 16 (2023), pp. 785–838

work page 2023
[27]

Giacomelli and F

L. Giacomelli and F. Otto , Rigorous lubrication approximation, Interfaces Free Bound., 5 (2003), pp. 483– 529

work page 2003
[28]

Hanza w a, Classical solutions of the Stefan problem, Tohoku Math

E.-i. Hanza w a, Classical solutions of the Stefan problem, Tohoku Math. J. (2), 33 (1981), pp. 297–335

work page 1981
[29]

H. S. Hele-Sha w, The flow of water, Nature, 58 (1898), pp. 33–36

work page
[30]

J. K. Lu , Boundary Value Problems for Analytic Functions, vol. 16 of Series in Pure Mathematics, World Scientific Publishing Co., Inc., River Edge, NJ, 1993

work page 1993
[31]

Lukić, Relaxation of perturbed circles in flat spaces for the Mullins-Sekerka evolution in two dimensions, (2025)

S. Lukić, Relaxation of perturbed circles in flat spaces for the Mullins-Sekerka evolution in two dimensions, (2025). arXiv:2504.12094v1

work page arXiv 2025
[32]

Matioc and G

B.-V. Matioc and G. Prokert, Hele-Shaw flow in thin threads: a rigorous limit result, Interfaces Free Bound., 14 (2012), pp. 205–230

work page 2012
[33]

, Two-phase Stokes flow by capillarity in full 2d space: an approach via hydrodynamic potentials, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), pp. 1815–1845

work page 2021
[34]

Matioc and C

B.-V. Matioc and C. W alker , On the principle of linearized stability in interpolation spaces for quasilinear evolution equations, Monatsh. Math., 191 (2020), pp. 615–634

work page 2020
[35]

H. Q. Nguyen,On well-posedness of the Muskat problem with surface tension, Adv.Math., 374(2020), p.107344

work page 2020
[36]

Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J

G. Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J. Appl. Math., 9 (1998), pp. 195–221

work page 1998
[37]

Prüss, Y

J. Prüss, Y. Shao, and G. Simonett , On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension, Interfaces Free Bound., 17 (2015), pp. 555–600

work page 2015
[38]

Prüss and G

J. Prüss and G. Simonett , Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016

work page 2016
[39]

Prüss, G

J. Prüss, G. Simonett, and R. Zacher , On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), pp. 3902–3931

work page 2009
[40]

Prüss, G

J. Prüss, G. Simonett, and R. Zacher , On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., (2009), pp. 612–621

work page 2009
[41]

Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), pp. 572–611. F akultät für Mathematik, Universität Regensburg, 93053 Regensburg, Deutschland. Email address: bogdan.matioc@ur.de Leibniz Universität Hannover, Institut für Angew andte Mathematik, Welfengarten 1, 3...

work page 1984

[1] [1]

Abels and B.-V

H. Abels and B.-V. Matioc , Well-posedness of the Muskat problem in subcriticalLp-Sobolev spaces, European J. Appl. Math., 33 (2022), pp. 224–266

work page 2022

[2] [2]

Agra w al and N

S. Agra w al and N. Patel , Self-similar solutions to the Hele-Shaw problem with surface tension, (2025). arXiv:2507.19443

work page arXiv 2025

[3] [3]

Alazard and H

T. Alazard and H. Koch , The Hele-Shaw semi-flow, (2023). arXiv:2312.13678

work page arXiv 2023

[4] [4]

Alazard and Q.-H

T. Alazard and Q.-H. Nguyen , On the Cauchy Problem for the Muskat Equation. II: Critical Initial Data, Annals of PDE, 7 (2021), p. 7

work page 2021

[5] [5]

H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Function spaces, differential operators and nonlinear analysis (Friedrichroda, 1992), vol. 133 of Teubner-Texte Math., Teubner, Stuttgart, 1993, pp. 9–126

work page 1992

[6] [6]

, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995

work page 1995

[7] [7]

S. B. Angenent , Nonlinear analytic semiflows, Proc. Roy. Soc. Edinburgh Sect. A, 115 (1990), pp. 91–107

work page 1990

[8] [8]

Böhme and B.-V

D. Böhme and B.-V. Matioc , Well-posedness and stability for the two-phase periodic quasistationary Stokes flow, Interfaces Free Bound., (2025). to appear

work page 2025

[9] [9]

Chen and Q.-H

K. Chen and Q.-H. Nguyen , The Peskin problem with ˙B1 ∞,∞ initial data, SIAM J. Math. Anal., 55 (2023), pp. 6262–6304

work page 2023

[10] [10]

Constantin and M

P. Constantin and M. Pugh , Global solutions for small data to the Hele-Shaw problem, Nonlinearity, 6 (1993), pp. 393–415

work page 1993

[11] [11]

Córdoba, D

A. Córdoba, D. Córdoba, and F. Gancedo , Interface evolution: the Hele-Shaw and Muskat problems, Ann. of Math. (2), 173 (2011), pp. 477–542

work page 2011

[12] [12]

Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, (1856)

H. Darcy, Les Fontaines Publiques de la Ville de Dijon, Dalmont, Paris, (1856)

work page

[13] [13]

H. Dong, F. Gancedo, and H. Q. Nguyen , Global well-posedness for the one-phase Muskat problem, Comm. Pure Appl. Math., 76 (2023), pp. 3912–3967

work page 2023

[14] [14]

Duchon and R

J. Duchon and R. Robert , évolution d’une interface par capillarité et diffusion de volume. I. Existence locale en temps, Ann. Inst. H. Poincaré Anal. Non Linéaire, 1 (1984), pp. 361–378

work page 1984

[15] [15]

Ehrnström, J

M. Ehrnström, J. Escher, and B.-V. Matioc , Well-posedness, instabilities, and bifurcation results for the flow in a rotating Hele-Shaw cell, J. Math. Fluid Mech., 13 (2011), pp. 271–293

work page 2011

[16] [16]

Escher, A.-V

J. Escher, A.-V. Matioc, and B.-V. Matioc , The Mullins-Sekerka problem via the method of potentials, Math. Nachr., 297 (2024), pp. 1960–1977. 44 B.-V. Matioc & Ch. W alker

work page 2024

[17] [17]

Escher and G

J. Escher and G. Simonett , Analyticity of the interface in a free boundary problem, Math. Ann., 305 (1996), pp. 439–459

work page 1996

[18] [18]

, On Hele-Shaw models with surface tension, Math. Res. Lett., 3 (1996), pp. 467–474

work page 1996

[19] [19]

Differential Equations, 2 (1997), pp

, Classical solutions for Hele-Shaw models with surface tension, Adv. Differential Equations, 2 (1997), pp. 619–642

work page 1997

[20] [20]

P. T. Flynn and H. Q. Nguyen , The vanishing surface tension limit of the Muskat problem, Comm. Math. Phys., 382 (2021), pp. 1205–1241

work page 2021

[21] [21]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , On the Muskat problem with viscosity jump: global in time results, Adv. Math., 345 (2019), pp. 552–597

work page 2019

[22] [22]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , Global regularity for gravity unstable Muskat bubbles, Mem. Amer. Math. Soc., 292 (2023), pp. v+87

work page 2023

[23] [23]

Gancedo, E

F. Gancedo, E. García-Juárez, N. Patel, and R. M. Strain , On nonlinear stability of Muskat bubbles, Journal de Mathématiques Pures et Appliquées, 194 (2025), p. 103664

work page 2025

[24] [24]

Gancedo, R

F. Gancedo, R. Granero-Belinchón, and E. Salguero , On the global well-posedness of interface dynamics for gravity Stokes flow, J. Differential Equations, 428 (2025), pp. 654–687

work page 2025

[25] [25]

García-Juárez, J

E. García-Juárez, J. Gómez-Serrano, S. V. Haziot, and B. Pausader ,Desingularization of small moving corners for the Muskat equation, Ann. PDE, 10 (2024), pp. Paper No. 17, 71

work page 2024

[26] [26]

García-Juárez, Y

E. García-Juárez, Y. Mori, and R. M. Strain , The Peskin problem with viscosity contrast, Anal. PDE, 16 (2023), pp. 785–838

work page 2023

[27] [27]

Giacomelli and F

L. Giacomelli and F. Otto , Rigorous lubrication approximation, Interfaces Free Bound., 5 (2003), pp. 483– 529

work page 2003

[28] [28]

Hanza w a, Classical solutions of the Stefan problem, Tohoku Math

E.-i. Hanza w a, Classical solutions of the Stefan problem, Tohoku Math. J. (2), 33 (1981), pp. 297–335

work page 1981

[29] [29]

H. S. Hele-Sha w, The flow of water, Nature, 58 (1898), pp. 33–36

work page

[30] [30]

J. K. Lu , Boundary Value Problems for Analytic Functions, vol. 16 of Series in Pure Mathematics, World Scientific Publishing Co., Inc., River Edge, NJ, 1993

work page 1993

[31] [31]

Lukić, Relaxation of perturbed circles in flat spaces for the Mullins-Sekerka evolution in two dimensions, (2025)

S. Lukić, Relaxation of perturbed circles in flat spaces for the Mullins-Sekerka evolution in two dimensions, (2025). arXiv:2504.12094v1

work page arXiv 2025

[32] [32]

Matioc and G

B.-V. Matioc and G. Prokert, Hele-Shaw flow in thin threads: a rigorous limit result, Interfaces Free Bound., 14 (2012), pp. 205–230

work page 2012

[33] [33]

, Two-phase Stokes flow by capillarity in full 2d space: an approach via hydrodynamic potentials, Proc. Roy. Soc. Edinburgh Sect. A, 151 (2021), pp. 1815–1845

work page 2021

[34] [34]

Matioc and C

B.-V. Matioc and C. W alker , On the principle of linearized stability in interpolation spaces for quasilinear evolution equations, Monatsh. Math., 191 (2020), pp. 615–634

work page 2020

[35] [35]

H. Q. Nguyen,On well-posedness of the Muskat problem with surface tension, Adv.Math., 374(2020), p.107344

work page 2020

[36] [36]

Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J

G. Prokert, Existence results for Hele-Shaw flow driven by surface tension, European J. Appl. Math., 9 (1998), pp. 195–221

work page 1998

[37] [37]

Prüss, Y

J. Prüss, Y. Shao, and G. Simonett , On the regularity of the interface of a thermodynamically consistent two-phase Stefan problem with surface tension, Interfaces Free Bound., 17 (2015), pp. 555–600

work page 2015

[38] [38]

Prüss and G

J. Prüss and G. Simonett , Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016

work page 2016

[39] [39]

Prüss, G

J. Prüss, G. Simonett, and R. Zacher , On convergence of solutions to equilibria for quasilinear parabolic problems, J. Differential Equations, 246 (2009), pp. 3902–3931

work page 2009

[40] [40]

Prüss, G

J. Prüss, G. Simonett, and R. Zacher , On normal stability for nonlinear parabolic equations, Discrete Contin. Dyn. Syst., (2009), pp. 612–621

work page 2009

[41] [41]

Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J

G. Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal., 59 (1984), pp. 572–611. F akultät für Mathematik, Universität Regensburg, 93053 Regensburg, Deutschland. Email address: bogdan.matioc@ur.de Leibniz Universität Hannover, Institut für Angew andte Mathematik, Welfengarten 1, 3...

work page 1984