Research Articles

  I am interested in the mathematical analysis of PDEs which arise in physical applications.  I have been particularly interested in problems involving interfaces and phase transitions. 

 My research is partially supported by NSF grant DMS-2452649. 

    Preprints

• The space-time geometry of Hele-Shaw flow, 

  with Carson Collins and Sebastian Munoz (2026)

   •  Solutions of the Bernoulli one-phase problem with a defect,

  with Will Feldman (2026).

    •  The pinning effect of dilute defects,

   with Will Feldman (2026) 

   •.  The Supercooled Stefan problem: fractal freezing and fine structure of maximal solutions, 

   with Raymond Chu and Sebastian Munoz (2025)

    •   Regularity of the free boundary in Supercooled Stefan problem in arbitrary dimensions,

  with Max Engelstein and Sebastian Munoz (2025)

    Publications 

•  On the geometry of rate independent droplet motion,

 with Will Feldman and Norbert Pozar. To appear in Cal.Var. PDE.

•  Existence for the supercooled Stefan problem in general dimensions, 

 with Sunhi Choi and Young-Heon Kim.  To appear in Duke Math J.

 •  On the Singular limit of Brinkman’s law to Darcy’s law,

  with Noemi David and Matt Jacobs.  Arch. Ration. Mech. Anal. vol. 250 no.3. (2026)

•   An obstacle approach to rate independent droplet evolution,

  with Will Feldman and Norbert Pozar.   Forum of Math, Sigma. vol. 14 (2026)

 •  The nonlocal Stefan problem via a Martingale transport

  with Raymond Chu, Young-heon Kim and Kyung-sik Nam.  Prob.Theory Related. Fields. vol.2 (2026)

 •    Mean field limit for Congestion dynamics in one dimension,  

  with Antoine Mellet and Jeremy SH Wu.  Comm. PDE vol. 49  (2024)

•  Regularity and nondegeneracy for tumor growth with nutrients

  with Carson Collins and Matt Jacobs, Arch. Ration. Mech. Anal vol. 249 no.1  (2025)

•  Regularity of Hele-Shaw Flow with source and drift

     with Yuming Zhang. Annals of PDE. vol. 10 no.2  (2024) 

 •  The sharp interface limit of an Ising game

   with Will Feldman and Aaron Palmer. ESAIM: COCV vol. 30 (2024)

   •  (survey) Aggregation-diffusion phenomena: from microscopic models to free boundary problems,

  With  Antoine Mellet and Jeremy S.H. Wu,  contributed chapter to Active Particles, vol. 4. (2024)

 •     The Stefan problem and Free Targets of Optimal brownian martingale Transport, 

   with Young-Heon Kim,  Annals of Applied Probability vol. 34 (2024).

 •   Tumor growth with nutrients: stability of the tumor patches

     with Jona Lelmi, SIMA, vol.55 (2023)

 •   Incompressible limit of PME with bistable and monostable reaction terms

     with Antoine Mellet, SIMA, vol. 55 (2023) 

 •    Well-posedness and Regularity for a Polyconvex Energy,  

       with Wilfrid Gangbo and Matt Jacobs.    ESAIM: COCV,  vol. 29 (2023)

• Tumor growth with Nutrients: Regularity and Stability

      with Matt Jacobs and Jiajun Tong,   Communications AMS, vol. 3 (2023)

  •   Density constrained Chemotaxis and Hele-Shaw flow

       with Antoine Mellet and Yijing Wu,   Transactions AMS, vol. 377 (2024)

•   A density-constrained model for Chemotaxis

       with Antoine Mellet and Yijing Wu,  Nonlinearity vol. 36, no.2 (2023).

 •    A Hele-Shaw limit without Monotonicity,  

    with Nestor Guillen and Antoine Mellet,  Arch. Ration. Mech. Anal. 243, no.2 (2022), 829-868.

  • L^1 contraction in Optimal Transport,   

    with Matt Jacobs and Jiajun Tong,   Ann. Sc. Norm. Super. Pisa Cl. Sci. (2022), p. 1817-1919.

  •     Porous Medium Equation with a Drift: Free boundary regularity,  

    with Yuming Zhang,  Arch. Ration. Mech. Anal. (2021), p 1-52.

  •   On Volume-preserving Crystaline Mean Curvature Flow, 

   with Dohyun Kwon and Norbert Pozar,  Mathematische Annalen.  384 no.1 (2022), 1-42.

   • Darcy’s Law with a Source Term,   

    with Matt Jacobs and Jiajun Tong, to appear,  Arch. Ration. Mech. Anal. 239 no.3 (2021), 1349 - 1393.

    • Interface Dynamics in a Two-phase Tumor Growth Model,   

     with Jiajun Tong.  Interfaces and Free boundaries. vol. 23, no. 2 (2021), 191-304.

   •  Weak solutions to the Muskat problem with surface tension via optimal transport, 

    with Matthew Jacobs and Alpár Mészáros.  Arch. Ration. Mech. Anal. 239 no.1 (2021), 389 -430.

    •    Volume preserving mean curvature flow for star-shaped sets,    

     with Dohyun Kwon.  Cal. Var. and PDE. 59. no.2 (2020),. 1-40.

      •    Geometry and Stability of normalized mean curvature flow,

           with Dohyun Kwon.  Comm. PDE. 45, no.5 (2020), 414-455.

    • Homogenization of oblique boundary value problems,  

         with Sunhi Choi, to appear in Adv. Nonlinear studies, for David Jerison’s  70th Birthday. 

     •   Head and tail speeds of mean curvature flow with forcing,  

          with Hongwei Gao,   Arch. Ration. Mech. Anal. 235, no.1 (2020), 287-354.

    • Singular limit of the Porous Medium Equation with a Drift,

        with Norbert Pozar and Brent Woodhouse, Advances in Mathematics 349 (2019), 682-732.

• Regularity Properties of Degenerate Diffusion Equations with Drifts, 

        with Yuming Zhang, SIAM J. Math. Anal. 50 (2018), no.4, 4371 - 4406.

   •    On Nonlinear Cross-Diffusion systems: an Optimal Transport approach, 

    with Alp\’ar R. M\’esz\’aros, Cal. Var. and PDE. 57 (2018), no.3.

• Uniform convergence for the incompressible limit of a tumor growth model,

       with Olga Turanova,   Ann. Inst. H. Poincaré Anal. Non Linéaire. 35 (2018), no.5, 1321 - 1354.

   •   Liquid Drops on a Rough surface,

       with Will Feldman,  Comm. Pure. Appl. Math. 71 (2018), no.12, 2429 - 2499.

    •   Congested Aggregation via Newtonian interaction, 

        with Katy Craig and Yao Yao,  Arch. Ration. Mech. Anal. 227 (2018), no. 1, 1 - 67.

   • Porous Medium Equation to Hele-Shaw flow with general initial density,

         with N. Pozar,  Transactions AMS 370 (2018), no.2,  873--909.

 •    Continuity and Discontinuity of the boundary layer tail,  

        with W. Feldman,  Ann. Sci. Éc. Norm. Supér. 50 (2017), no. 4, 1017--1064.

 •   A drift approximation for parabolic PDEs with Oblique boundary data ,

       with D. Alexander, Transactions of AMS (2016),  no.8, 5753-5781.

  •    Free boundary problems for Tumor Growth: a Viscosity solutions approach, 

        with B. Perthame and P. Souganidis,  Nonlinear Analysis 138 (2016) (a Special Issue on J.L Vazquez's 70th Birthday), 207 - 228.

  •    Quantitative Homogenization of Elliptic PDEs with Random Oscillatory boundary data,

        with W. Feldman and P. Souganidis, J. Math. Pures Appl. (9) 103 (2015), no. 4, 958–1002.  

  • Quasi-static droplet percolation, 

        with N. Guillen,  Arch. Ration. Mech. Anal. Vol. 215 (2015), no. 1,  211-281.

• Quasi-static evolution and congested crowd motion,
• 
  with D. Alexander and Y. Yao,  Nonlinearity vol. 27, (2014) no.4. 823- 858. 

• Dynamic stability of Capillary drops,
• 
  with W. Feldman, Arch. Ration. Mech. Anal. Vol. 211 (2014), no.3, 819-878. 

• Homogenization for nonlinear PDEs in general domains with oscillatory Neumann data,
• 
  with S. Choi , J. Math. Pures Appl. (2014)  no. 2, 419-448. This article has an accompanying survey article below:

• (survey article)  Homogenization with oscillatory Neumann boundary data in general domain,

       with S. Choi,   Geometric partial differential equations, 105–118, CRM Series, 15, Ed. Norm., Pisa, 2013.

• An aggregation equation with degenerate diffusion: qualitative property of solutions,
• 
  with L. Chayes and Y. Yao,  SIAM. J. Math. Anal. (2013), no.5, 2995 - 3018.

• Liquid Drops sliding down an inclined plane, 
• 
  with A. Mellet,  Trans. Amer. Math. Soc. 366 (2014), no. 11, 6119 - 6150.

• Nonlinear Elliptic-Parabolic Problems,
• 
  with N. Pozar, Arch. Ration. Mech. Anal. vol. 210 (2013), no.3, 975 - 1020.

• Homogenization of Neumann boundary data with fully nonlinear PDEs,
•         with S. Choi and K. Lee, Analysis and PDE. vol. 6 (2013), no. 4, 951 - 972. 

• (survey article) Homogenization and error estimates of free boundary velocities in periodic media,
  ``Analysis and Approximation of Microstructure Models", Applicable analysis vol. 91 (2012), 1177 - 1187 . 

• Global Existence and Finite Time Blow-Up for Critical Patlak-Keller-Segel Models with Inhomogeneous Diffusion,
  with J. Bedrossian, SIAM. J. Math. Anal. vol. 45 (2013), no.3, 934 - 964. 

• The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle,
  with Y. Yao, SIAM J. Math. Anal. vol. 44 (2012) no. 2, 568-602. 

• Two phase Stefan-type problem: Regularization near initial data by phase dynamics,
  with S. Choi, , Analysis/PDE vol. 5 (2012) no.5, 1063-1103. 

• Viscosity solutions for the two-phase Stefan problem,
  with N. Pozar, Comm. PDE. vol 36 (2011), no.1, 42-66.

• A gradient flow approach to a free boundary droplet model,
  with N. Grunewald, Calc. Var. and PDE vol. 41(2011), no. 1-2, 1-19. 

• Regularity of one-phase Stefan problem near Lipschitz initial data,
  with S. Choi, Amer. J. Math. vol. 132 (2010) no. 6. 

• Degenerate diffusion with a drift potential: a viscosity solutions approach,
  with H. K. Lei, DCDS-A vol 27. (2010). Erratum DCDS-A vol. 30 (2011).
•  
• Homogenization of one-phase Stefan-type problems in periodic and random media,
  with A. Mellet, Transactions of AMS. vol. 362 (2010) no. 8. 

• Global existence and uniqueness of solutions to a model of price formation,
  with L. Chayes, M. Guandani and M. Gonzalez, SIAM J. Math. Anal. vol. 41(2009) no.5. 

• The Supercooled Stefan Problem in One Dimension,
  with L. Chayes, CPAA, no. 2 (2012) 845-859. 

• Two-sided Contracting Stefan problem,
  with L. Chayes, Comm. PDE. vol. 33(2008) no.12, 2225-2256. 

• Homogenization of a Hele-Shaw type problem in periodic and random media,
  with A. Mellet, Arch. Ration. Mech. Anal. vol. 194 (2009) no. 2. 

• Error estimate on homogenization of free boundary velocities,
  Ann. Inst. Henri Poincare. vol. 26 (2009) no. 3. 

• A local regularization theorem on one-phase Hele-Shaw flow,
  with S. Choi and D. Jerison, Indiana Univ. Math. Journal. vol. 58 (2009) no. 6. 

• Locating the first nodal set in higher dimensions,
  with S. Choi and D. Jerison, to appear in Transactions of AMS. vol. 361 (2009) no. 10. 

• Homogenization of a model problem on contact angle dynamics,
  Comm. PDE. vol.33 (2008), no. 7, 1235-1271. 

• Global-time solutions for a model of contact line motion,
  with K. Glasner, Interfaces and Free boundaries. vol. 11 (2009), no.1, 37--60. 

• Homogenization of free boundary velocities,
  Arch. Ration. Mech. Anal. vol. 185 (2007), 69--103. 

• Regularity properties of initially Lipschitz free boundarys in One-Phase Hele-Shaw problem
  with S. Choi and D. Jerison, Amer. J. Math. 129 (2007), no. 2, 527--582. 

• Waiting time phenomena for Hele-Shaw and Stefan problems
  with S. Choi, Indiana Univ. Math Journal, vol. 55 (2006) 525-552. 

• The one-phase Hele-Shaw problem with singularities
  with D. Jerison, J. Geom. Anal., vol. 15 (2005), 641-667.
•  
• (survey article) Regularity and speed of the Hele-Shaw flow
  Viscosity Solution Theory of Differential Equations and its Developments , RIMS Kokyuroku (Proceedings) vol.1428 (2005), 19-27. 

• A free boundary problem with curvature
  Comm. PDE. vol. 30 (2005), 121-138. 

• Long time regularity of solutions of the Hele-Shaw problem
  Nonlinear Anal. TMA, vol.64, (2006), 2817-2831. 

• Regularity of the one-phase Hele-Shaw problem,
  J. Diff. Equations, vol. 223 (2006) 161-184. 

• Nonlocal front propagation problems in bounded domains with Neumann boundary conditions and applications
  with D. Slepcev and F. Da Lio, Asymptotic analysis, vol.37 (2004) 257-292 

• A free boundary problem arising in flame propagation
  J. Diff. Equations, vol 191 (2003) 470-489 

• Uniqueness and Existence of Hele-Shaw and Stefan problem
  Arch. Rat. Mech. Anal. vol 168 (2003) 299-328 

• Singular limits of Chematoxis model
  Nonlinear Anal. TMA vol.46 (2001), 817-834 

Inwon C. Kim