• Work supported by the National Science Foundation

    • Compressible Fluids

    1. I. NEAL, S. SHKOLLER and V. VICOL, Gradient catastrophes and an infinite hierarchy of Hölder cusp singularities for 1D Euler, J. London Math. Soc., 112(2), e70261, (2025) DOI: 10.1112/jlms.70261or arXiv:2412.21040

    2. J. CHEN, G. CIALDEA, S. SHKOLLER and V. VICOL, Vorticity blowup in 2D compressible Euler equations, (2024), arXiv:2407.06455

    3. S. SHKOLLER and V. VICOL, The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions, Invent. Math., 237, 871--1252, (2024), https://doi.org/10.1007/s00222-024-01269-x or arXiv:2310.08564

    4. I. NEAL, C. RICKARD, S. SHKOLLER and V. VICOL, A new type of stable shock formation in gas dynamics, Communications on Pure and Applied Analysis, 23, 1423--1447, (2024). Doi: 10.3934/cpaa.2023118 or arXiv:2303.16842

    5. I. NEAL, S. SHKOLLER and V. VICOL, A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy, Communications in Analysis and Mechanics, 17, 188--236, (2025). 10.3934/cam.2025009 or arXiv:2302.01289

    6. T. BUCKMASTER, T. DRIVAS, S. SHKOLLER and V. VICOL, Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data, Annals of PDE, 8:26, 1--199, (2022), arXiv:2106.02143

    7. T. BUCKMASTER, S. SHKOLLER and V. VICOL, Shock formation and vorticity creation for 3d Euler, Comm. Pure Appl. Math., 76 , 1965--2072, (2023), https://doi.org/10.1002/cpa.22067, arXiv:2006.14789

    8. T. BUCKMASTER, S. SHKOLLER and V. VICOL, Formation of point shocks for 3D compressible Euler, Comm. Pure Appl. Math., 76 , 2069--2120, (2023), https://doi.org/10.1002/cpa.22068, arXiv:1912.04429

    9. T. BUCKMASTER, S. SHKOLLER and V. VICOL, Formation of shocks for 2D isentropic compressible Euler, Comm. Pure Appl. Math., 75 , 2069--2120, (2022), https://doi.org/10.1002/cpa.21956, arXiv:1907.03784

    10. S. SHKOLLER and T. SIDERIS, Global existence of near-affine solutions to the compressible Euler equations, Arch. Rational Mech. Anal., 234, 115--180, (2019), ArXiv.

    11. M. HADZIC, S. SHKOLLER, and J. SPECK, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, Comm. Partial Differential Equations, 44, 859--906, (2019), ArXiv.

    12. D. COUTAND, J. HOLE and S. SHKOLLER, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45, 3690--3767, (2013), PDF.

    13. D. COUTAND and S. SHKOLLER, Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum, Arch. Rational Mech. Anal., 206 , 515--616, (2012), PDF.

    14. D. COUTAND and S. SHKOLLER, Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 , 328--366, (2011), PDF.

    15. D. COUTAND, H. LINDBLAD, and S. SHKOLLER, A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum, Commun. Math. Phys., 296, (2010), 559--587. PDF.

    Numerical methods and asymptotic models for fluid interfaces, Rayleigh-Taylor instabilities, shocks, and contact discontinuities

    1. R. RAMANI and S. SHKOLLER, A fast dynamic smooth adaptive meshing scheme with applications to compressible flow, Journal of Computational Physics, 490, 112280, (2023), PDF.

    2. G. PANDYA and S. SHKOLLER, Interface models for three-dimensional Rayleigh-Taylor instability, Journal of Fluid Mechanics, 959, A10, (2023), ArXiv:2201.04538, DOI

    3. R. RAMANI and S. SHKOLLER, A multiscale model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities, Journal of Computational Physics, 405, 109177, (2020), ArXiv.

    4. A. CHENG, R. GRANERO-BELINCHON, S. SHKOLLER, and J. WILKENING, Rigorous asymptotic models of water waves, Water Waves, 1 , 71--130, (2019), ArXiv.

    5. R. RAMANI, J. REISNER, AND S. SHKOLLER, A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: the 1-D case, Journal of Computational Physics, 387, (2019), 81--116, ArXiv.

    6. R. RAMANI, J. REISNER, AND S. SHKOLLER, A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 2: the 2-D case, Journal of Computational Physics, 387, (2019), 45--80, ArXiv.

    7. R. GRANERO-BELINCHON and S. SHKOLLER, A model for Rayleigh-Taylor mixing and interface turn-over, Multiscale Model. Simul., 15 , 274--308, (2017), PDF.

    8. J. REISNER, J. SERENCSA, AND S. SHKOLLER, A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws , Journal of Computational Physics, 235, (2013), 912--933, PDF.

    Convex integration and nonuniqueness

    1. T. BUCKMASTER, S. SHKOLLER, and V. VICOL, Nonuniqueness of weak solutions to the SQG equation, Comm. Pure Appl. Math., 72(9), 1809--1874, (2019), ArXiv.

    Elliptic systems on Sobolev-class domains

    1. A. CHENG and S. SHKOLLER, Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains, J. Math. Fluid Mech. 19, 375--422, (2017), PDF.

    Incompressible Euler and Navier-Stokes Free-Boundary Problems

    1. J. ROBERTS, S. SHKOLLER, and T. SIDERIS, Affine motion of 2d incompressible fluids and flows in SL(2,R), Commun. Math. Phys., 375, 1003--1040, (2020), ArXiv.

    2. D. COUTAND and S. SHKOLLER, On the splash singularity for the free-surface of a Navier-Stokes fluid, Ann. I.H.Poincare--AN, 36 , 475--503, (2019), PDF.

    3. D. COUTAND and S. SHKOLLER, Regularity of the velocity field for Euler vortex patch evolution, Trans. AMS, 370 , 3689--3720, (2018), PDF.

    4. D. COUTAND and S. SHKOLLER, On the impossibility of finite-time splash singularities for vortex sheets, Arch. Rational Mech. Anal., 221 , 987--1033, (2016), PDF.

    5. D. COUTAND and S. SHKOLLER, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325 , 143--183, (2014), PDF.

    6. D. COUTAND and S. SHKOLLER, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3, 429--449, (2010), PDF.

    7. A. CHENG, D. COUTAND, and S. SHKOLLER, On the limit as the density ratio tends to zero for two perfect incompressible 3-D fluids separated by a surface of discontinuity, Comm. Partial Differential Equations, 35, 817--845, (2010). PDF.
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    8. A. CHENG, D. COUTAND AND S. SHKOLLER, On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity, Comm. Pure Appl. Math., 61(12), (2008), 1715--1752. PDF.

    9. D. COUTAND AND S. SHKOLLER, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20(3), (2007), 829--930. PDF.

    Stefan Problem

    1. M. HADZIC, G. NAVARRO, and S. SHKOLLER, Local well-posedness and global stability of the two-phase Stefan problem, SIAM J. Math. Anal. 49 , 4942--5006, (2017), PDF.

    2. M. HADZIC AND S. SHKOLLER, Global stability of steady states in the classical Stefan problem for general boundary shapes, Philos. Trans. Roy. Soc. London Ser. A, 373 , 20140284, (2015), http://dx.doi.org/10.1098/rsta.2014.0284, PDF.

    3. M. HADZIC AND S. SHKOLLER, Global stability and decay for the classical Stefan Problem, Comm. Pure Appl. Math, 68 , 689--757, (2015), PDF.

    4. M. HADZIC AND S. SHKOLLER, Well-posedness for the classical Stefan problem and the zero surface tension limit, , Arch. Rational Mech. Anal., 223 , 213--264, (2017), PDF.

    Muskat and Hele-Shaw Problems

    1. R. GRANERO-BELINCHON and S. SHKOLLER, Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability , Trans. AMS, 372, 2255--2286, (2019), ArXiv.

    2. A. CHENG, R. GRANERO-BELINCHON AND S. SHKOLLER, Well-posedness of the Muskat problem with H2 initial data, Adv. Math., 286 , 32--104, (2016), PDF.

    3. A. CHENG, D. COUTAND, AND S. SHKOLLER, Global existence and decay for solutions of the Hele-Shaw flow with injection, Interfaces and Free Boundaries, 16 , 297--338, (2014), PDF.

    Fluid-Structure Interaction Problems

    1. A. CHENG AND S. SHKOLLER, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal., 42 , (2010), 1094--1155. PDF.

    2. A. CHENG, D. COUTAND AND S. SHKOLLER, Navier-Stokes equations interacting with a nonlinear elastic biofluid shell, SIAM J. Math. Anal., 39 , (2007), 742--800. PDF.

    3. D. COUTAND AND S. SHKOLLER, On the interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch. Rational Mech. Anal. 179(3), (2006), 303--352. PDF.

    4. D. COUTAND AND S. SHKOLLER, Motion of an elastic solid inside of an incompressible viscous fluid, Arch. Rational Mech. Anal. 176(1), (2005), 25--102. PDF.

    Analysis of Friction, Liquid crystals, and non-Newtonian fluids

    1. A. CHENG, L. KELLOG, S. SHKOLLER, AND D. TURCOTTE, A liquid-crystal model for friction, Proc. Natl. Acad. Sci. USA, 105, (2008), 7930--7935. PDF.

    2. D. COUTAND AND S. SHKOLLER, Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals, C. R. Acad. Sci. Paris Ser. I Math., 333, (2001), 919-924. PDF.

    3. S. SHKOLLER,Well-posedness and global attractors for liquid crystals on Riemannian manifolds, Comm. Partial Differential Equations, 27, (2002), 1103-1137. PDF.

    4. M. OLIVER AND S. SHKOLLER, The vortex blob method as a second-grade non-Newtonian fluid, Comm. Partial Differential Equations, 26, (2001), 295-314. PDF.

    Lagrangian averaged Navier-Stokes and Euler equations

    1. D. COUTAND AND S. SHKOLLER, Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-α) equations, Commun. Pure Appl. Anal., 3, (2004), 1--23. PDF.

    2. J.E. MARSDEN AND S. SHKOLLER, The anisotropic Lagrangian averaged Euler and Navier-Stokes equations, Arch. Rational Mech. Anal., 166, (2003), 27-46. PDF.

    3. J.E. MARSDEN AND S. SHKOLLER, Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains, Phil. Trans. R. Soc. Lond. A, 359, (2001), 1449-1468. PDF.

    4. K. MOHSENI, B. KOSOVIC, S. SHKOLLER, AND J.E. MARSDEN, Numerical simulations of the Lagrangian averaged Navier-Stokes (LANS-α) equations for homogeneous isotropic turbulence, Physics of Fluids, 15, (2003), 524--544. PDF.

    5. S. SHKOLLER, The Lagrangian averaged Euler (LAE-α) equations with free-slip or mixed boundary conditions, Geometry, Mechanics, and Dynamics, eds. P. Holmes, P. Newton, A. Weinstein, Special Volume, Springer-Verlag, 2002, 169--180. PS.

    Analysis on diffeomorphism groups

    1. S. SHKOLLER, Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid, J. Differential Geom., 55, (2000), 145-191. PDF.

    2. J.E. MARSDEN, T. RATIU, AND S. SHKOLLER, The geometry and analysis of the averaged Euler equations and a new diffeomorphism group, Geom. Funct. Anal., 10, (2000), 582-599. PDF.

    3. S. SHKOLLER, Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics, J. Funct. Anal., 160, (1998), 337-365. PDF.

    Multisymplectic geometry and geometric integrators

    1. J. MARSDEN, S. PEKARSKY, S. SHKOLLER, AND M. WEST, On a multisymplectic approach to continuum mechanics, J. Geom. Phys., 38, (2001), 253-284. PDF.

    2. S. KOURANBAEVA AND S. SHKOLLER, A variational approach to second-order multisymplectic field theory, J. Geom. Phys., 35, (2000), 333-366. PDF.

    3. M. CASTRILLON, T. RATIU AND S. SHKOLLER, Reduction in principal fiber bundles: covariant Euler-Poincaré equations, Proc. Amer. Math. Soc. 128 (2000), 2155-2164. PDF.

    4. J.E. MARSDEN, S. PEKARSKY, AND S. SHKOLLER, Symmetry reduction of discrete Lagrangian mechanics on Lie groups, J. Geom. Phys., 36, (2000), 139-150. PDF.

    5. J.E. MARSDEN, S. PEKARSKY, AND S. SHKOLLER, Discrete Euler-Poincaré and Lie-Poisson Algorithms, Nonlinearity, 12, (1999), 1647-1662. PDF.

    6. J. MARSDEN AND S. SHKOLLER, Multisymplectic geometry, covariant Hamiltonians, and water waves, Math. Proc. Camb. Phil. Soc., 125, (1999), 553-575. PDF.

    7. J. MARSDEN, G. PATRICK AND S. SHKOLLER, Multisymplectic geometry, variational integrators, and nonlinear PDEs, Comm. Math. Phys., 199, (1998), 351-391. PDF.

    Dynamical systems

    1. D.A. JONES AND S. SHKOLLER, Persistence of invariant manifolds for nonlinear PDEs, Studies in Appl. Math, 102, (1999), 27-67. PDF.

    2. S. SHKOLLER AND J.B. MINSTER, Reduction of Dieterich-Ruina attractors to unimodals maps, J. Nonlinear Processes in Geophysics, 4, (1997), 63-69. PDF.

    Homogenization theory in material science

    1. S. SHKOLLER, On an approximate homogenization scheme for nonperiodic materials, Comp. Math. Appl., 33, (1997), 15-34. PDF.

    2. S. SHKOLLER AND A. MAEWAL, A model for defective fibrous composites, J. Mech. Phys. Solids, 44, (1996), 1929-1951. PDF.

    3. S. SHKOLLER AND G. HEGEMIER, Homogenization of Plain Weave Composites Using Two-Scale Convergence, Int. J. Sol. Str., 32, (1995), 783-794. PDF.