• Work supported by the National Science Foundation

    • Compressible Fluids

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

    2. 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

    3. 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, (2023), Doi: 10.3934/cpaa.2023118 or arXiv:2303.16842

    4. I. NEAL, S. SHKOLLER and V. VICOL, A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy, (2023), arXiv:2302.01289

    5. 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

    6. 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

    7. 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

    8. 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

    9. 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.

    10. 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.

    11. 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.

    12. 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.

    13. 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.

    14. 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.