Classical Euler flows generate the strong Guderley imploding shock wave
G. Cialdea, S. Shkoller, and V. Vicol
(2025) arXiv:2510.19688
Gradient catastrophes and an infinite hierarchy of Hölder cusp singularities for 1D Euler
I. Neal, S. Shkoller, and V. Vicol
J. London Math. Soc., 112(2), e70261, (2025)
A characteristics approach to shock formation in 2D Euler with azimuthal symmetry and entropy
I. Neal, S. Shkoller, and V. Vicol
Communications in Analysis and Mechanics, 17, 188--236, (2025)
Vorticity blowup in 2D compressible Euler equations
J. Chen, G. Cialdea, S. Shkoller, and V. Vicol
(2024), to appear in Duke Math J.
The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions
S. Shkoller and V. Vicol
Invent. Math., 237, 871--1252, (2024)
A new type of stable shock formation in gas dynamics
I. Neal, C. Rickard, S. Shkoller, and V. Vicol
Communications on Pure and Applied Analysis, 23, 1423--1447, (2024)
Shock formation and vorticity creation for 3d Euler
T. Buckmaster, S. Shkoller, and V. Vicol
Comm. Pure Appl. Math., 76 , 1965--2072, (2023)
Formation of point shocks for 3D compressible Euler
T. Buckmaster, S. Shkoller, and V. Vicol
Comm. Pure Appl. Math., 76 , 2069--2120, (2023)
A fast dynamic smooth adaptive meshing scheme with applications to compressible flow
R. Ramani and S. Shkoller
Journal of Computational Physics, 490, 112280, (2023)
Interface models for three-dimensional Rayleigh-Taylor instability
G. Pandya and S. Shkoller
Journal of Fluid Mechanics, 959, A10, (2023)
Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data
T. Buckmaster, T. Drivas, S. Shkoller, and V. Vicol
Annals of PDE, 8:26, 1--199, (2022)
Formation of shocks for 2D isentropic compressible Euler
T. Buckmaster, S. Shkoller, and V. Vicol
Comm. Pure Appl. Math., 75 , 2069--2120, (2022)
Affine motion of 2d incompressible fluids and flows in SL(2,R)
J. Roberts, S. Shkoller, and T. Sideris
Commun. Math. Phys., 375, 1003--1040, (2020)
A comparison of interface growth models applied to Rayleigh–Taylor and
Richtmyer–Meshkov instabilities
J. Canfield, N. Denissen, M. Francois, R. Gore, R. Rauenzahn, J. Reisner, and S. Shkoller
Journal of Fluids Engineering, 142, 121108-1, (2020)
A multiscale model for Rayleigh-Taylor and Richtmyer-Meshkov instabilities
R. Ramani and S. Shkoller
Journal of Computational Physics, 405, 109177, (2020)
Global existence of near-affine solutions to the compressible Euler equations
S. Shkoller and T. Sideris
Arch. Rational Mech. Anal., 234, 115--180, (2019)
A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary
M. Hadzic, S. Shkoller, and J. Speck
Comm. Partial Differential Equations, 44, 859--906, (2019)
Rigorous asymptotic models of water waves
A. Cheng, R. Granero-Belinchon, S. Shkoller, and J. Wilkening
Water Waves, 1 , 71--130, (2019)
A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 1: the 1-D case
R. Ramani, J. Reisner, and S. Shkoller
Journal of Computational Physics, 387, (2019), 81--116
A space-time smooth artificial viscosity method with wavelet noise indicator and shock collision scheme, Part 2: the 2-D case
R. Ramani, J. Reisner, and S. Shkoller
Journal of Computational Physics, 387, (2019), 45--80
Nonuniqueness of weak solutions to the SQG equation
T. Buckmaster, S. Shkoller, and V. Vicol
Comm. Pure Appl. Math., 72(9), 1809--1874, (2019)
On the splash singularity for the free-surface of a Navier-Stokes fluid
D. Coutand and S. Shkoller
Ann. I.H.Poincare--AN, 36 , 475--503, (2019)
Well-posedness and decay to equilibrium for the Muskat problem with discontinuous permeability
R. Granero-Belinchon and S. Shkoller
Trans. AMS, 372, 2255--2286, (2019)
Regularity of the velocity field for Euler vortex patch evolution
D. Coutand and S. Shkoller
Trans. AMS, 370 , 3689--3720, (2018)
A model for Rayleigh-Taylor mixing and interface turn-over
R. Granero-Belinchon and S. Shkoller
Multiscale Model. Simul., 15 , 274--308, (2017)
Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains
A. Cheng and S. Shkoller
J. Math. Fluid Mech. 19, 375--422, (2017)
Local well-posedness and global stability of the two-phase Stefan problem
M. Hadzic, G. Navarro, and S. Shkoller
SIAM J. Math. Anal. 49 , 4942--5006, (2017)
Well-posedness for the classical Stefan problem and the zero surface tension limit
M. Hadzic and S. Shkoller
Arch. Rational Mech. Anal., 223 , 213--264, (2017)
On the impossibility of finite-time splash singularities for vortex sheets
D. Coutand and S. Shkoller
Arch. Rational Mech. Anal., 221 , 987--1033, (2016)
Well-posedness of the Muskat problem with H2 initial data
A. Cheng, R. Granero-Belinchon and S. Shkoller
Adv. Math., 286 , 32--104, (2016)
Global stability of steady states in the classical Stefan problem for general boundary shapes
M. Hadzic and S. Shkoller
Philos. Trans. Roy. Soc. London Ser. A, 373 , 20140284, (2015)
Global stability and decay for the classical Stefan Problem
M. Hadzic and S. Shkoller
Comm. Pure Appl. Math, 68 , 689--757, (2015)
On the finite-time splash and splat singularities for the 3-D free-surface Euler equations
D. Coutand and S. Shkoller
Commun. Math. Phys., 325 , 143--183, (2014)
Global existence and decay for solutions of the Hele-Shaw flow with injection
A. Cheng, D. Coutand, and S. Shkoller
Interfaces and Free Boundaries, 16 , 297--338, (2014)
Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit
D. Coutand, J. Hole and S. Shkoller
SIAM J. Math. Anal., 45, 3690--3767, (2013)
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
J. Reisner, J. Serencsa, and S. Shkoller
Journal of Computational Physics, 235, (2013), 912--933
Well-posedness in smooth function spaces for the moving-boundary 3-D compressible Euler equations in physical vacuum
D. Coutand and S. Shkoller
Arch. Rational Mech. Anal., 206 , 515--616, (2012)
Well-posedness in smooth function spaces for the moving-boundary 1-D compressible Euler equations in physical vacuum
D. Coutand and S. Shkoller
Comm. Pure Appl. Math., 64 , 328--366, (2011)
A priori estimates for the free-boundary 3-D compressible Euler equations in physical vacuum
D. Coutand, H. Lindblad, and S. Shkoller
Commun. Math. Phys., 296, (2010), 559--587
A simple proof of well-posedness for the free-surface incompressible Euler equations
D. Coutand and S. Shkoller
Discrete Contin. Dyn. Syst. Ser. S, 3, 429--449, (2010)
On the limit as the density ratio tends to zero for two perfect incompressible 3-D fluids separated by a surface of discontinuity
A. Cheng, D. Coutand, and S. Shkoller
Comm. Partial Differential Equations, 35, 817--845, (2010)
The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell
A. Cheng and S. Shkoller
SIAM J. Math. Anal., 42 , (2010), 1094--1155
On the Motion of Vortex Sheets with Surface Tension in the 3D Euler Equations with Vorticity
A. Cheng, D. Coutand and S. Shkoller
Comm. Pure Appl. Math., 61(12), (2008), 1715--1752
A liquid-crystal model for friction
A. Cheng, L. Kellog, S. Shkoller, and D. Turcotte
Proc. Natl. Acad. Sci. USA, 105, (2008), 7930--7935
Well-posedness of the free-surface incompressible Euler equations with or without surface tension
D. Coutand and S. Shkoller
J. Amer. Math. Soc., 20(3), (2007), 829--930
Navier-Stokes equations interacting with a nonlinear elastic biofluid shell
A. Cheng, D. Coutand and S. Shkoller
SIAM J. Math. Anal., 39 , (2007), 742--800
The interaction between quasilinear elastodynamics and the Navier-Stokes equations
D. Coutand and S. Shkoller
Arch. Rational Mech. Anal. 179(3), (2006), 303--352
Motion of an elastic solid inside of an incompressible viscous fluid
D. Coutand and S. Shkoller
Arch. Rational Mech. Anal. 176(1), (2005), 25--102
Turbulent channel flow in weighted Sobolev spaces using the anisotropic Lagrangian averaged Navier-Stokes (LANS-α) equations
D. Coutand and S. Shkoller
Commun. Pure Appl. Anal., 3, (2004), 1--23
The anisotropic Lagrangian averaged Euler and Navier-Stokes equations
J.E. Marsden and S. Shkoller
Arch. Rational Mech. Anal., 166, (2003), 27-46
Numerical simulations of the Lagrangian averaged Navier-Stokes (LANS-α) equations for homogeneous isotropic turbulence
K. Mohseni, B. Kosovic, S. Shkoller, and J.E. Marsden
Physics of Fluids, 15, (2003), 524--544
Well-posedness and global attractors for liquid crystals on Riemannian manifolds
S. Shkoller
Comm. Partial Differential Equations, 27, (2002), 1103-1137
The Lagrangian averaged Euler (LAE-α) equations with free-slip or mixed boundary conditions
S. Shkoller
Geometry, Mechanics, and Dynamics, Springer-Verlag, 2002, 169--180
Well-posedness of the full Ericksen-Leslie model of nematic liquid crystals
D. Coutand and S. Shkoller
C. R. Acad. Sci. Paris Ser. I Math., 333, (2001), 919-924
The vortex blob method as a second-grade non-Newtonian fluid
M. Oliver and S. Shkoller
Comm. Partial Differential Equations, 26, (2001), 295-314
Global well-posedness for the Lagrangian averaged Navier-Stokes (LANS-α) equations on bounded domains
J.E. Marsden and S. Shkoller
Phil. Trans. R. Soc. Lond. A, 359, (2001), 1449-1468
Variational methods, multisymplectic geometry and continuum mechanics
J. Marsden, S. Pekarsky, S. Shkoller, and M. West
J. Geom. Phys., 38, (2001), 253-284
Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid
S. Shkoller
J. Differential Geom., 55, (2000), 145-191
The geometry and analysis of the averaged Euler equations and a new diffeomorphism group
J.E. Marsden, T. Ratiu, and S. Shkoller
Geom. Funct. Anal., 10, (2000), 582-599
A variational approach to second-order multisymplectic field theory
S. Kouranbaeva and S. Shkoller
J. Geom. Phys., 35, (2000), 333-366
Reduction in principal fiber bundles: covariant Euler-Poincaré equations
M. Castrillon, T. Ratiu and S. Shkoller
Proc. Amer. Math. Soc. 128 (2000), 2155-2164
Symmetry reduction of discrete Lagrangian mechanics on Lie groups
J.E. Marsden, S. Pekarsky, and S. Shkoller
J. Geom. Phys., 36, (2000), 139-150
Discrete Euler-Poincaré and Lie-Poisson Algorithms
J.E. Marsden, S. Pekarsky, and S. Shkoller
Nonlinearity, 12, (1999), 1647-1662
Multisymplectic geometry, covariant Hamiltonians, and water waves
J. Marsden and S. Shkoller
Math. Proc. Camb. Phil. Soc., 125, (1999), 553-575
Persistence of invariant manifolds for nonlinear PDEs
D.A. Jones and S. Shkoller
Studies in Appl. Math, 102, (1999), 27-67
Geometry and curvature of diffeomorphism groups with H1 metric and mean hydrodynamics
S. Shkoller
J. Funct. Anal., 160, (1998), 337-365
Multisymplectic geometry, variational integrators, and nonlinear PDEs
J. Marsden, G. Patrick and S. Shkoller
Comm. Math. Phys., 199, (1998), 351-391
Reduction of Dieterich-Ruina attractors to unimodals maps
S. Shkoller and J.B. Minster
J. Nonlinear Processes in Geophysics, 4, (1997), 63-69
On an approximate homogenization scheme for nonperiodic materials
S. Shkoller
Comp. Math. Appl., 33, (1997), 15-34