Steve Shkoller

About

Professor Steve Shkoller is a mathematician whose research is extensively focused on the analysis of partial differential equations, fluid dynamics, and mathematical physics. His work encompasses a broad range of topics including the Euler and Navier-Stokes equations, compressible and incompressible fluid flows, free-boundary problems, shock formation, and the dynamics of interfaces in fluids. He has contributed to the rigorous mathematical understanding of complex phenomena such as shock waves, vorticity creation, and instabilities like Rayleigh-Taylor and Richtmyer-Meshkov. His research also addresses the well-posedness and stability of classical problems in fluid mechanics, including the Stefan problem and the Muskat problem, often involving sophisticated techniques in geometric analysis and nonlinear PDEs. Throughout his career, Professor Shkoller has collaborated with numerous researchers to develop new mathematical models and analytical methods for studying fluid interfaces, elastic solids interacting with fluids, and liquid crystal dynamics. His work includes the development of artificial viscosity methods for nonlinear conservation laws and the study of singularities in fluid flows. He has also contributed to the mathematical theory of Lagrangian averaged Euler and Navier-Stokes equations, providing insights into turbulence modeling and the geometry of diffeomorphism groups. His research has been supported by the National Science Foundation and published in leading…

Research topics

• Mechanics
• Mathematics
• Mathematical analysis
• Physics

Selected publications

• Formation of Shocks for<scp>2D</scp>Isentropic Compressible Euler

  Communications on Pure and Applied Mathematics · 2020 · 45 citations

  Abstract We consider the 2D isentropic compressible Euler equations, with pressure law p ( ρ ) = (1 γ ) ρ γ , with γ &gt; 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1 ε , for ε &gt; 0 taken sufficiently small relative to the amplitude, there exist smooth solutions to the Euler equations which form a shock in time . The bl…

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• Shock Formation and Vorticity Creation for 3d Euler

  Communications on Pure and Applied Mathematics · 2022 · 39 citations

  Abstract We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direc…

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• Simultaneous Development of Shocks and Cusps for 2D Euler with Azimuthal Symmetry from Smooth Data

  Annals of PDE · 2022-11-19 · 29 citations

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• Formation of Point Shocks for 3D Compressible Euler

  Communications on Pure and Applied Mathematics · 2022-05-27 · 20 citations

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  We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions . We prove that for an open set of Sobolev‐class initial data that are a small L ∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic sta…

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• The geometry of maximal development and shock formation for the Euler equations in multiple space dimensions

  Inventiones mathematicae · 2024-06-03 · 9 citations

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Recent grants

• Analysis of moving interface problems in fluid dynamics

  NSF · $217k · 2013–2017

• Well-posedness of moving interface problems in perfect fluids

  NSF · $275k · 2010–2014

• Well-posedness of moving interface problems in perfect fluids

  NSF · $126k · 2007–2011

Frequent coauthors

• Daniel Coutand

  48 shared

• Vlad Vicol

  29 shared

• C. H. Arthur Cheng

  National Central University

  20 shared

• Mahir Hadžić

  18 shared

• Jerrold E. Marsden

  17 shared

• Tristan Buckmaster

  14 shared

• Rafael Granero-Belinchón

  13 shared

• Tudor S. Raţiu

  7 shared

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• Xin Liuh-101
• Alan Hastingsh-97
• James Crutchfieldh-63