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 journals, reflecting his significant impact on the field of mathematical fluid dynamics and applied analysis.

Research topics

• Mechanics
• Mathematics
• Mathematical analysis
• Physics

Selected publications

• Sub-cell Wave Reconstruction from Differentiated Riemann Variables

  SSRN Electronic Journal · 2026-01-01

  preprintOpen access1st authorCorresponding
  PublisherDOI

• Intermittent Sub-grid Wave Correction from Differentiated Riemann Variables

  SSRN Electronic Journal · 2026-01-01

  preprintOpen access1st authorCorresponding
  PublisherDOI

• Sub-cell Wave Reconstruction from Differentiated Riemann Variables

  arXiv (Cornell University) · 2026-03-17

  preprintOpen access1st authorCorresponding

  We introduce a postprocessing procedure that recovers sub-cell wave geometry from a standard one-dimensional Euler shock-capturing computation using differentiated Riemann variables (DRVs) -- characteristic derivatives that separate the three wave families into distinct localized spikes. Filtered DRV surrogates detect the waves, plateau sampling extracts the local states, and a pressure-wave-function Newton closure completes the geometry. The entire pipeline adds less than $0.25\%$ to the cost of a baseline WENO--5/HLLC solve. For Sod, a severe-expansion problem, and the LeBlanc shock tube, wave locations are recovered to within roundoff or $O(10^{-4})$ and the contact is sharpened to one cell width; a pattern-agnostic extension handles all four Riemann configurations with errors at the $10^{-6}$--$10^{-8}$ level. Direct comparison with MUSCL--THINC--BVD and WENO-Z--THINC--BVD shows that neither reproduces the combination of sharp contacts, small contact-window internal-energy error, and elimination of the LeBlanc positive overshoot achieved by the DRV reconstruction.

  PublisherDOI

• Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold

  ArXiv.org · 2026-03-11

  articleOpen access1st authorCorresponding

  We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,α}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0<α<\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|ω(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3α)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $α=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted Hölder topology.

  PublisherOA PDF

• Intermittent Sub-grid Wave Correction from Differentiated Riemann Variables

  arXiv (Cornell University) · 2026-03-23

  articleOpen access1st authorCorresponding

  We introduce a low-cost every-$K$-step correction for one-dimensional Euler computations. The correction uses differentiated Riemann variables (DRVs) -- characteristic derivatives that isolate the left acoustic wave, the contact, and the right acoustic wave -- to locate the current wave packet, sample the surrounding constant states, perform a short Newton update for the intermediate pressure and contact speed, and conservatively remap a sharpened profile back onto the grid. The ingredients are elementary -- filtered centered differences, local state sampling, a single Newton step, and conservative cell averaging -- yet the effect on accuracy is disproportionate. On a long-time severe-expansion benchmark ($N=900$, $t=0.4$), intermittent correction drives the intermediate-state errors from $O(10^{-2})$ to $O(10^{-13})$, i.e. to machine precision. On a long-time LeBlanc benchmark ($N=800$, $t=1$), the method crosses a qualitative threshold: one-shot final-time reconstruction fails entirely (shock location error $2.7\times 10^{-1}$), whereas correction every three steps recovers an almost exact sharp solution with contact and shock positions accurate to machine precision. The same detector-and-Newton mechanism handles two-shock and two-rarefaction packets without case-specific logic, with plateau improvements of four to sixteen orders of magnitude. In an unoptimized Python prototype the wall-clock overhead is below a factor of two even on the most aggressively corrected benchmark. To our knowledge, no comparably lightweight fixed-grid add-on has been shown to recover this level of coarse-grid accuracy on the long-time LeBlanc and related near-vacuum problems.

  PublisherOA PDF

• Smooth and stable Euler implosions

  arXiv (Cornell University) · 2026-05-01

  preprintOpen access

  We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

  PublisherDOI

• Sub-cell Wave Reconstruction from Differentiated Riemann Variables

  ArXiv.org · 2026-03-17

  articleOpen access1st authorCorresponding

  We introduce a postprocessing procedure that recovers sub-cell wave geometry from a standard one-dimensional Euler shock-capturing computation using differentiated Riemann variables (DRVs) -- characteristic derivatives that separate the three wave families into distinct localized spikes. Filtered DRV surrogates detect the waves, plateau sampling extracts the local states, and a pressure-wave-function Newton closure completes the geometry. The entire pipeline adds less than $0.25\%$ to the cost of a baseline WENO--5/HLLC solve. For Sod, a severe-expansion problem, and the LeBlanc shock tube, wave locations are recovered to within roundoff or $O(10^{-4})$ and the contact is sharpened to one cell width; a pattern-agnostic extension handles all four Riemann configurations with errors at the $10^{-6}$--$10^{-8}$ level. Direct comparison with MUSCL--THINC--BVD and WENO-Z--THINC--BVD shows that neither reproduces the combination of sharp contacts, small contact-window internal-energy error, and elimination of the LeBlanc positive overshoot achieved by the DRV reconstruction.

  PublisherOA PDF

• Incompressible Euler Blowup at the $C^{1,\frac{1}{3}}$ Threshold

  arXiv (Cornell University) · 2026-03-11

  preprintOpen access1st authorCorresponding

  We prove finite-time Type-I blowup for the three-dimensional incompressible Euler equations in the axisymmetric no-swirl class, with initial velocity in $C^{1,α}(\mathbb{R}^3)\cap L^2(\mathbb{R}^3)$, odd symmetry in $z$, and $0&lt;α&lt;\tfrac13$, for an explicit class of finite-energy initial data. The singularity forms at a stagnation point on the symmetry axis. The on-axis axial strain and the global vorticity norm blow up at the Type-I rates $-\partial_z u_z(0,0,t)\sim (T^*-t)^{-1}$ and $\|ω(\cdot,t)\|_{L^\infty}\sim (T^*-t)^{-1}$, while the meridional Jacobian collapses according to $J(t)\sim (T^*-t)^{1/(1-3α)}$. The proof introduces a Lagrangian clock-and-strain framework that replaces the Eulerian self-similar ansatz used in prior work with a Lagrangian flow decomposition. The collapse dynamics are governed by a Riccati law for the on-axis axial strain, coupled to a clock ODE for the meridional Jacobian. The decisive step is a non-perturbative strain-pressure comparison showing that the pressure Hessian cannot cancel the quadratic compressive strain responsible for collapse. This gives a dynamical explanation of the threshold $α=\tfrac13$. The blowup mechanism is structurally stable and persists for an open set of admissible angular profiles in a weighted Hölder topology.

  PublisherDOI

• Smooth and stable Euler implosions

  arXiv (Cornell University) · 2026-05-01

  articleOpen access

  We construct a new class of self-similar implosion profiles for the multi-dimensional compressible Euler equations. These profiles are smooth, genuinely non-isentropic, radially/spherically symmetric, and have explicit (closed-form) similarity exponents. We prove that the exact Euler solution corresponding to the ground state implosion profile is stable to radially symmetric perturbations, as a solution to the full nonlinear compressible Euler equations, modulo a one-dimensional compatibility condition on the initial data. For perturbations of the Euler solution corresponding to the ground state implosion profile of a monatomic or diatomic gas, that do not obey any symmetry assumptions, we provide a complete characterization of the set of initial data that yield nonlinear stability.

  PublisherOA PDF

• Gradient catastrophes and an infinite hierarchy of Hölder cusp‐singularities for 1D Euler

  Journal of the London Mathematical Society · 2025-08-01

  article

  Abstract We establish an infinite hierarchy of finite‐time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with nonconstant entropy. Specifically, for all integers , we prove that there exist classical solutions, emanating from smooth, compressive, and nonvacuous initial data, which form cusp‐type gradient singularities in finite time, in which the gradient of the solution has precisely Hölder‐regularity. We show that such Euler solutions are codimension‐ stable in the Sobolev space .

  PublisherDOI

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

• ITR: Analysis and Simulation of Interface Dynamics in Multiphase Fluids and Solids

  NSF · $300k · 2003–2007

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