home

research

teaching

notes

I study mathematical aspects of fluid motion.

"When a continuous medium is deprived of its physical properties (elasticity, thermal and electrical conductivity, and so on) its property of occupying a definite position in space remains, as do elementary interactions through the mutual pressure of its parts, due to Aristotle’s principle that it is impossible for two bodies to occupy the same space. It is amazing that it is these elementary interactions that cause the most complicated effects, including turbulence (viscosity of course plays an essential role in generating it)." -- V. I. Yudovich

Incompressible Fluid Dynamics
  1. Singular vortex pairs follow magnetic geodesics, with D. Glukhovskiy and B. Khesin submitted (2024), [arXiv].
  2. The Feynman-Lagerstrom criterion for boundary layers, with S. Iyer and T. T. Nguyen submitted (2023), [arXiv].
  3. Islands in stable fluid equilibria, with D. Ginsberg, submitted (2023), [arXiv].
  4. Twisting in Hamiltonian Flows and Perfect Fluids, with T.M. Elgindi, I-J Jeong, submitted (2023), [arXiv].
  5. On the Support of Anomalous Dissipation Measures, with L. De Rosa and M. Inversi, submitted (2023), [arXiv].
  6. On maximally mixed equilibria of two-dimensional perfect fluids, with M.Dolce, Archive for Rational Mechanics and Analysis (2022), [Article], [arXiv].
  7. Singularity formation in the incompressible Euler equation in finite and infinite time, with T.M.Elgindi, submitted (2022), [arXiv].
  8. Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids, with T.M.Elgindi, J.La, Mathematische Annalen (2022), [Article], [arXiv].
  9. Self-Regularization in turbulence from the Kolmogorov 4/5-Law and Alignment, Phil. Trans. R. Soc. A. (2022), [Article], [arXiv].
  10. Conjugate and cut points in ideal fluid motion, Annales Mathématiques du Québec, in honor of Alexander Shnirelman’s 75th birthday, (2021), with G. Misiołek, B. Shi, T. Yoneda [Article], [arXiv].
  11. Flexibility and rigidity in steady fluid motion, Communications in Mathematical Physics, (2021), with P. Constantin and D. Ginsberg, [Article], [arXiv].
  12. Inviscid limit of vorticity distributions in Yudovich class, Communications on Pure and Applied Mathematics (2020), with P. Constantin and T.M. Elgindi, [Article], [arXiv].
  13. Boundary conditions and polymeric drag reduction for the Navier-Stokes equations, Archive for Rational Mechanics and Analysis (2021), with J. La, [arXiv].
  14. Turbulent Cascade Direction and Lagrangian Time-Asymmetry, J of Nonlinear Sci (2019) 29: 65, [Article], [arXiv].
  15. Onsager's conjecture and anomalous dissipation on domains with boundary, SIAM J. Math. Anal., 50(5), 4785–4811 (2018), with H.Q. Nguyen, [Article], [arXiv].
  16. Remarks on the emergence of weak Euler solutions in the vanishing viscosity limit, J Nonlinear Sci (2019) 29: 709, with H.Q. Nguyen, [Article], [arXiv].
  17. An Onsager Singularity Theorem for Leray Solutions of Incompressible Navier-Stokes, Nonlinearity 32 4465 (2019), with G.L. Eyink, [Article], [arXiv].
  18. On the large-scale sweeping of small-scale eddies in turbulence: A filtering approach, Phys. Rev. Fluids, 2, 104603 (2017) , with P.L. Johnson, C.C. Lalescu and M. Wilczek, [Article].
  19. Triad Resonance between Gravity and Vorticity Waves in Vertical Shear, Ocean Modelling 103 87-97 (2016), with S.E. Wunsch, [Article].
  20. Internal tide energy transfer by nonlinear refraction, VIIIth International Symposium on Stratified Flows. Vol. 1. No. 1. (2016)., with S.E. Wunsch, [Article].

Compressible Fluid Dynamics
  1. Simultaneous development of shocks and cusps for 2D Euler with azimuthal symmetry from smooth data, Annals of PDE (2022), with T. Buckmaster, S. Shkoller, V. Vicol [Article], [arXiv].
  2. Entropy Hierarchies for equations of compressible fluids and self-organized dynamics, SIAM J. Math. Anal. (2019), with P. Constantin and R. Shvydkoy, [arXiv].
  3. Compressible Fluids and Active Potentials,
  4. Annales de l'Institut Henri Poincare C, Analyse Non Lineaire, (2019), with P. Constantin, H.Q. Nguyen and F. Pasqualotto, [Article], [arXiv].
  5. An Onsager Singularity Theorem for Turbulent Solutions of Compressible Euler Equations,
  6. Commun. Math. Phys. (2017), with G.L. Eyink, [Article], [arXiv].
  7. Cascades and Dissipative Anomalies in Relativistic Fluid Turbulence,
  8. Phys. Rev. X 8, 011022 (2018), with G.L. Eyink, [Article], [arXiv].
  9. Cascades and Dissipative Anomalies in Compressible Fluid Turbulence,
  10. Phys. Rev. X 8, 011023 (2018), with G.L. Eyink, [Article], [arXiv].
  11. Spontaneous Stochasticity and Anomalous Dissipation in Burgers Equation, J Stat Phys, 158, 2, 386-432 (2014), with G.L. Eyink, [Article].

Mixing, Enhanced Diffusion and Convection
  1. Statistically self-similar mixing by Gaussian random fields, submitted, (2023) with M. Coti Zelati and R.S. Gvalani, [arXiv].
  2. Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries, Philosophical Transactions of the Royal Society A, 380.2225 (2022): 20210025 with H.Q. Nguyen, C. Nobili, [arXiv].
  3. The statistical geometry of material loops in turbulence, Nature Communications (2021), with L. Bentkamp, C. Lalescu, and M. Wilczek, [arXiv], [Article].
  4. A stochastic approach to enhanced diffusion, Annali della Scuola Normale Superiore di Pisa (2020e), with M. Coti Zelati, [arXiv].
  5. Anomalous Dissipation in Passive Scalar Transport, Archive for Rational Mechanics and Analysis (2021), with T.M. Elgindi, G. Iyer and I-J. Jeong, [arXiv].
  6. A Lagrangian fluctuation-dissipation relation for scalar turbulence, III. Rayleigh-Bénard convection, J. Fluid Mech, 836, 560-598 (2017), with G.L. Eyink, [Article], [arXiv].
  7. A Lagrangian fluctuation-dissipation relation for scalar turbulence, II. Wall-bounded flows, J. Fluid Mech, 829, 236-279 (2017), with G.L. Eyink, [Article], [arXiv].
  8. A Lagrangian fluctuation-dissipation relation for scalar turbulence, I. Flows with no bounding walls, J. Fluid Mech, 829, 153-189 (2017), with G.L. Eyink, [Article], [arXiv].
  9. Backwards Two-Particle Dispersion in a Turbulent Flow, Phys. Rev. E 89, 041003(R) (2014), with D. Benveniste, [Article], [arXiv].

Non-Smooth Dynamical Systems
  1. Statistical determinism in non-Lipschitz dynamical systems, submitted (2020), with A.A. Mailybaev and A. Raibekas [arXiv].
  2. "Life after death" in ordinary differential equations with a non-Lipschitz singularity, Nonlinearity, (2020), with A.A. Mailybaev, [arXiv].
  3. Quantum Spontaneous Stochasticity, submitted, with G.L. Eyink, [arXiv].

Plasma Physics
  1. On the distribution of heat in fibered magnetic fields, submitted, with D. Ginsberg and H. Grayer II [arXiv].
  2. Flexibility and rigidity of free boundary MHD equilibria, Nonlinearity, 35(5) 2363 (2022) with P. Constantin and D. Ginsberg, [arXiv].
  3. On quasisymmetric plasma equilibria sustained by small force, J. Plasma Physics (2020), with P. Constantin and D. Ginsberg, [arXiv].
  4. Inertial-Range Reconnection in Magnetohydrodynamic Turbulence and in the Solar Wind, Phys. Rev. Lett. 115, (2015), with C.L. Lalescu, Y-K. Shi, G.L. Eyink, E. Vishniac and A. Lazarian, [Article].

Stochastic Partial Differential Equations
  1. Invariant measures for stochastic conservation laws on the line, submitted (2022), with A. Dunlap, C. Graham, J. La, L. Ryzhik, [arXiv].
  2. Lagrangian averaged stochastic advection by Lie transport for fluids, J Stat Phys (2020), with D.D. Holm and J-M. Leahy, [Article], [arXiv].
  3. Energy and Circulation Theorem Preserving Stochastic Fluids, Proc. Royal Soc. of Edinburgh: Section A Mathematics, 1-39, (2019), with D.D. Holm, [Article], [arXiv].

General Relativity
  1. Caustics and wave propagation in curved spacetimes, Phys. Rev. D 85, 124039 (2012), with A.I. Harte, [Article], [arXiv].
  2. Dependence of Self-Force on Central Object, Class. Quantum Grav. 28, 145025 (2011), with S.E. Gralla, [Article], [arXiv].

PhD thesis:
Anomalous Dissipation, Spontaneous Stochasticity & Onsager’s Conjecture


Most of my works can be found on:
Google Scholar, arXiv, MathSciNet, GitHub, Youtube.

Current Support: NSF CAREER award #2235395 and NSF #2106233
Past Support: NSF #1703997