Research

My daughter and I jumping on a trampoline.

My recent research has been motivated by the concerning prospect of non-uniqueness in modeling and forecasting. There is strong evidence for non-uniqueness in the natural energy class for the 3D Navier-Stokes equations, a system which models viscous incompressible fluid flow. Taking the viewpoint that non-uniqueness holds, an important question is: How severe is it? Ideally all possible solutions remain close together, indicating they are predictable from a single flow. This can be viewed as a sort of stability. On the other hand, the possibility that two solutions can separate explosively raises problems for modeling. A major research goal is to understand how rapidly solutions can and, possibly, must separate as well as how this separation is affected by properties of the initial data. This has led to work investigating bounds on separation rates, uniqueness criteria in terms of the error, higher order time regularity at t=0, etc.

I have also recently written about the long time behavior of flows in terms of both eventual regularity, where the goal is to find minimal integrability conditions on the data which lead to eventual regularity, and data assimilation, which is concerned with recovering a flow globally based on coarse, localized measurements.

Self-similarity has played a recurring role in my work. During my post-doctoral work, Tai-Peng Tsai and I constructed self-similar solutions for large, rough data and explored generalizations. Related results for the critical SQG problem were explored by Dallas Albritton and I. Working with my former PhD student, Patrick Phelps, we improved asymptotic results for self-similar and discretely self-similar solutions.

I have studied regularity through a variety of lenses. Several papers have explored the necessary role small scales play in possible singularity development.

Preprints for most of my papers can be found on arXiv.

The image appearing at the top of this page is a visualization of a Karman vortex street. (Jürgen Wagner – Self-photographed, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=37610334).

Publications and preprints

  1. Asymptotic properties of discretely self-similar Navier-Stokes solutions with rough data. Submitted.
  2. Asymptotic stability for the 3D Navier-Stokes {equations} in $L^3$ and nearby spaces. Submitted.
  3. Remarks on the separation of Navier-Stokes flows. Accepted to Nonlinearity. ArXiv link
  4. (With Misha Chernobai and Tai-Peng Tsai) Global Navier-Stokes flows in intermediate spaces, submitted. ArXiv link
  5. (With Anmikh Biswas and Michael Jolly) Convergence of a mobile data assimilation scheme for the 2D Navier-Stokes equations, Discrete and Continuous Dynamical Systems (2024)
  6. (With Patrick Phelps [PhD student]) Estimation of non-uniqueness and short-time asymptotic expansions for Navier-Stokes flows, Annales de l’Institut Henri Poincar\’e, Analyse Non Lineaire (2024)
  7. (With Chen-Chih Lai and Tai-Peng Tsai) Mild solutions and spacetime integral bounds for Stokes and Navier-Stokes flows in Wiener amalgam spaces, Mathematische Annalen (2023)
  8. (With Patrick Phelps [Phd Student]) Spatial decay of discretely self-similar solutions to the Navier-Stokes equations, Pure and Applied Analysis (2023)
  9. (With Dallas Albritton) Remarks on sparseness and regularity of Navier-Stokes solutions. Nonlinearity (2023)
  10. (With Igor Kukavica and Wojciech Ozanski) Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space, Arch. Ration. Mech. Anal. (2022)
  11. (With Igor Kukavica and Tai-Peng Tsai) Existence of global weak solutions to the Navier-Stokes equations in weighted spaces. Indiana Univ. Math. J. (2022)
  12. (With Dallas Albritton) Non-decaying solutions to the critical surface quasi-geostrophic equations with symmetries, Trans. Amer. Math. Soc. (2022)
  13. (With Tai-Peng Tsai) On the local pressure expansion for the Navier-Stokes equations, J. Math. Fluid Mech. (2022)
  14. (With Animikh Biswas and Michael Jolly) Data Assimilation for the Navier–Stokes Equations Using Local Observables. SIAM J. Appl. Dyn. Syst. (2021)
  15. Local energy solutions to the Navier-Stokes equations in Wiener amalgam spaces. SIAM J. Math. Anal. (2021)
  16. (With Tai-Peng Tsai) Global existence, regularity and uniqueness of infinite energy solutions to the Navier-Stokes equations. Comm. Partial Differential Equations (2020)
  17. (With Igor Kukavica) Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data. J. Math. Fluid Mech. (2020)
  18. (With Tai-Peng Tsai) Discretely self-similar solutions to the Navier-Stokes equations with data in L^2_{loc} satisfying the local energy inequality, Anal. PDE (2019)
  19. (With Aseel Farhat and Zoran Gruji\’c) An algebraic reduction of the `scaling gap’ in the Navier-Stokes regularity problem. Arch. Ration. Mech. Anal. (2019)
  20. (With Tai-Peng Tsai) Self-similar solutions to the Navier-Stokes equations with Besov space data. Arch. Ration. Mech. Anal (2018)
  21. (With Tai-Peng Tsai) Self-similar solutions to the Navier-Stokes equations: a survey of recent results. Nonlinear Analysis in Geometry and Applied Mathematics, Harvard CMSA Series in Mathematics, Volume 2, 2018.
  22. (With Tai-Peng Tsai) Rotationally corrected scaling invariant solutions to the Navier-Stokes equations. Comm.~Partial Differential Equations. 42 (2017)
  23. (With Zoran Gruji\’c) Frequency Localized Regularity Criteria for the 3D Navier-Stokes Equations. Arch. Ration. Mech. Anal. 224 (2017)
  24. (With Tai-Peng Tsai) Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations II. Ann. Henri Poincar\’e 18 (2017)
  25. (With Zoran Gruji\’c and Igor Kukavica) Analyticity radii and the Navier-Stokes equations:~recent results and applications. Recent progress in the theory of the Euler and Navier-Stokes equations, 22–36, London Math. Soc. Lecture Note Ser., 430, Cambridge Univ. Press, Cambridge, 2016.
  26. (With Zoran Gruji\’c and Igor Kukavica) Local analyticity radii of solutions to the 3D Navier-Stokes equations with locally analytic forcing. J. Differential Equations 259 (2015)
  27. (With Zoran Gruji\’c) A spatially localized $L \log L$ estimate on the vorticity in the 3D NSE. Indiana Univ. Math. J. 64 (2015)
  28. (With Zoran Gruji\’c) A note on the surface quasi-geostrophic temperature variance cascade. Comm.~Math.~Sci.~13 (2015)
  29. (With Zoran Gruji\’c) Blow-up scenarios for the 3D Navier-Stokes equations exhibiting sub-criticality with respect to the scaling of one-dimensional local sparseness. J. Math. Fluid Mech. 16 (2014)
  30. (With Zoran Gruji\’c) Energy cascades in physical scales of 3D incompressible magnetohydrodynamic turbulence. J. Math. Phys. 54 (2013)
  31. (With Zoran Gruji\’c) On the transport and concentration of enstrophy in 3D magnetohydrodynamic turbulence. Nonlinearity 26 (2013)
  32. Geometric measure-type regularity criteria for the 3D magnetohydrodynamical system. Nonlinear Anal. 75 (2012)
  33. (With Richard Hammack) Minimum cycle bases of direct products of graphs with cycles. Ars Math. Contemp. 2 (2009)
  34. (With Mohammed Jaradat) Minimum cycle bases for direct products of K2 with complete graphs. Australas. J. Combin. 43 (2009)