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MATH 745 [Fall 2025] »

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Research Activities

[Emerging Research Directions]

  • Applications of the Riemann-Hilbert Problem in Fluid Mechanics
  • Computational Nonequilibrium Thermodynamics
  • Bayesian Inversion and Uncertainty Quantification

    • [Main Research Themes (past & present)]

    • Systematic Search for Extreme and Singular Behavior in Hydrodynamics -- This research program, summarized in the review paper Protas (2022), has opened a "computational window" into some basic questions of the mathematical fluid dynamics. The main problem motivating it concerns the possibility of finite-time blow-up in solutions of the 3D Navier-Stokes and Euler systems, an issue recognized by the Clay Mathematics Institute as one of its "Millennium Problems". Related questions also arise in the context of globally well-posed systems where the key issue has to do with the most "extreme" behaviour, as measured by the growth of suitable norms, that can be realized by solutions and how it compares to rigorous a priori estimates. To address such questions we have developed a versatile framework wherein the most singular solutions are sought systematically by numerically solving suitably-defined variational optimization problems. Although even in such worst-case scenarios we have found no evidence of singularity formation in Navier-Stokes flows, this allowed us to characterize flow evolutions realizing the most extreme behavior (Kang & Protas, 2022; Kang, Yun & Protas, 2020). On the other hand, such search did yield solutions of the 3D Euler system that appear to form a singularity in finite time (Zhao & Protas, 2023). While computational studies of potential singularity formation have had a long history in hydrodynamics, what sets our approach apart is that we methodically search for such elusive solutions using optimization methods. In Matharu, Yoneda & Protas (2022) we considered the question of the dissipation anomaly in 2D Navier-Stokes flows and showed that solutions found by maximizing the enstrophy dissipation saturate the best rigorous estimates describing how this quantity depends on viscosity. This showed these estimates are sharp and cannot be fundamentally improved, thereby establishing a first of its kind bridge between the rigorous PDE analysis and physics. Our earlier work on the extreme growth in the 1D Burgers system (Ayala & Protas, 2011) inspired others to seek to refine the corresponding a priori estimates, with a breakthrough achieved only in 2023 when Albritton & De Nitti finally closed the gap. We made a good progress on these and related problems in the course of the semester-long thematic program on "Mathematical aspects of turbulence: where do we stand?" at the Newton Institute in Cambridge, UK, in 2022.

    • Existence and Stability of Vortex Equilibria -- Problems of vortex dynamics are among the most classical in fluid mechanics and have been studied for a long time. My research has offered a new perspective on a number of basic questions concerning the existence and stability of vortex equilibria in inviscid flows. The main difficulty is that these problems are often of the free-boundary type and to address this issue we developed a general framework based on modern tools from differential geometry known as the "shape calculus". This approach was applied to resolve long-standing open problems concerning the stability of Hill's and Norbury's vortices with respect to axisymmetric perturbations (Protas, 2019; Protas & Elcrat, 2016). Interesting insights into vortex dynamics in 2D can be obtained using methods of complex analysis and in Protas & Sakajo (2020) we established a new connection between relative equilibria of vortex sheets, defined as vorticity concentrated on planar curves, and the Riemann-Hilbert problem. It led to the discovery of a new family of equilibria which has since come to be known as the "Protas-Sakajo class", with other groups already working to generalize these solutions and using them as benchmarks. The presence of endpoints complicates the problem and the question of the linear stability of finite vortex sheets was taken up in Protas, Llewellyn Smith & Sakajo (2021) where, remarkably, closed-form solutions were obtained with an analogous structure as in the classical spatially periodic case. More recently we focused on problems in which instabilities of vortex equilibria are controlled by an interplay of the discrete and essential spectrum of the linearized Euler operator and therefore cannot be understood without properly accounting for the infinite-dimensional nature of the problem (Protas, 2024; Zhao, Protas & Shvydkoy, 2024). In particular, this allowed us to uncover a new mechanism of a linear instability of the 2D Taylor-Green vortex where an exponential growth is realized by perturbations other than eigenfunctions. Some of these developments occurred during the semester-long thematic program on Applied and Computational Complex Analysis I co-organized at the Newton Institute in 2019. I am among the organizers of a follow-up thematic program scheduled to take place at ICERM in fall 2027.

    • Optimal Closures for PDE Models -- In this effort we have made contributions to a problem important when modelling real-life systems, namely, the question of using data to derive suitable "closures" for PDE models. They can have the form of unknown constitutive relations appearing in conservation equations or "inertial manifolds" mapping unresolved degrees of freedom to the resolved ones in simplified models. As an alternative to recently popular methods based on machine learning, our approach uses techniques of the calculus of variations allowing one to infer the closures in the continuous (infinite-dimensional) setting subject to only minimum assumptions. As regards the first class of applications, this approach has was employed to solve inverse problems in electrochemitry (Ahmadi et al., 2025; Daniels et al., 2023) where key characteristics of battery electrodes were inferred from actual measurement data provided by our chemistry collaborators. As the second class of applications, we considered the question of deducing optimal turbulence closures in Large Eddy Simulations which are filtered versions of the Navier-Stokes system (Matharu & Protas, 2020, 2022). Obtaining such closures remains an open problem in fluid mechanics and our approach makes it possible to find key elements of closures optimally, bringing a degree of mathematical rigour to a field that has traditionally been dominated by empiricism. This, in turn, allows one to establish fundamental performance limitations of different classes of closure models (Matharu & Protas, 2022). These techniques are currently used by researchers at the University of Calgary to improve reduced-order models of turbulent flows of engineering interest.

    • Numerical Optimization of Partial Differential Equations -- The advances reported above would not have been possible without continued research aiming to improve methods for the numerical solution of PDE optimization problems. Most of the problems described above are Riemannian, in the sense that optimization is performed subject to constraints defining manifolds in the respective function spaces. Efficient numerical methods to solve large-scale optimization problems of this type were developed based on techniques of differential geometry in Zhao, Protas & Shvydkoy (2024), Zhao & Protas (2023), Kang & Protas (2022), Kang, Yun & Protas (2020) and were instrumental in the success of these studies. As a key enabler, gradients of the objective functional are conveniently computed by solving a suitably-defined adjoint system and in Matharu & Protas (2024) we extended this approach so that it can be used to enforce complicated state constraints. I have been invited to present graduate-level tutorials on the topic of PDE optimization, most recently at the Newton Institute in Cambridge in January 2022 and at IIT Hyderabad in India in April 2025.

    • Mathematical Modelling in Electrochemistry -- Fueled by advances made through my basic research, these studies have focused on applied problems arising in industry such as modeling mechanical processes and transport of charges in materials used in modern Li-ion batteries. A hallmark of this research is elegant integration of experimental data with mathematical modeling, asymptotic analysis and scientific computation, which made it possible for us to obtain some of the first systematic solutions of inverse problems in electrochemistry featuring careful quantification of uncertainties (Morales Escalante et al., 2020, 2021; Sethurajan et al., 2019a). These advances in turn allowed us to obtain new insights about the validity of key mathematical models used in this field (Morales Escalante et al., 2020; Sethurajan et al., 2019b). This interdisciplinary research involved close collaboration with chemists and engineers who typically formulated the research problems and provided data. The results were used by the industrial partners (General Motors, Pulsenics) in their modelling efforts.

    [Current Research Collaborations]

    • Miguel Bustamante (School of Mathematics and Statistics, University College Dublin, Ireland)
    • S. Jon Chapman (Mathematical Institute University of Oxford, UK)
    • Ionut Danaila (Laboratoire de Math�matiques Rapha�l Salem, Universit� de Rouen, France)
    • Jamie Foster (School of Mathematics & Physics, University of Portsmouth, UK)
    • Gillian Goward (Department of Chemistry & Chemical Biology, McMaster University, Canada)
    • Nicholas Kevlahan (Department of Mathematics and Statistics, McMaster University, Canada)
    • Stefan Llewellyn Smith (Department of Mechanical and Aerospace Engineering at the University of California, San Diego, USA)
    • Arian Novruzi (Department of Mathematics and Statistics, University of Ottawa, Canada)
    • Takashi Sakajo (Department of Mathematics, Kyoto University, Japan)
    • Roman Shvydkoy (Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, USA)
    • Tsuyoshi Yoneda (Graduate School of Economics, Hitotsubashi University, Japan)

    [Research Funding]