My current interests run along two lines: 1. rational continuum mechanics, 2. stochastic analysis.
More specifically, goal to the first line is to rigorously derive continuum level equations of hydrodynamics (and thermodynamics next) through first principles. I take an approach that relies upon mathematical tools such as Hamilton-Jacobi theory in calculus of variations, optimal control and PDEs; averaging theory for Hamiltonian systems; Stochastics; Mass transport theory and first order calculus in metric spaces. Regarding the second line, I am interested in a number of singular stochastic PDEs with only first order derivatives (hence no regularization effects in the usual sense) or with totally non-linearity in the equation (hence regularization effect, if any, is expressed in more subtle ways). These are usually equations modeling physical systems under random influences. In recent years, I entered a very critical stage of development for the first topic, hence have not been able to spend enough time with the second one.
A common theme in all these works is that I explore the maximum principle in either direct or hidden abstract ways.
Three lines of significant works in past research:
- Developed a Hamilton-Jacobi method for large deviation of metric space valued Markov process [Feng-Kurtz 2006].
- Introduced an idea of formulating Hamilton-Jacobi in the space of probability measures, and developed well-posedness theories for a class of such equations [Chapters 9.1, 9.4 and 13.3 of Feng-Kurtz 2006, Feng-Katsoulakis 2009 which was written earlier and explained more in detail in Feng-Nguyen 2012 (J. of Math. Pures Appl.), and Ambrosio-Feng 2014 for models directly applicable to variational formulation in continuum mechanics].
- Found a successful implementation of renormalized solution idea from deterministic nonlinear PDE theory to stochastic settings, illustrated through stochastic scalar conservation laws. Such approach has a satisfactory well-posedness theory [Feng-Nualart 2008].
- A book: Large Deviations for Stochastic Processes (with T.G. Kurtz), Mathematical Surveys and Monographs Vol. No. 131, American Mathematical Society, (2006) 410 pages. Large deviations from a nonlinear semigroup convergence and variational point of view. Starting from martingale-problem formulation of Markov processes and viscosity solution for Hamilton-Jacobi equations, this book develops both stochastic as well as PDE methods in generic metric spaces which have potential to solve concrete problems.
Part one (on Large deviation) of June 2015 Marc Kac seminar on Probability and Physics in Utrecht, the Netherlands.
Part two (on Hamilton-Jacobi equations) of the Kac seminar.
- A Hamilton-Jacobi PDE associated with hydrodynamic fluctuations from a nonlinear diffusion Arxiv: 1903.00052 (with T. Mikami and J. Zimmer), Communications in Mathematical Physics. Vol. 385, (2021), 1-54. https://doi.org/10.1007/s00220-021-04110-1 Another title of this can be large deviation for diffusive scaling limit of the stochastic Carleman models. Section 4 of this work introduces a new method for the averaging step of stochastic hydrodynamic derivations. The new method is most naturally viewed from perspective of Hamiltonian dynamical systems through the weak KAM theory.
- On a class of first order Hamilton-Jacobi equations in metric spaces (with L. Ambrosio), J. Diff. Equations. Vol. 256, No. 7. (2014), 2194-2245. For certain class of equations, it is possible to define viscosity solution and get well-posedness theory for Hamilton-Jacobi PDE in generic geodesic metric spaces (free of any curvature assumption). The results are applied in Section 4 to understand equations arising from action minimization formulation of compressible flows (in this case the space is non-negatively curved in Alexandrov sense).
- A comparison principle for Hamilton-Jacobi equations related to controlled gradient flows in infinite dimensions (with M. Katsoulakis), Archive for Rational Mechanics and Analysis, Vol. 192 (2009), 275-310. This paper showed that we can develop a uniqueness viscosity theory for Hamilton-Jacobi equations in “very bad” spaces such as the space of probability measures. Such space does not have Radon-Nikodyn property as previous literature pioneered by Crandall-Lions required. The new method relies upon modern mass transport theory. Illustrations using three examples gradually move from good (Hilbert spaces) to the very bad spaces ( probability measures). The examples show how a physically motivated structure regarding fluctuations and entropy-entropy dissipation relation induced some inequalities, making the above claims became theorems.
- Stochastic scalar conservation laws (with D. Nualart), Journal of Functional Analysis, Vol. 55, No. 2 (2008), 313-373. Shows that Kruzkov’s entropy solution (and later renormalized solution) ideas have a complete analogy in stochastic situations once the stochastic equations are properly re-formulated. A companion stochastic compensated-compactness method is also developed.
- Large deviation for diffusions and Hamilton-Jacobi equation in Hilbert spaces, The Annals of Probability, Vol. 34, No.1 (2006), 321-385. The paper treats models of stochastic evolution equations with examples including stochastic Allen-Cahn equation. An earlier method by Tataru-Crandall-Lions is adapted to ensure uniqueness of the corresponding Hamilton-Jacobi equations in Hilbert spaces.
- Large deviation for stochastic Cahn-Hilliard equation, Methods of Functional Analysis and Topology, Vol. 9, No. 4 (2003), 333-356. An illustration of the Hamilton-Jacobi approach to large deviation applied to stochastic Cahn-Hilliard equation, with a detailed treatment on uniqueness theory for Hamilton-Jacobi PDEs with field-valued variables modeled in Hilbert space H-1 .
- Hydrodynamic limit derivations of asymptotically non-interacting deterministic particles through Hamilton-Jacobi theory. An outline of the program is announced in a four-page publication in Oberwolfach Reports Vol. 17, Issue 2/3, 2020. DOI: 10.4171/OWR/2020/29
- A change of coordinate methods for Hamilton-Jacobi equations in geodesic metric spaces.