Jin Feng

Jin Feng standing in front of round metal object

 

Research

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:

  1. Developed a Hamilton-Jacobi method for large deviation of metric space valued Markov process [Feng-Kurtz 2006].
  2. 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].
  3. 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].

Current focus

  • 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.