The proposed topic of the grant was two-fold; the study of anomalous dissipation in hydrodynamic models (Onsager's conjecture) and of non-uniqueness of orbits in non-smooth dynamical systems (spontaneous stochasticity).

In the first direction, the PI established sharp (as demonstrated by exact shock solutions) criteria for anomalous entropy production in the fully compressible Euler/Navier-Stokes system. The PI then extended the framework used to study Onsager's conjecture in order to allow for weak dependence of the dissipation on Reynolds number and accommodate the effects of physical boundaries.  Finally, in an effort to understand drag reduction by polymers, the PI derived a model which incorporated the effect of polymeric interaction at the wall and rigorously showed how it can mollify strong dissipation events. 

In the second direction, at the intersection of both themes, the PI established novel Lagrangian representation formulae for the dissipation anomaly involving relative tracer particle dispersion forward and backward in time. The PI also introduced a set of singular ODE models in arbitrary dimensions for which non-uniqueness can be studied in-depth and connected to "hidden" chaotic properties of the system. 

In addition to this work, the PI collaborated with other researchers at Princeton in a research program on plasma confinement fusion.  In this direction, the PI worked towards the understanding flexibility and rigidity properties of equilibrium configurations in ideal hydrodynamic models. Additional work was performed towards understanding stochastic fluid models, Arnold's geometric picture of fluid dynamics, and the dynamics of one-dimensional compressible fluid systems.

The work conducted during the tenure of this grant resulted in sixteen publications in top journals. Additionally, the work was disseminated at twenty-four seminars and colloquia talks, and fifteen conference talks. Additional broader impacts included: (1) Designed and taught a graduate course  "Introduction to Stochastic Differential Equations with Applications to Fluid Dynamics" for Brazilian summer school short course (Jan. 2019) on the second theme of the proposal, (2) co-organized the "Analysis of Fluids and Related Topics", a weekly seminar series at Princeton University (2017–2020) and (3) co-organized the special session "Regularity, Singularity and Turbulence in Fluids", SIAM Conference on Analysis of Partial Differential Equations, La Quinta, CA (Dec. 2019).

Last Modified: 08/12/2021
Modified by: Theodore D Drivas