Diffusive Regularization in Kinetic and Fluid Equations

Kinetic equations form the mathematical basis for modeling and understanding high-energy gases with large-scale interactions and are used to predict the motion and radiation of plasmas, e.g., in industry and astronomy, as well as fluid flows at high speed and low density, for example, supersonic flows. Despite their complexity, these models often see manifestations of the second law of thermodynamics, which push the gas or fluid towards a state of maximum entropy, which is, statistically, easier to predict.

This project explores the finer details that determine whether such models remain in a chaotic regime, manifesting as turbulence, shocks, and plasma echoes, or thermalize, becoming smoother and converging to an equilibrium. These phenomena are explored in two main contexts: in the regularity properties, continuation criteria, potential shock formation of the Boltzmann and Landau equations, and versions with large-scale electromagnetic interactions; and in the enhanced diffusivity of fluid equations where effective viscosity grows with local turbulence, a family of models originated by Kolmogorov and used in oceanography.

The project also provides training and research opportunities for graduate, undergraduate, and high school students.

This research examines the construction and implementation of novel regularizing mechanisms in two important contexts. First, the investigator will apply their recent discoveries in kinetic mass spreading to probe the current frontier of the regularity program for the Boltzmann and Landau equations. For these models of high-energy gases and plasmas, the collision interaction is known to behave roughly like a fractional Laplacian operator with highly nonlocal and possibly degenerate coefficients.

These intricacies are major impediments to the well-posedness theory. Nevertheless, the current state-of-the-art grants that smooth unique solutions exist for as long as certain macroscopic quantities remain under control a priori. The investigator's recent work establishes that half of these quantities are in fact controlled dynamically, yielding more precise estimates for the solution.

This project extends these results to wider scopes, domains with boundary, rotationally symmetric configurations, and settings with electromagnetic interactions, and pairs them with existing estimates from the regularity theory for fluid equations. Second, the project will investigate novel a priori bounds that can be derived from non-isothermal fluid equations where the local temperature influences the viscosity.

The investigator's prior work has demonstrated a unique mechanism for enhanced dissipation arising from thermal viscosity and in developing maximum principles for coupled non-isothermal models. These effects are examined in the Navier-Stokes-Fourier system and in models of porous media type and of turbulent dissipation.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.