Diffusion and Regularity

The investigator will study evolutionary equations which exhibit some sort of diffusive behavior, understood in a broad sense. The most classical diffusion equation is the heat equation, describing the evolution of the distribution of temperature in some homogeneous media. One remarkable characteristic of the heat equation is that, even for very irregular initial data, the solutions become immediately smooth.

This behavior is observed in many other partial differential equations that are classified as "parabolic". In general, the regularization effect of the equation plays a key role in our understanding of the nature of solutions and in obtaining a priori bounds for them. The investigator seeks to understand the regularization effect in equations whose diffusion appears in a nonstandard form.

For example, in equations with integral diffusion, like the Boltzmann equation from statistical mechanics, in fully nonlinear parabolic equations, or in entropy solutions of inviscid conservation laws. The project provides research training opportunities for graduate students.

The study of integro-differential equations has been a very active area of research in the last twenty years. The principal investigator played a central role in the development of regularity results. The equations are motivated, for the most part, from probabilistic models involving jump processes.

More recently, the methods are also finding applications in deterministic models. The investigator has been studying the Boltzmann equation from this perspective. It is a model from statistical mechanics describing the evolution of the density of particles in a dilute gas.

When the particles are assumed to repel each other in small scales by a power law potential, the equation exhibits a subtle regularization effect, driven by a nonlinear integro-differential diffusion that acts like an elliptic operator of fractional order. In the most singular scenario, the Boltzmann equation turns into the Landau equation, which is a kinetic equation with a more classical second order diffusion.

The principal investigator is also interested in studying more general and abstract fully nonlinear parabolic equations, in order to understand their regularity and possible singularity formation. Even in conservation law equations, without any explicit diffusion term, there is a subtle regularization effect in low order Sobolev spaces that can be arguably understood as a remnant of the vanishing viscosity approximation.

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.