Schedule, Notes/Handouts & Videos

In these three lectures I will give an overview of the DiPerna-Lions theory for transport equations and ODEs with Sobolev coefficients, including the developments of the last two decades, namely Ambrosio's extension to the case of BV coefficients, Lagrangian estimates and Bressan's mixing conjecture, near incompressibility of the flow versus bounds on the divergence of the field, the role of summability and the impact of convex integration techniques. I will try to cover the general ideas of several results, without going into the technical details, and I will give a special emphasis on open problems.

In these three lectures I will give an overview of the DiPerna-Lions theory for transport equations and ODEs with Sobolev coefficients, including the developments of the last two decades, namely Ambrosio's extension to the case of BV coefficients, Lagrangian estimates and Bressan's mixing conjecture, near incompressibility of the flow versus bounds on the divergence of the field, the role of summability and the impact of convex integration techniques. I will try to cover the general ideas of several results, without going into the technical details, and I will give a special emphasis on open problems.

In this talk we will introduce some concepts of random dynamical systems and stochastic PDEs which make it possible to study the prove that Lagrangian trajectories in a variety of stochastically forced fluid equations are chaotic and lead to exponentially fast mixing of any passive scalar transported by the flow. We will briefly discuss how to use these facts to make mathematiclaly rigorous some of the classical predictions of the statistical theory of "passive scalar turbulence" in a certain, especially simple regime. These predictions were made in the 1940s and 1950s and observed to be consistent with experimental evidence, but previously lacked any mathematical justification. The work discussed is all joint with Alex Blumenthal and Sam Punshon-Smith. 

In these three lectures I will give an overview of the DiPerna-Lions theory for transport equations and ODEs with Sobolev coefficients, including the developments of the last two decades, namely Ambrosio's extension to the case of BV coefficients, Lagrangian estimates and Bressan's mixing conjecture, near incompressibility of the flow versus bounds on the divergence of the field, the role of summability and the impact of convex integration techniques. I will try to cover the general ideas of several results, without going into the technical details, and I will give a special emphasis on open problems.

For any positive regularity parameter $\beta < 1/2$, we construct infinitely many weak solutions of the 3D incompressible Euler equations on the periodic box, which lie in $C^0_t H^\beta_x$. 

In particular, these solutions may be taken to have an $L^2$-based regularity index strictly larger than $1/3$, thus deviating from the scaling of the Kolmogorov-Obhukov $5/3$ power spectrum in the inertial range.

This is a joint work with T. Buckmaster, N. Masmoudi, and M. Novack. 

We will give a broad overview of the problem of finite-time singularity in the 3d Euler equation. We will highlight some of the main difficulties in the problem and different ways to overcome them.

The water wave equations describe the motion of the free surface of an incompressible, irrotational fluid. The aim of these lectures will be to provide an introduction to this research direction, and to outline some of the most interesting developments during the last several years.

The water wave equations describe the motion of the free surface of an incompressible, irrotational fluid. The aim of these lectures will be to provide an introduction to this research direction, and to outline some of the most interesting developments during the last several years.

In this talk I will explain some of the basic ideas involved in proving the well-posedness of the Cauchy problem of
a class of fluid free boundary problems. I will then present a result on the wellposedness of a barotropic
fluid free boundary problem in Minkowski spacetime. This is a joint work with Shuang Miao and Sohrab Shahshahani.

The water wave equations describe the motion of the free surface of an incompressible, irrotational fluid. The aim of these lectures will be to provide an introduction to this research direction, and to outline some of the most interesting developments during the last several years.

Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. We first review the numerical evidence of finite time singularity for 3D axisymmetric Euler equations by Luo and Hou. The singularity is a ring like singularity that occurs at a stagnation point in the symmetry plane located at the boundary of the cylinder. We then present a novel method of analysis and prove that the 1D HL model and the original De Gregorio model develop finite time self-similar singularity. We will also report our recent progress in analyzing the finite time singularity of the axisymmetric 3D Euler equations with initial data considered by Luo and Hou. Finally, we present some recent numerical results on singularity formation of the 3D axisymmetric Euler equation along the symmetry axis.