Georgia Institute of Technology College of Sciences

The Guderley problem describes the behavior of a strong self-similar shock wave propagating radially in an ideal gas. A spherical shock converges radially inwards to the spatial origin, strengthening as it collapses. At the collapse point, the shock's strength becomes infinite, leading to the formation of a new outgoing shock wave of finite strength, which then propagates outwards to infinity. 

In this talk, I will present recent work on the rigorous construction of the self-similar converging and diverging shock solutions for $\gamma \in (1,3]$. These solutions are analytic away from the shock interfaces and the blow-up point. The proof relies on continuity arguments, nonlinear invariances, and barrier functions.

We study causal optimal transport problems with Markovian cost and prescribed Markovian marginal laws. We show that the associated value function solves a fully nonlinear parabolic PDE, for which we establish a comparison principle and, consequently, uniqueness of its viscosity solution. This PDE characterization allows us to identify the value with that of a constrained version of the control problem for the Kushner–Stratonovich equation. We also obtain a third equivalent optimal control formulation with a state constraint, which leads to implementable numerical schemes for causal optimal transport. This is joint ongoing work with Julio Backhoff, Erhan Bayraktar, and Antonios Zitridis.

Fluid models are used to make predictions about critical real-world systems arising in diverse fields including but not limited to meteorology, climate science, mechanical engineering, and geophysics. Simulations based on fluid models can, for example, be used to make predictions about the strength of a tornado or the stresses on an aircraft wing passing through turbulent air. The possibility that a mathematical model does not capture the full range of possible real-world scenarios is concerning if the predictions do not account for extreme events. It has been confirmed by computer assisted proof that the 3D Navier-Stokes equations possess non-unique solutions. The existence of such solutions can, in principle, pose a challenge to forecasters. This talk explores mathematical work aiming to quantify the rate at which non-unique solutions can separate.

Hydrodynamic bores are front-type traveling wave solutions to the two-layer free boundary Euler equations in two dimensions. We  prove that there exists a family of bores that starts at trivial laminar flow where the interface is flat and continues until the interface develops a vertical tangent. This type of behavior was first observed over 45 years ago in computations of internal gravity waves and gravity water waves with vorticity via numerical continuation. Despite considerable progress over the past decade in constructing global families of water waves that potentially overturn, a rigorous proof that overturning definitively occurs has been stubbornly elusive.  

Gravity currents arise when a heavier fluid intrudes into a region of lighter fluid. Typical examples are muddy water flowing into a cleaner body of water and haboobs (dust storms). We give the first rigorous proof of a conjecture of von Kármán on the structure of gravity currents near the rigid boundary. 

This is joint work with Ming Chen (Pittsburgh) and Miles Wheeler (Bath)