Georgia Institute of Technology College of Sciences

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)

This talk concerns the Benjamin-Ono (BO) equation of internal wave theory, and properties of the solution of the Cauchy initial-value problem in the situation that the initial data is fixed but the coefficient of the nonlocal dispersive term in the equation is allowed to tend to zero (i.e., the zero-dispersion limit). It is well-known that existence of a limit requires the weak topology because high-frequency oscillations appear even though they are not present in the initial data.  Physically, this phenomenon corresponds to the generation of a dispersive shock wave. In the setting of the Korteweg-de Vries (KdV) equation, it has been shown that dispersive shock waves exhibit a universal form independent of initial data near the two edges of the dispersive shock wave, and also near the gradient catastrophe point for the inviscid Burgers equation from which the shock wave forms. In this talk, we will present corresponding universality results for the BO equation. These have quite a different character than in the KdV case; while for KdV one has universal wave profiles expressed in terms of solutions of Painlevé-type equations, for BO one instead has expressions in terms of classical Airy functions and Pearcey integrals. These results are proved for general rational initial data using a new approach based on an explicit formula for the solution of the Cauchy problem for BO. This is joint work with Elliot Blackstone and Matthew Mitchell, based on other work with Blackstone, Louise Gassot, and Patrick Gérard.

It has long been conjectured that the Klein-Gordon equation on a Schwarzschild black hole behaves very differently from the wave equation at late-time, due to the presence of stable (timelike) trapping and the manifestation of long-range scattering. We will present our recent resolution of this problem, establishing that, contrary to previous expectations, solutions with sufficiently localized initial data decay polynomially in time. Time permitting, we will explain how the proof uses, at a crucial step, results from analytic number theory for bounding exponential sums.