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Sept 29

Matthieu Cadiot, École Polytechnique Paris

Overhanging waves in the water wave equations with constant vorticity

Abstract. In this talk, we present a computer-assisted methodology for proving the existence of periodic traveling gravity waves at the free surface of water, in a flow of constant vorticity over a flat bed. Using conformal mappings, the free-boundary problem is reformulated as a quasilinear pseudo-differential equation for a periodic function of a single variable. We then expand the solution in a Fourier series, reducing the problem to an infinite-dimensional system of equations for the Fourier coefficients.

To establish existence, we employ a Newton–Kantorovich type argument, proving the existence of a true solution in a neighborhood of a numerically computed approximation. The verification of the hypotheses of this fixed-point argument relies on a careful combination of analytical estimates and rigorous numerical computations. This approach enables us to rigorously prove the existence of an overhanging wave, that is, wave profiles that are no longer graphs of functions. Moreover, it allows for the construction of branches of solutions parameterized by the relative mass flux, bifurcating from the trivial flat solution. In particular, we rigorously construct a branch that originates at the flat solution and extends to a wave with a self-intersecting profile, passing through overhanging configurations along the way.

Joint work with Susanna Haziot (Princeton).

Oct 13

Florian Schaefer, NYU

Abstract. Shock waves in high-speed gas dynamics cause severe numerical difficulties for classical solvers and scientific machine learning. They are fundamentally a multiscale problem: While viscous effects ensure smoothness on microscopic scales, shocks manifest as macroscopic discontinuities. This talk begins with the observation that shock formation arises from the flow map reaching the boundary of the manifold of diffeomorphisms. We modify its geometry such that geodesics approach but never reach the boundary. The resulting information geometric regularization (IGR) has smooth solutions while avoiding the excessive dissipation of viscous regularizations, accelerating and simplifying the simulation of flows with shocks. We prove the existence of global strong IGR solutions in the unidimensional pressureless case and illustrate its practical utility on multidimensional examples with complex shock interactions. With S. Bryngelson and other collaborators, we use IGR to conduct the first compressible flow simulation exceeding a quadrillion degrees of freedom. The modified geometry of the diffeomorphism manifold is the information geometry of the mass density. The last part of the talk explains how this observation motivates information geometric mechanics that views the solutions of continuum mechanical PDEs as parameters of probability distributions originating from statistical physics. Replacing the Euclidean geometry of individual particles with the information geometry of statistical families promises performant numerical methods that preserve the positivity of densities and energies and readily integrate with scientific machine learning.