Potomac Region
PDE Seminar

Variational principle and Lagrangian formulations of hydrodynamic equations

The seminal work by Arnold and Ebin-Marsden back in 60-70s uncovered the geodesic interpretation of incompressible Euler equation. This geometric framework has since been extensively developed, and the variational nature of inviscid incompressible hydrodynamic models are now well understood. However, existing frame work fails to extend to viscous hydrodynamics. Based on Hamilton-Pontryagin action principle in geometric mechanics, we developed a framework to realize many viscous hydrodynamic models as critical points of stochastic action functionals. This variational principle also echoes Constantin-Iyer's stochastic Lagrangian formulation of Navier-Stokes equation. We'll also discuss analysis of local well-posedness and Lagrangian analyticity of fluid PDEs in this Lagrangian framework. This talk is based on joint work with A.Mazzucato.

Dispersive shock waves in the discrete nonlinear Schrödinger equation

In conservative media, the dispersive regularization of gradient catastrophe gives rise to dispersive shock waves (DSWs). Unlike classical viscous shocks, a DSW is a highly oscillatory nonlinear wavetrain whose leading edge propagates faster than the long-wave speed, while the entire structure expands over time. A powerful framework for describing DSWs is Whitham modulation theory (WMT), a nonlinear WKB-type approach that captures the slow evolution of wave parameters such as amplitude, wavelength, and frequency.

In this talk, we study DSWs in the one-dimensional discrete, defocusing nonlinear Schrödinger equation (DNLS), with a particular focus on strongly discrete regimes approaching the anti-continuum limit (ACL), as well as intermediate regimes bridging the ACL and the continuum limit. Using WMT in combination with asymptotic reductions, we analyze the long-time evolution of step initial data and elucidate how lattice-induced dispersion alters shock structure. Our analysis reveals a sharp discretization threshold beyond which continuum DSW dynamics are recovered, as well as a rich variety of intermediate shock morphologies unique to the discrete setting. Finally, we apply these results to shock wave formation in ultracold atomic gases confined in optical lattices, within the framework of the tight-binding approximation.

Non-minimizing and min-max solutions to Bernoulli problems

Bernoulli type free boundary problems have a well-developed existence and regularity theory. Much of this, however, is restricted to the case of minimizers of the natural energy (the Alt-Caffarelli functional). I will describe a compactness and regularity theorem that applies to any critical point instead, based on a nonlinear frequency formula and Naber-Valtorta estimates. Then I will explain, via an example involving gravity water waves, how to use this theorem to find min-max type (mountain pass) solutions. This is based on joint work with Georg Weiss.

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Global well-posedness of the stochastic Abelian-Higgs equations in two dimensions

There has been much recent progress on the local solution theory for geometric singular SPDEs. However, the global theory is still largely open. In this talk, we discuss the global well-posedness of the stochastic Abelian-Higgs model in two dimension, which is a geometric singular SPDE arising from gauge theory. The proof is based on a new covariant approach, which consists of two parts: First, we introduce covariant stochastic objects, which are controlled using covariant heat kernel estimates. Second, we control nonlinear remainders using a covariant monotonicity formula, which is inspired by earlier work of Hamilton. 

This is joint work with S. Cao.

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