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.

On well-posedness of generalized surface quasi-geostrophic equations in borderline Sobolev spaces

Generalized surface quasi-geostrophic equations (gSQG) are a family of active scalar equations that interpolate between the 2D incompressible Euler equations and the surface quasi-geostrophic equations (SQG) and extrapolate beyond SQG to more singular equations. In this talk, we present a collection of results on fractionally dissipative gSQG equations in the most singular regime where the order of dissipation is small relative to the order of the velocity. For this family, we establish well-posedness and smoothing of the solutions in borderline Sobolev spaces. We also discuss corresponding results in the case of a mildly dissipative counterpart where the fractional Laplacian is replaced by a logarithmic Laplacian in the dissipative term. This is based on joint work with M.S Jolly and V. Martinez.

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.

Bifurcations with cyclic symmetries in partial differential equations models in biology and materials science

In the study of pattern forming systems of partial differential equations, the bifurcation structure of the equilibrium solutions serves as an organizing structure of the dynamics. Werner and Spence (1984) developed the theory of symmetry-breaking pitchfork  bifurcation structures for dynamical systems with even and odd symmetries. In recent work with P. Rizzi and T. Wanner, we were able to extend these results to cases with dihedral symmetries, giving a computer-assisted proof of such bifurcations in the case of the Ohta-Kawasaki model for diblock copolymers. In current work with M. Breden and T. Wanner, we extend these results beyond pitchfork bifurcations to symmetry-breaking transcritical bifurcations. Additionally, we extend our set of examples to higher dimensions and also to the Shigesada-Kawasaki-Teramoto model, a partial differential reaction-diffusion system for spatial segregation in the coexistence of two competing species.

Optimal transport and Monge-Ampère type equations in the design of freeform optical surfaces

A freeform optical surface, simply stated, refers to an optical surface (lens or mirror) whose shape lacks rotational symmetry. The use of such surfaces allows generation of complex, compact and highly efficient imaging systems. Mathematically, the design of freeform optical surfaces is an inverse problem that can be studied by using variational technique of optimal transportation theory and nonlinear partial differential equations of Monge-Ampère type. In this talk we will focus on the problem of design of refracting lenses and describe some of the approaches used to solve these problems.

Optimal transport as a transform for scalar conservation laws

We present a framework in which optimal transport provides a nonlinear coordinate system for scalar conservation laws. In one dimension, this is realized through the Cumulative Distribution Transform (CDT), which recasts transport-dominated dynamics into a representation where evolution becomes simpler.

From this perspective, nonlinear solution features, such as translations and dilations, are captured by a low-dimensional structure in transform space. We show that, for one-dimensional conservation laws, the dynamics can be accurately approximated using a small number of transport-based modes, offering an alternative to classical linear representations such as Fourier or Proper Orthogonal Decomposition (POD) expansions.

These results suggest new directions for the analysis of nonlinear PDEs and for the design of efficient reduced-order models tailored to transport-dominated regimes. This is ongoing joint work with the groups of Prof. Gustavo Rohde (University of Virginia) and Prof. Harbir Antil (George Mason University).

Recovering the nonlinearity from the scattering map

We will discuss the problem of recovering an unknown gauge-invariant nonlinearity from the (small-data) scattering map in the setting of nonlinear Schrödinger equations.  After reviewing several results concerning local nonlinearities, we will discuss a recent preliminary result for nonlocal (Hartree-type) nonlinearities.  The talk with cover joint works with L. Campos, G. Chen, R. Killip, and M. Visan.

Convergence of semilinear parabolic flows with general initial data

We study the long-time behavior of solutions to semilinear parabolic equations in Euclidean space that arise as gradient flows of an energy functional J. Under fairly general assumptions, this problem reduces to analyzing the behavior of Palais–Smale sequences (i.e., almost critical points) of J. In this unbounded setting, Palais–Smale sequences are generally non-compact, as they may asymptotically decompose into superpositions of two or more critical points drifting apart to infinity. This phenomenon is commonly referred to as bubbling in the parabolic literature.

In this talk, we present a method to rule out bubbling for gradient flows associated with a certain class of semilinear parabolic equations. Our approach is based on a sharp stability estimate for almost critical points of J, which yields a flexible framework for proving convergence of gradient flows arising from constrained minimization problems.

As applications, we establish convergence for a diffuse model of volume-preserving mean curvature flow, as well as convergence to a unique ground state for a class of semilinear equations within the framework of Berestycki–Lions.

TBA

TBA