Spring 2024 Schedule
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FROM EULER FLOWS WITH FRICTION TO GRADIENT FLOWS
I will review some works on the high-friction limit (or small mass approximation) from Euler flows to advection-diffusion systems that are gradient flows, and related asymptotic problems in fluid me- chanics. The formulation at an abstract level exploits the variational structure of compressible Euler flows and is connected to the interpreta- tion of nonlinear Fokker-Planck systems as gradient flows in Wasserstein distance. The technical tool is relative entropy formula for abstract Eu- ler flows induced by the variational structure. Examples that fit into the framework is the emergence of porous media as the high friction limit from the compressible Euler equations with friction, or the Cahn- Hilliard as a limit of the Euler-Korteweg system. Others examples in- clude the high-friction limit form bipolar Euler-Poisson models to the bipolar drift-diffusion equations, as well as other asymptotic limit prob- lems for electromechanical models like the zero-electron mass limit in plasmas. Finally, I discuss high-friction limits in multicomponent sys- tems and the emergence of the Maxwell-Stefan diffusion system from systems describing multicomponent flows of compressible gases.
1:00 pm, On Zoom
Extreme Superposition: Models for Large-Amplitude Rogue Waves
Rogue waves, often referred to as freak waves or monster waves, are spatially localized disturbances of a background field that are also temporally localized, making them appear seemingly out of nowhere. These waves have long captured the attention of seafarers and scientists alike due to their unpredictable and destructive nature. Focusing nonlinear Schrödinger equation serves as a universal model for the amplitude of a wave packet in a general one-dimensional weakly-nonlinear and strongly-dispersive setting that includes water waves and nonlinear optics as special cases. In the setting of this model, a special exact solution exhibiting rogue-wave character was found by D. H. Peregrine in 1983, and since then, Peregrine’s solution has been generalized to a family of solutions of arbitrary “order”, involving more parameters as the order increases. These parameters can be adjusted to obtain rogue wave solutions of maximal amplitude for a given order. In this colloquium, we will describe several recent results concerning such maximal-amplitude rogue wave solutions in the asymptotic regime of large order, which effectively is a large-amplitude regime. For instance, it turns out that there is a limiting structure in a suitable near-field scaling of the peak of the rogue wave and this structure is a novel exact solution of the focusing nonlinear Schrödinger equation — the “rogue wave of infinite order” — that is also connected with the hierarchy of the third Painlevé equation. This is joint work with Liming Ling and Peter D. Miller.
1:00 pm, On Zoom
From Nonlinear Optics to Ultra-Cold Atomic Physics and Rogue Waves: Adventures in Applied and Computational Mathematics
Complex systems are ubiquitous in nature and human-designed environments. The overarching goal of our research is to leverage advanced computational methods with fundamental theoretical analysis to model the nonlinear behavior of systems that are not otherwise amenable to integrable systems techniques. Examples include: Studies of superfluidity and superconductivity in ultra-cold atomic physics (e.g., Bose-Einstein condensation), extreme and rare events (e.g., tsunamis and rogue waves), and collapse phenomena in optics (e.g., light propagation through a medium without diffraction). We have developed computational methods for bifurcation analysis that explain the structure of the parameter space of these systems and continuation methods (pseudo-arclength and Deflated Continuation Method (DCM)) for efficient tracking of solution branches and connecting them to physical observations. The objective is to enable technological innovations, such as the discovery of new materials and development of devices for precision measurements (e.g., interferometers), or to predict extreme phenomena based on the features of the eigenvalue spectra of the system. Inconspicuous solutions of the Nonlinear Schrödinger (NLS) equation were discovered by developing DCM specifically for NLS to uncover previously unknown behavior and weakly nonlinear unstable solutions that are potential targets for experimental verification. Furthermore, a novel Kuznetsov-Ma breather (time-periodic) solution to the discrete and non-integrable NLS equation relevant to predicting periodic extreme and rare events in optical systems was discovered by employing pseudo-arclength continuation. The combination of perturbation methods with pseudo-arclength continuation enabled the elucidation of collapsing waveforms associated with the NLS. Future research will focus on the development of computational tools for numerical simulation of complex nonlinear systems with the ultimate goal being the dissemination of an open-source library that can be used to study bifurcations and perform stability analysis of such systems.
1:00 pm, 306 Snow Hall
Water waves: Theory, computation and applications
Surface water waves of significant interest such as tsunamis, solitary waves and undular bores are nonlinear and dispersive waves. Unluckily, the equations describing the propagation of surface water waves known as Euler’s equations are immensely hard to solve. For this reason, several simplified systems of PDEs have been proposed as alternative approximations to Euler’s equations. In this presentation we review the theoretical properties of such systems. We show that only some of the asymptotically derived systems obey to the laws of mathematics and physics while there is only one with complete mathematical theory for physically sound initial-boundary value problems. We also discuss the numerical modeling of such problems. In particular, we focus on Galerkin / Finite element methods, which is a class of high-order methods that has been proved convergent to certain initial-boundary value problems of physical interest (perhaps the only one). We close this presentation with conditions for the existence of special solutions and validation with laboratory data.
1:00 pm, On Zoom
Optimal Control of Moving Sets
We consider a controlled reaction-diffusion equation, modeling the spreading of an invasive population, and derive a simpler model describing the controlled evolution of a contaminated set. We first analyze the optimal control of 1-dimensional traveling wave profiles. Using Stokes' formula, explicit solutions are obtained, which in some cases require measure-valued optimal controls. Then we introduce a family of optimization problems for a moving set and show how these can be derived from the original parabolic problems, by taking a sharp interface limit. Assuming that the initial contaminated set is convex, we prove that an eradication strategy is optimal if and only if at each given time the control is active along the portion of the boundary where the curvature is maximal. This is a joint work with Stefano Bianchini, Alberto Bressan and Najmeh Salehi.
1:00 pm, 306 Snow Hall