NSF-FRG Conference on Fluids and Computer Assisted Proofs

Anima Anandkumar: AI Accelerating Science: Neural Operators for Learning on Function Spaces

Abstract : I will present exciting developments in the use of AI for scientific applications such as solving PDEs. Neural operators yield 4-5 orders of magnitude speedups over traditional simulations. They learn mappings between function spaces that makes them ideal for capturing multi-scale processes. Our recent work shows that both theoretically and empirically they are superior to standard approaches of closure modeling for learning turbulent flows and other chaotic systems.

Matthieu Cadiot: Computer-assisted Proofs in PDEs on $\mathbb{R}^n$

Abstract : In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on $\mathbb{R}^n$. Such solutions arise naturally in various models, in the form of travelling waves, self-similar solutions or localized patterns. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for existence proofs. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the Whitham equation will be exposed.

Gonzalo Cao Labora: Nonradial Self-similar Singularity Formation for Compressible Euler and NLS

Abstract: In the talk, we will discuss recent techniques, both analytical and computer-assisted regarding self-similar singularity formation. We will focus on non-radial stability, both for the compressible Euler equations and in the context of the NLS equations. We will also mention recent developments in the self-similar singularity formation literature and how computer assisted techniques can be used to address stability and non-uniqueness problems.

Ryan Creedon: A Proof of the Transverse Instability of Stokes Waves

Abstract: Over forty years ago, McLean showed via numerical methods that periodic water waves are unstable to transverse perturbations, even those of small amplitude. In this presentation, I give the first rigorous proof of these instabilities for small-amplitude waves in water of almost any depth, including infinite depth. Key ideas behind the proof will be illustrated, including connections to previous formal asymptotic and numerical investigations. As will be seen, some steps require computer assistance to complete.

Joel Dahne: Self-similar Singular Solutions for the Nonlinear Schrödinger Equation and the Complex Ginzburg-Landau Equation

Abstract: In 2001, Plecháč and Sverak gave strong numerical evidence for the existence of branches of backwards self-similar singular solutions to the complex Ginzburg-Landau equation. We now present a rigorous proof of these branches' existence, which in particular includes the 3D cubic Schrödinger equation.

Our proof follows the same strategy as Plecháč and Sverak, which reduces the problem to proving the existence of a solution to a certain ODE with prescribed behaviour at zero and infinity. Near zero the solution is constructed using a rigorous numerical ODE solver and near infinity by carefully analysing the asymptotic expansion. These two solutions are then glued together to form the full solution.

Camillo de Lellis: Besicovitch's $\frac{1}{2}$ Problem and Linear Programming

Abstract: In 1928 Besicovitch formulated the following conjecture, probably the oldest open problem in geometric measure theory. Let $E$ be a closed subset of the plane with finite length (more precisely finite Hausdorff 1-dimensional measure) and assume its length is more than half of the diameter in all sufficiently small disks centered at a.a. its points. Then E is rectifiable, i.e. it lies in a countable union of $C^1$ arcs with the exception of a set of points with zero length. $\frac{1}{2}$ cannot be lowered, while Besicovitch himself showed that the statement holds if it is replaced by $\frac{3}{4}$. His bound was improved only once by Preiss and Tiser in the nineties to an (algebraic) number which is approximately 0.735. In this talk I will report on further progress stemming from a joint work with Federico Glaudo, Annalisa Massaccesi, and Davide Vittone. Besides improving the bound of Preiss and Tiser to a substantially lower number ($0.7$), our work proposes a family of variational methods to find and improve the latter bound. We can improve Preiss and Tiser bounds both with a pen-and-paper proof (recovering an unpublished result of Schechter) and with the assistance of a computer (which is used to examine a very large, but finite, number of cases). The latter is in fact a feasible computation because a part of the variational problems can be formulated as a linear programming task.

Susanna Haziot: On Global Regularity Theory for the Peskin Problem

Abstract: The Peskin problem describes the flow of a Stokes fluid through the heart valves. We begin by presenting the simpler 2D model and investigate its small data critical regularity theory, with initial data possibly containing small corners. We then present the 3D problem and describe the challenges that arise to proving global well-posedness.

The first part is joint work with Eduardo Garcia-Juarez, and the second is on-going work with Eduardo Garcia-Juarez and Yoichiro Mori.

Jonathan Jaquette: Uncovering Unstable Blowup in PDEs through an Exploration of Invariant Manifolds

Abstract: When a PDE that generates an analytic semiflow blows up, its solutions may be continued in complex time around the singularity potentially producing a branched Riemann surface. The work Cho et al. [*Jpn. J. Ind. Appl. Math.* 33 (2016): 145-166] investigated this phenomena for the quadratic heat equation $ u_t = u_{xx} + u^2$. When solutions are continued for purely imaginary time, a nonlinear Schrödinger equation (NLS) $i u_t = u_{xx} + u^2$ for $ x \in \mathbb{T}$ is obtained, and the authors conjectured that this NLS is globally well-posed for real initial data. Using a mix of analytical and computer-assisted techniques, we have shown that this equation exhibits rich dynamical structure punctuated by (presumably unstable) blowup solutions. It is also of note that the nonlinearity here, a complex quadratic, is essentially the same nonlinearity as in the Constantin-Lax-Majda equation, a 1D model for incompressible fluids.

In recent work we have identified real initial data whose numerical solution blows up, in contradiction of the conjecture by Cho et al. Furthermore, the set of real initial data which blows up under the NLS dynamics appears to occur on a codimension-1 manifold, and we conjecture that it is precisely the stable manifold of the zero equilibrium for the nonlinear heat equation. We apply the parameterization method to study the internal dynamics of this manifold, offering a heuristic argument in support of our conjecture. The solution exhibits self-similar blowup and potentially nontrivial self-similar dynamics, however the proper scaling ansatz remains elusive.

Hao Jia: Relaxation Mechanisms in Incompressible Fluid Flows and Related Models

Abstract: In this talk we will present several mechanisms for asymptotic stability in the two dimensional incompressible fluid equations, including inviscid damping, vorticity depletion, and enhanced dissipation. These mechanisms are essentially connected to the spectral regularity of the continuous spectrum of the linearized operators, and its interaction with the singular perturbation by the viscous term. We will also discuss an important stability mechanism due to the balance between convection and vorticity stretching in the De Gregorio model for the three dimensional incompressible Euler equations. These stability mechanisms play an essential role in the long time behavior of smooth solutions to incompressible fluid equations such as the reverse cascade of energy in 2d flows, and have been used to prove asymptotic stability of important steady solutions. Numerical results and open questions about dynamical behavior of general solutions will also be presented.

Aditi Krishnapriyan: Bridging Numerical Methods and Deep Learning with Differentiable Solvers

Abstract: Machine learning (ML) is increasingly playing a pivotal role in spatiotemporal modeling. A number of open questions remain on the best learning strategies to maximize the utility of machine learning while ensuring the validity of such predictions at test time (i.e., "deployment"), particularly in limited data scenarios. This talk will focus on machine learning methods for neural PDE solvers, with an emphasis on broad learning strategies that are applicable across a wide variety of systems and neural network architectures. Some topics I will discuss include: developing expressive neural network architectures, inspired by spectral methods, for turbulent fluid flow, using self-supervised learning to change the basis of learning with spectral methods to solve fluid dynamics and transport PDE problems, and “simulation-in-the-loop” approaches via incorporating PDE-constrained optimization as a layer in neural networks. In each of these settings, I will discuss how ML methods can be used with numerical methods through fully differentiable settings.

Stan Palasek: Non-uniqueness for the Navier-Stokes Equations from Critical Data

Abstract: A fundamental problem in the theory of the Navier-Stokes equations is the uniqueness of solutions of the Cauchy problem. After discussing some of the recent progress in this area, we will describe a new approach to constructing solutions that exhibit non-uniqueness. As an application, we will show an example of non-unique Leray-Hopf solutions in a dyadic model of the 3D Navier-Stokes, with initial data in a sharp regularity class. Then we will present recent work with M. Coiculescu that uses a similar mechanism to construct non-unique solutions to the full Navier-Stokes whose data lies in a critical space.

Anastassiya Semenova: Instabilities of Large Amplitude Stokes Waves

Abstract: The study of ocean waves, especially surface gravity waves, is essential for understanding the formation of rogue waves and whitecaps within ocean swell. Waves that propagate from the epicenter of a storm can be treated as unidirectional. In this presentation, we will examine periodic traveling waves on the free surface of an ideal two-dimensional fluid of infinite depth. Specifically, we will introduce surface waves of permanent shape, also known as Stokes waves and discuss their stability.