Fluid dynamics.

Organizer: Javier Gómez Serrano, jg27@math.princeton.edu

The study of the fluid dynamics has been historically linked to the development of fascinating mathematical theories and tools to tackle the challenges presented by even the most basic questions in the field. It is not strange then that the qualitative and quantitative description of fluid dynamics often involves the most sophisticated analytic, variational and asymptotic techniques, among many others. In order to promote deeper understanding of these evolution problems it is paramount to create opportunities where researchers working on this and related areas meet to follow from close the most recent results and techniques obtained by different groups with different perspectives.
The objective of this session is to bring together researchers working on problems related to the dynamics of fluids. It turns out that in many physically relevant situations the evolution of a fluid can be reduced to a simpler system. This instance provides an excellent opportunity to address the questions that have to do with analysis, partial differential equations, fluid mechanics and numerical computation.
This session will see a healthy interaction engaging different experts in developing new and interesting areas of research and will most likely lay the foundation for future discoveries that will bring us closer to a thorough understanding of the most fundamental open problems in the area.

Confirmed speakers

* Nathan Glatt-Holtz, Tulane U
* Nastasia Grubic, ICMAT, Madrid
* Hao Jia, U. Minnesota
* Benoit Pausader, Brown
* Ian Tice, Carnegie Mellon
* Bruno Vergara, U. Zurich

If you are interested in giving a talk in this session please contact the corresponding organizer and Renato Iturriaga (renato@cimat.mx), Pablo Padilla (pabpad@gmail.com)

Abstracts

Uniqueness of Whitham’s highest cusped wave.

Bruno Vergara
The Whitham equation is a weakly dispersive, non-homogeneous and non-local model for shallow water waves. Like in the case of the Stokes wave for Euler, non-smooth traveling waves with greatest height between crest and trough have been shown to exist for this model. In this talk I will discuss uniqueness of solutions to the Whitham equation in the class of monotone functions and show that there exists a unique, even and periodic traveling wave of greatest height, which is moreover convex between consecutive cusps. This is joint work with A. Enciso and J. G\'omez-Serrano.

Recent progress in nonlinear inviscid damping for two dimensional incompressible Euler equation

Hao Jia
Inviscid damping is a fundamental relaxation mechanism for two dimensional Euler equation. Recently there have been significant advances in understanding linear inviscid damping in Sobolev spaces for shear flows and vortices. Extending the linear analysis to nonlinear analysis is challenging since the rate of stabilization is slow, convergence of vorticity field only holds in weak distributional sense and the effect of nonlinearity is strong. So far there are only very few results proving full nonlinear inviscid damping, which include Bedrossian and Masmoudi's breakthrough work on Couette flow, and Ionescu and J.'s recent work on axi-symmetrization of vorticity near point vortex solutions. In this talk, we will discuss some recent progress, that aims to study more general flows. Based on joint work with Alex Ionescu.

A Bayesian Approach to Quantifying Uncertainty in Divergence Free Flows.

Nathan Glatt-Holtz
We treat the statistical regularization of the ill-posed inverse problem of estimating a divergence free flow field $u$ from the partial and noisy observation of a passive scalar $\theta$. Our solution is a Bayesian posterior distribution, that is a probability measure $\mu$ which precisely quantifies uncertainties in $u$ once one specifies models for measurement error and a prior knowledge for $u$. We present some of our recent work which analyzes $\mu$ both analytically and numerically. In particular we discuss a
posterior contraction (consistency) result as well as some Markov Chain Monte Carlo (MCMC) algorithms which we have developed and refined and rigorously analyzed to effectively sample from $\mu$. This is joint work with Jeff Borggaard, Justin Krometis and Cecilia Mondaini.

Derivation of the Ion equation

Benoit Pausader
(Joint work with E. Grenier, Y. Guo and M. Suzuki). We consider the two fluid Euler-Poisson system and show that when the ratio of the mass of electron/mass of ion goes to zero, the solutions can be decomposed into a main term that converges strongly to a solution of the ion equation and an initial layer which converges weakly to 0 but carries some energy to infinity for ill-prepared initial data. This is related to the Low-Mach number limit for compressible fluids.

On an instability in anisotropic micropolar fluids

Ian Tice
Many real-world fluids possess a nontrivial microstructure that plays an important role in the dynamics. Examples include aeorsols, colloidal suspensions, biological fluids such as blood, lubricants, liquid crystals, and ferromagnetic fluids. The micropolar fluid model describes the microstructure as an infinitesimal rigid body at each fluid point, rotating with an unknown micro-angular velocity. The model allows for the exchange of angular momentum between the bulk and microstructures. In this talk we will survey recent work with Antoine Remond-Tiedrez on the nonlinear instability of a class of equilibrium solutions to the anisotropic micropolar equations. Here anisotropy means that the microstructure is not inertially spherical and can be thought of as being rod-like or pancake-like.

On the fluid-squeezing singularities for the 2D incompressible Euler equations

Nastasia Grubic
We first discuss a recent result on the existence of stationary solutions to the free boundary Euler equations with two fluids which feature a splash singularity. The proof is based on a set of weighted estimates for the self-intersecting interfaces that squeeze an incompressible fluid. We then consider the application of these estimates to the interface evolution problems, i.e. how to show a local existence result starting from a fluid-squeezing singularity.

Schedule

Thursday
15:30-17:30 Nathan Glatt-Holtz
                    Nastasia Grubic
                    Benoit Pausader

Friday
12:00-13:00 Hao Jia
                    Ian Tice
                    Bruno Vergara