Since the beginning of the program, results on long time existence for solutions to gravity/capillary wave equations with small data have been described, in particular in the series of lectures of Daniel Tataru during the Introductory workshop. In this talk, we aim at giving a review of a set of results concerning the possibility of going past the "cubic life span" for solutions to the gravity-capillary system with small data in the periodic setting.

This is a report on joint work with Friedrich Klaus and Baoping Liu. We prove wellposedness for all equations of the KdV hierarchy in H^{-1}. The proof follows the outline of Killip and Visan for KdV: We prove tightness, and convergence of the difference flow simultaneously for all equations of the hierarchy.

In contrast to Killip and Visan we reduce the problem via a modified Miura map to the Gardner hierarchy.

I will describe an implosion mechanism for three dimensional viscous compressible fluids, and will make connections with highly oscillatory blow up mechanisms for energy super critical defocusing non linear wave models.

A classical model to describe the dynamics of Newtonian stars is the gravitational Euler-Poisson system. The Euler-Poisson system admits a wide range of star solutions that are in equilibrium or expand for all time or collapse in a finite time or rotate. In this talk, I will discuss some recent progress on those star solutions with focus on expansion and collapse. The talk is based on joint works with Yan Guo and Mahir Hadzic.

In this talk, we will consider the low regularity well-posedness problem for the two dimensional gravity water waves. This quasilinear dispersive system admits an interesting structure which we exploit to prove a new class of energy estimates, which we call balanced cubic estimates. This yields a key improvement over the earlier energy estimates of Hunter-Ifrim-Tataru. These results allow us to significantly lower the regularity threshold for local well-posedness, even without using dispersive properties. Combined with nonlinear vector field Sobolev inequalities, an idea first introduced by the last two authors in the context of the Benjamin-Ono equations, these improvements extend to global solutions for small and localized data. This is joint work with Mihaela Ifrim and Daniel Tataru.

In a joint work with Valeria Banica, we make a connection between a famous analytical object introduced in the 1860s by Riemann, as well as some variants of it, and a nonlinear geometric PDE, the binormal curvature flow. As a consequence this analytical object has a non-obvious non- linear geometric interpretation. We recall that the binormal flow is a standard model for the evolution of vortex filaments. We prove the existence of solutions of the binormal flow with smooth trajectories that are as close as desired to curves with a multifractal behavior. Finally, we show that this behavior falls within the multifractal formalism of Frisch and Parisi, which is conjectured to govern turbulent fluids. If time permits, I will also mention some recent results on intermittency in the context of the linear Schrödinger equation.

It is known since the work of Dyachenko \& Zakharov in 1994 that for the weakly nonlinear 2d infinite depth water waves, there are no 3-wave interactions and all of the 4-wave interaction coefficients vanish on the non-trivial resonant manifold. In this talk I will present a recent result that proves this partial integrability from a different angle. We construct a sequence of energy functionals $\mathfrak E_j(t)$, directly in the physical space, which are explicit in the Riemann mapping variable and involve material derivatives of order $j$ of the solutions for the 2d water wave equation, so that $\frac d{dt} \mathfrak E_j(t)$ is quintic or higher order. We show that if some scaling invariant norm, and a norm involving one spacial derivative above the scaling of the initial data are of size no more than $\varepsilon$, then the lifespan of the solution for the 2d water wave equation is at least of order $O(\varepsilon^{-3})$, and the solution remains as regular as the initial data during this time. If only the scaling invariant norm of the data is of size $\varepsilon$, then the lifespan of the solution is at least of order $O(\varepsilon^{-5/2})$. Our long time existence results do not impose size restrictions on the slope of the initial interface and the magnitude of the initial velocity, they allow the interface to have arbitrary large steepnesses and initial velocities to have arbitrary large magnitudes.

In recent years there has been a lot of progress on two-dimensional steady water waves with vorticity and three-dimensional waves with zero vorticity (irrotational flow). The three-dimensional case with vorticity remains largely unexplored. Part of the reason is that the problem (generally) doesn't reduce to an elliptic free boundary problem. In my talk I will discuss recent progress on this topic. I will mainly focus on an ongoing work where we consider doubly periodic, symmetric gravity-capillary waves with small vorticity, by treating the problem as elliptic-hyperbolic and using ideas from the theory of magnetohydrostatic equilibria.

This is joint work with D. S. Seth and K. Varholm.

The two-dimensional incompressible Euler equations have solutions for vorticity discontinuities (planar analogs of vortex patches) that separate two half-spaces with constant but distinct vorticities. Vorticity discontinuities are linearly stable but support unidirectional waves that propagate along the discontinuity and decay exponentially into the interior of the fluid. The linearized frequency of these waves is independent of their wave numbers, which leads to interesting and unusual weakly nonlinear, nondispersive dynamics for discontinuities with small slope. In particular, we formulate contour dynamics equations for the motion of vorticity discontinuities before they filament, and prove that a quadratically nonlinear Burgers-Hilbert equation provides an asymptotic description of the motion of small-slope vorticity discontinuities on cubically nonlinear time-scales. This is joint work with Joseph Biello, Ryan Moreno-Vasquez, Jingyang Shu, and Qingtian Zhang.

I will present recent results concerning global existence for the Kuramoto-Sivashinsky equation (KSE) in 2 space dimensions with and without growing modes. The KSE is a model of long-wave instability in dissipative systems. This is joint work with David Ambrose (Drexel).

This talk addresses the mathematical analysis of fluid models including a maximum packing constraint. These equations arise naturally for instance in the modeling of mixtures like suspensions or in the modeling of collective motion. I will present recent existence and stability results on two classes of PDEs systems corresponding to two modeling approaches. The first one, called the "soft approach", is based on compressible equations complemented with constitutive laws that are singular close to the maximal packing constraint. The second one, the "hard approach", is based on a free boundary problem between a congested domain with incompressible dynamics and a free domain with compressible dynamics.

We consider the 3d incompressible Euler equation with a Coriolis force and show that small axially symmetric solutions lead to solutions which exist for a long time. This is a joint work with Yan Guo and Klaus Widmayer.

We will discuss stability properties of stationary solutions to the 2D Euler equation.

We consider solutions of the Navier-Stokes equations in 3d with vortex filament initial data of arbitrary circulation, that is, initial vorticity given by a divergence-free vector-valued measure of arbitrary mass supported on a smooth curve. First, we prove global well-posedness for perturbations of the Oseen vortex column in scaling-critical spaces. Second, we prove local well-posedness (in a sense to be made precise) when the filament is a smooth, closed, non-self-intersecting curve. Besides their physical interest, these results are the first to give well-posedness in a neighborhood of large self-similar solutions of 3d Navier-Stokes, as well as solutions which are locally approximately self-similar. This is joint work with Pierre Germain and Benjamin Harrop-Griffiths.

There is a substantial literature on horizontally propagating waves in stratified flows. The vast majority of this work, particularly when it concerns nonlinear structures and solitary waves, focuses on “mode one”, that is, the "fast" waves of the system whereby all the pycnoclines (density jumps) are deflected with the same polarity. The simplest model for mode one waves is the two-layer flow of a lighter fluid above a heavier one bounded above and below by rigid boundaries. In that case the mode one wave is the only wave in the system. Mode two waves require (at least) one additional layer in order for the two interfaces to deflect with opposite polarity (mode two), or the same polarity (mode one). We shall consider the three-layer problem in this talk, considering KdV and MCC-like models, and the full Euler equations. We shall describe the problem and consider the question: do mode two solitary waves exist in the Euler equations?

Optimization and optimal dynamical control can be used to investigate the precision of rigorous analytical estimates (inequalities) for solutions of partial differential equations. Even when constituent estimates are demonstrably sharp, the result of a sequence of applications of such inequalities need not be, leaving uncertainty in the accuracy of the ultimate result of the analysis. We examine classical bounds on enstrophy and palinstrophy amplification in the 2D and 3D Navier-Stokes and Burgers’ equations and discover that the best known instantaneous growth rates estimates for these quantities are indeed sharp. Integrating instantaneous estimates on growth rates in time may – as in the 2D Navier-Stokes case – but does not always – as in the Burgers case – produce sharp estimates in which case optimal control techniques must be brought to bear to determine the actual extreme behavior of the nonlinear dynamics. The relevant question for 3D Navier-Stokes equations, i.e., the $1M Millennium Prize Problem, remains unresolved although work is in progress to apply these tools to address it.  Some of the research reported here is joint with Lu Lu, Indiana University Mathematics Journal 57, 2693-2727 (2008) and some with Diego Ayala & Thilo Simon, Journal of Fluid Mechanics 837, 839-857 (2018).

In this talk I shall consider the binormal flow equation, which is a model for the dynamics of vortex filaments in Euler equations. Geometrically it is a flow of curves in three dimensions, explicitly connected to the 1-D Schrödinger map with values on the 2-D sphere, and to the 1-D cubic Schrödinger equation. Although these equations are completely integrable we show the existence of an unbounded growth of the energy density. The density is given by the amplitude of the high frequencies of the derivative of the tangent vectors of the curves, thus giving information of the oscillation at small scales. In the setting of vortex filaments the variation of the tangent vectors is related to the one of the direction of the vorticity, that according to the Constantin-Fefferman-Majda criterion plays a relevant role in the possible development of singularities for Euler equations. This is a joint work with Luis Vega.