# Registered Data

## [01152] Recent trends in the mathematical theory for incompressible fluids

**Session Date & Time**:- 01152 (1/3) : 4E (Aug.24, 17:40-19:20)
- 01152 (2/3) : 5B (Aug.25, 10:40-12:20)
- 01152 (3/3) : 5C (Aug.25, 13:20-15:00)

**Type**: Proposal of Minisymposium**Abstract**: Models for incompressible fluid flows are omnipresent in a.o. (geo-)physical, biological and engineering applications. Nonetheless, the intrinsic lack of regularity of solutions to systems such as the incompressible Euler and Navier-Stokes equations constitutes a central challenge in developing further their mathematical theory. The mini-symposium approaches these regularity questions from both a deterministic and stochastic perspective with a focus on most recent results on singularity formation, regularization procedures, the emergence of quasi-periodic solutions and hydrodynamic stability. We bring together speakers from diverse backgrounds in terms of region, gender, and specific research methods to foster and encourage scientific exchange between different communities.**Organizer(s)**: Gennaro Ciampa, Lars Eric Hientzsch**Classification**:__35Q35__,__35Q31__,__76B03__,__76F02__,__76B47__**Speakers Info**:- Siddhant Agrawal (ICMAT)
- Gennaro Ciampa (Università di Milano)
- Luigi De Rosa (University of Basel)
- Michele Dolce (EPFL Lausanne)
- Luca Franzoi (NYU Abu Dhabi)
- Lucio Galeati (EPFL Lausanne)
- Zineb Hassaina (NYU Abu Dhabi)
- Shulamit Terracina (University of Milan)
- Riccardo Montalto (Università di Milano)
- Evan Miller (University of British Columbia)
- Quoc-Hung Nguyen (Chinese Academy of Sciences Beijing)
- Andre Schenke (Bielefeld University)

**Talks in Minisymposium**:**[02208] Nonuniqueness in Law for Stochastic Hypodissipative Navier--Stokes Equations****Author(s)**:**Andre Schenke**(Courant Institute of Mathematical Sciences at New York University)

**Abstract**: We study the incompressible hypodissipative Navier--Stokes equations with dissipation exponent $0 < \alpha < \frac{1}{2}$ on the three-dimensional torus perturbed by an additive Wiener noise term and prove the existence of an initial condition for which distinct probabilistic weak solutions exist. To this end, we employ convex integration methods to construct a pathwise probabilistically strong solution, which violates a pathwise energy inequality up to a suitable stopping time. This paper seems to be the first in which such solutions are constructed via Beltrami waves instead of intermittent jets or flows in a stochastic setting.

**[02509] Restoration of well-posedness of 2D fluid dynamics equations by transport noise****Author(s)**:**Lucio Galeati**(EPFL)- Dejun Luo (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)

**Abstract**: A longstanding problem in fluid dynamics is whether solution to 2D Euler with $L^p$-valued vorticity are unique, for some $p<\infty$. A related question on the probabilistic side is whether one can find a physically meaningful noise that can restore such uniqueness. Here I will present some recent progress, concerning other closely related 2D equations, for which we can provide a positive answer. Based on a joint work with Dejun Luo.

**[03123] Flows with lower dimensional dissipations****Author(s)**:**Luigi De Rosa**(University of Basel)

**Abstract**: In my talk I will describe how to put in a rigorous framework the study of turbulent solutions, i.e. rough fluid flows, whose energy cascade accumulates on lower dimensional sets. This naturally connects to intermittency phenomena which have been playing a major role in the current mathematical research.

**[03363] Uniform in gravity estimates for 2D water waves****Author(s)**:**Siddhant Agrawal**(ICMAT)

**Abstract**: We consider the 2D gravity water waves equation on an infinite domain. We prove a local wellposedness result which allows interfaces with corners and cusps as initial data, such that the time of existence of solutions is uniform even as the gravity parameter $g \to 0$. As an application of the new energy estimate, we prove a blow up result for the water waves model where the fluid is homeomorphic to the disc.

**[03417] Quasi-periodic invariant structures in incompressible fluids****Author(s)**:**Luca Franzoi**(New York University Abu Dhabi)- Nader Masmoudi (New York University Abu Dhabi)
- RICCARDO MONTALTO (University of Milan)

**Abstract**: In this talk, I present a recent result about the existence of nontrivial steady flows near the Couette flow in the channel $\mathbb{R}\times [-1,1]$ that are quasi-periodic in space and solve the incompressible Euler equations. First, I recall the result of Lin $\&$ Zeng and their construction of periodic flows. Then, I state the main result for space quasi-periodic flows. Finally, I show what are the main issues in our construction and how to solve them.

**[03663] Euler and Navier-Stokes equations. Quasi-periodic solutions and inviscid limit****Author(s)**:**RICCARDO MONTALTO**(University of Milan)

**Abstract**: In this talk I will discuss some recent results on Euler and Navier Stokes equations concerning the construction of quasi-periodic solutions. In particular, I will focus on the construction of vanishing viscosity quasi-periodic solutions for the Navier-Stokes equation in the inviscid limit. The key step of the analysis is to implement Normal Form techniques which allow to prove sharp estimates (uniform in time) w.r. to the viscosity.

**[03906] Geometric structures in incompressible fluids: vortex and magnetic reconnection****Author(s)**:**Gennaro Ciampa**(University of Milan)

**Abstract**: The goal of this talk is to provide examples of smooth solutions of the Navier-Stokes equations such that the topology of the vortex lines changes during the evolution without any loss of regularity. This phenomenon is known as vortex reconnection. We will also discuss the applications to Magnetohydrodynamics: we will construct smooth solutions of the MHD equations such that the topology of the magnetic field lines changes during evolution, providing analytical examples of magnetic reconnection.

**[04219] On maximally mixed equilibria of two-dimensional perfect fluid****Author(s)**:**Michele Dolce**(EPFL)

**Abstract**: The motion of a 2D perfect fluid can be described as an area-preserving rearrangement of the initial vorticity that conserves the kinetic energy. In the infinite time limit, vorticity mixing can occur and is conjectured to be a generic phenomenon. We offer a new perspective on the ``maximally mixed states" introduced by Shnirelman by proving that many of them can be obtained as minimizers of a variational problem and we discuss some of their properties.

**[04625] Reducibility of a class of quasi-linear wave equation on the torus****Author(s)**:**Shulamit Terracina**(Università degli Studi di Milano)

**Abstract**: We discuss the reducibility of a linear wave equation on the torus perturbed by a pseudo-differential potential of order 2 depending quasi-periodically on time. Under suitable conditions on the frequency vector, we develop a general strategy, combining Egorov theory with straightening of vector fields, to reduce to constant coefficients a class of weakly dispersive operators. Finally, we discuss generalizations to operators arising from the linearization of fluid models such as pure gravity Water Waves.

**[04740] Invariant KAM tori around annular vortex patches for 2D Euler equations****Author(s)**:**Zineb Hassainia**(NYUAD)- Taoufik Hmidi (New York University Abu Dhabi)
- Emeric Roulley (SISSA International School for Advanced Studies)

**Abstract**: We shall discuss the emergence of quasi periodic vortex patch solutions with one hole for the 2D-Euler equations. We prove the existence of such structures close to any annular vortex patch provided that its modulus belongs to a Cantor set with almost full Lebesgue measure. The proof is based on Nash-Moser implicit function theorems and KAM theory.

**[04785] Finite-time blowup for a 3D hypo-dissipative Navier-Stokes model equation****Author(s)**:**Evan Miller**(University of British Columbia)- Johannes Haubner (University of Graz)
- Bastian Zapf (University of Oslo)

**Abstract**: In this talk, I will discuss a new blowup result for a model equation for the 3D hypo-dissipative Navier-Stokes equation based on considering a restricted constraint space. When imposing the right geometric conditions on initial data, involving planar stretching at the origin, this allows a forward energy cascade that generates finite-time blowup. This model equation respects both the energy equality and the identity for enstrophy growth.