Special Session 48: Fluid dynamics and KAM theory

Unstable vortices and non-uniqueness for 2D Euler and $\alpha-$SQG

Angel Castro ICMAT-CSIC Spain Co-Author(s):    Daniel Faraco, Francisco Mengual, Marcos Solera

Abstract: In the sixties Yudovich proved global existence and uniqueness of solutions for the 2D Euler incompressible equation in $L^1\cap L^\infty$. This result extends to the case with a force in $L^1_t( L^1\cap L^\infty)$. Although global existence in $L^1\cap L^p$, $1

On the vanishing viscosity limit and propagation of regularity for the 2D Euler equations

Gennaro Ciampa University of L`Aquila Italy Co-Author(s):

Abstract: The goal of this talk is to analyze the Cauchy problem for the 2D Euler equations under very low regularity assumptions on the initial datum. We prove propagation of regularity of logarithmic order in the class of weak solutions with $L^p$ initial vorticity, provided that $p\geq 4$. We also study the inviscid limit from the 2D Navier-Stokes equations for vorticity with logarithmic regularity in the Yudovich class, showing a rate of convergence of order $|\log\nu|^{-\alpha/2}$ with $\alpha>0$.

Asymptotically full measure sets of almost-periodic solutions for the NLS equation

Livia Corsi University "Roma Tre" Italy Co-Author(s):

Abstract: In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behaviour of `typical` solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are in fact typical in the integrable case. In this talk I shall consider a family of NLS equations parametrized by a smooth convolution potential and prove that for `most` choices of the parameter there is a full measure set of Gevrey initial data that give rise to almost-periodic solutions whose hulls are invariant tori. As a consequence the elliptic fixed point at the origin turns out to be statistically stable in the sense of Lyapunov. This is a joint work with L.Biasco, G.Gentile and M.Procesi.

Onsager conjecture for SQG

Mimi Dai University of Illinois at Chicago USA Co-Author(s):    Vikram Giri, Razvan-Octavian Radu

Abstract: The Hamiltonian of the surface quasi-geostrophic (SQG) equation is an invariant quantity for regular enough solutions. It is postulated that the critical Holder regularity required to have the Hamiltonian conserved is C^{0}, known as the Onsager type of conjecture for SQG. We give a proof of this conjecture using a two-step scheme of convex integration.

Large amplitude quasi-periodic waves in rotating fluids

Luca Franzoi University of Milan Italy Co-Author(s):    Roberta Bianchini, Riccardo Montalto, Shulamit Terracina

Abstract: The $\beta$-plane equation is a 2D approximation model of the 3D Euler-Coriolis equations for rotating fluid. The goal of this short talk is to give an overview on these equations and to quickly present a recent result that studies the dynamics under the effect of a highly oscillating forcing term of substantial large size, therefore not perturbative. Moreover, we will assume the external forcing term to be traveling quasi-periodic in order to resolve a natural degeneracy arising from the rotation of the fluid. This is a joint work with Roberta Bianchini, Riccardo Montalto and Shulamit Terracina.

Desingularization of corners in the Muskat and Peskin problems

Eduardo Garcia-Juarez Universidad de Sevilla Spain Co-Author(s):    Javier Gomez-Serrano, Susanna V. Haziot, Benoit Pausader

Abstract: The Muskat and Peskin problems model very different physical phenomena, but both are described by quasilinear and nonlocal parabolic partial differential equations. The former describes the movement of two immiscible and incompressible fluids filtrating a porous medium, while the latter corresponds to an elastic filament immersed in a Stokesian fluid. We will study the small data critical regularity theory for these two models and show that interfaces with corners desingularize in time.

Asymptotic behavior of perturbations of the Euler equations in Yudovics`s class

Haroune HH Houamed New York University Abu Dhabi United Arab Emirates Co-Author(s):

Abstract: We illustrate the state of art of a simple, but robust, procedure to study the asymptotic behavior of perturbations of a vorticity solving the Euler equations in Yudovich`s class. The perturbation can be with/without a vanishing source term and/or a vanishing viscosity parameter (Navier--Stokes equations, for instance), and the setup of the problem can be within the entire two-dimensional space or torus. Broadly speaking, we show how the rate of convergence of the approximate vorticity can be improved by understanding the evanescence of some appropriately post-determined high frequencies. We also comment on another application of our method in the asymptotic analysis of a Plasma model within the non-relativistic regime.

One dimensional energy cascade in a quasi-linear dispersive equation

Federico Murgante University of Milan Italy Co-Author(s):    Alberto Maspero

Abstract: We investigate the transfer of energy to high frequencies in a quasi-linear Schr\{o}dinger equation with sublinear dispersion relation on the one-dimensional torus. Specifically, we construct initial data that undergo finite but arbitrarily large Sobolev norm explosions: starting with arbitrarily small norms in Sobolev spaces of high regularity, these norms become arbitrarily large at later times. Our analysis introduces a novel instability mechanism. By applying para-differential normal forms, we derive an effective equation that governs the dynamics, whose leading term is a non-trivial transport operator with non-constant coefficients. Using a positive commutator method, inspired by Mourre`s commutator theory, we demonstrate that this operator drives the energy cascade, leading to the observed instability.

Vortex patch motion in bounded domains

Emeric Roulley SISSA Italy Co-Author(s):    Zineb Hassainia and Taoufik Hmidi

Abstract: We consider the Euler equations within a simply-connected bounded domain. The dynamics of a single point vortex are governed by a Hamiltonian system, with most of its energy levels corresponding to time-periodic motion. We show that for the single point vortex, under certain non-degeneracy conditions, it is possible to desingularize most of these trajectories into time-periodic concentrated vortex patches. We provide concrete examples of domains where these non-degeneracies are satisfied, in particular convex ones. We also present a duplication method to construct synchronized motion of several vortices.

Large amplitude traveling waves for the nonresistive MHD system

Shulamit Terracina SISSA Italy Co-Author(s):    G. Ciampa, R. Montalto

Abstract: The goal of this talk is to discuss the existence of large amplitude traveling waves of the two-dimensional nonresistive Magnetohydrodynamics (MHD) system with a traveling wave external force. More precisely, we assume that the force is a smooth bi-periodic traveling wave propagating in the direction $\omega=(\omega_{1}, \omega_{2})\in\mathbb{R}^{2}$, with large amplitude of order $O(\lambda^{1+})$ and with large velocity speed $\lambda\omega$. Then, for most values of $\omega$ and for $\lambda\gg1$ large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude. Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes with respect to the large parameter $\lambda$. \noindent This is a joint work with G. Ciampa and R. Montalto.

Infinitely many isolas of modulational instability for Stokes waves

Paolo Ventura Universita` degli Studi di Milano Italy Co-Author(s):    Massimiliano Berti, Livia Corsi, Alberto Maspero

Abstract: Stokes waves are the simplest nontrivial form of water waves, featuring a periodic profile moving steadily in one direction. In the `70s, Benjamin and Feir discovered through experiments that the steady profile is unstable under long-wave perturbations, i.e. disturbances that, although small in amplitude, have much longer period than the initial wave. In recent years, significant mathematical progress has been made on this `modulational` instability, particularly in the linear approximation, which involves understanding the $L^2(\mathbb R)$-spectrum of the water wave operator linearized along a Stokes wave. I will present our latest results on the topic, specifically the full description of arbitrary portions of the unstable spectrum. As long conjectured by numerical investigations, this spectrum consists of infinitely many isolated elliptical branchings, called `isolas`, centered on the imaginary axis that become exponentially small as one moves away from the origin of the complex plane. This work was done in collaboration with M. Berti, L. Corsi, and A. Maspero.

Doubly connected V-states for the active scalar equations

Liutang Xue Beijing Normal University Peoples Rep of China Co-Author(s):    Taoufik Hmidi, Liutang Xue, Zhilong Xue

Abstract: This talk is concerned with the existence of doubly connected V-states (i.e. rotating patches) close to an annulus for the active scalar equations with completely monotone kernels. This provides a general way to unify various results on this topic related to geophysical flows. As some applications, the existence of doubly connected V-states for the gSQG equation and QGSW equation on radial domains are obtained, which are completely new.