Speakers and Abstracts

Tuesday, December, 10, 2024

All times are in Gulf Standard Time (GST)

• Hatem Zaag — Various blow-up regimes for the Complex Ginzburg-Landau equation

  Speaker: Hatem Zaag, Université Sorbonne Paris Nord
  

  Abstract:

  The cubic Complex Ginzburg-Landau (CGL) equation is one of the most studied nonlinear equations in the physics community, where it attracted a lot of attention in the 1970’. It was first derived by Newell and Whitehead in 1969 as a model for instability in fluid convection problems, and used by Stewartson and Stuart in 1971, for the description of unstable plane Poiseuille flows. 

  As for the math community, the first results were given around the year 2000. This delay is mainly due to the lack of structure in this parabolic system, where no variational structure exists and no maximum principle is available.

  Introducing new tools, we were able to rigorously construct some blow-up solutions for CGL, in different parameters’ regimes. This talk is dedicated to the proof of those results, in relation with the earlier physicists’ predictions. 

  This is a joint work with G.K. Duong and N. Nouaili.

• Kai Yang — Numerical and analytical approaches for the blow-up dynamics for some nonlinear dispersive equations.

  Speaker: Kai Yang

  Abstract:

  We discuss some results about the existence and stable blow-up solutions and their dynamics for the L2 critical and super-critical nonlinear Schrodinger type and the KdV type equations. These results are obtained from both numerical and analytical approaches. This is joint work with Luiz Farah, Justin Holmer, Annie Millet, Svetlana Roudenko, and Yanxiang Zhao.

• Slim Ibrahim, University of Victoria

  Speaker: Slim Ibrahim, University of Victoria

  Abstract:

  TBA

• Marta Lewicka — Flexibility results for the Monge-Ampere system

  Speaker: Marta Lewicka, University of Pittsburgh

  Abstract:

   We study the Monge-Ampere system (MA), naturally arising from the prescribed curvature problem and related to the isometric immersions of d-dimensional metrics in d+k dimensional ambient space. When d=2 and k=1, (MA) reduces to the classical Monge-Ampere equation.

  Our main results achieve density in the set of subsolutions, of the Holder solutions to the Von Karman system which is the weak formulation of (MA), used in the mathematical elasticity in the aforementioned special case d=2, k=1. 

  We will present a panorama of the recent results in this context, exhibiting regularity dependence on the arbitrary dimension d and codimension k of the problem.

• Maurizio Grasselli — Long-term behavior of a diffuse interface model for two-phase magnetohydrodynamic flows

  Speaker: Maurizio Grasseli, Politecnico di Milan

  Abstract:

  We introduce a diffuse interface model which describes the interaction between a magnetic field and two partly miscible, conducting, incompressible fluids. The model consists of the Cahn-Hilliard equation for the relative difference of the fluid concentrations coupled with the equations of resistive magnetohydrodynamics for the volume-averaged velocity and for the magnetic field. The resulting evolution system is endowed with suitable initial and boundary conditions. We show that, in dimension two, we can define a dissipative dynamical system on a finite energy phase space and that system has the global attractor. Moreover, the backward uniqueness property holds on the global attractor and any weak solution does convergence to a single equilibrium. We also discuss some issues in dimension three.

Wednesday, December 11, 2024

• Manuel del Pino — Overhanging solitary waves in the Water Wave Problem

  Speaker:  Manuel del Pino, University of Bath
  

  Abstract:
  
  In the classical Water Wave Problem, we construct new overhanging solitary waves by a procedure resembling desingularization of the gluing of constant mean curvature surfaces by tiny catenoidal necks. The solutions here predicted have long been numerically detected. This is joint work with Juan Davila, Monica Musso, and Miles Wheeler.

• Anna Mazzucato — On the Euler equations with in-flow and out-flow boundary conditions

  Speaker: Anna Mazzucato, Pennsylvania State University

  Abstract:

  I will discuss recent results concerning the well-posedness and regularity for the incompressible Euler equations when in-flow and out-flow boundary conditions are imposed on parts of the boundary, motivated by applications to boundary layers. This is joint work with Gung-Min Gie (U. Louisville, USA) and James Kelliher (UC Riverside, USA). If time permits, I will also discuss energy dissipation and enstrophy production in the zero-viscosity limit at outflow, joint work with (Jincheng Yang, U Chicago and IAS), Vincent Martinez (CUNY, Hunter College), and Alexis Vasseur (UT Austin).

• Michele Dolce — The long way of a viscous vortex dipole

  Speaker: Michele Dolce, EPFL

  Abstract:

  The evolution of two counter-rotating point vortices in a 2D inviscid fluid is governed by the Helmholtz-Kirchhoff system, resulting in a translation at a constant speed. However, at large but finite Reynolds numbers, vortex core sizes increase due to diffusion, making the point vortex approximation unjustified over long times for the resulting viscous dipole. This talk aims to systematically define an asymptotic expansion accounting for streamline deformation from vortex interactions and understand the finite size effects on the dipole's translation speed. We then prove that the exact solution remains close to our approximation over a very long time interval, which extends unboundedly as the Reynolds number approaches infinity. The proof relies on energy estimates inspired from the Arnold’s variational characterization of the steady states of the 2D Euler equation, as recently revised by Gallay and Šverák for viscous fluids as well. This work is a collaboration with T. Gallay.

• Eliot Pacherie — Orbital stability of the vortex pair for the Gross-Pitaevskii equation

  Speaker:  Eliot Pacherie, Université de Cergy

  Abstract:

  The Gross-Pitaevskii equation describes the movement of superfluids, and has stationary solution called vortices. When two such vortices are present, they move together at a constant speed. In this talk, we will show the orbital stability in a metric space of this traveling wave solution. We will explain how to adapt the classical scheme of proof of orbital stability in such a space, and why the proof fails in more classical settings. This is a joint work with Philippe Gravejat and Frederic Valet.

Thursday, December 12, 2024

• Livia Corsi — Asymptotically full measure sets of almost-periodic solutions for the NLS equation

  Speaker: Livia Corsi, Università di Roma Tre

  Abstract:

  I shall discuss the existence of infinite-dimensional invariant tori in a mechanical system made of infinitely many rotators weakly interacting with each other. I shall concentrate on interactions depending only on the angles, with the aim of discussing in a simple case the analyticity properties to be required on the perturbation of the integrable system in order to ensure the persistence of a large measure set of invariant tori with finite energy.

• Vahang Nersesyan — On the chaotic behavior of the Lagrangian flow of the 2D Navier–Stokes system with bounded degenerate noise

  Speaker: Vahagn Nersesyan, NYU Shanghai

  Abstract:

  In this talk, we will consider a fluid governed by the randomly forced 2D Navier–Stokes system. We will assume that the force is bounded, acts directly only on a small number of Fourier modes, and satisfies some natural decomposability and observability properties. Under these assumptions, we will show that the Lagrangian flow associated with the random fluid exhibits chaotic behavior characterized by the strict positivity of the top Lyapunov exponent. To achieve this, we will introduce a new abstract result that allows to derive positivity of the top Lyapunov exponent from the controllability properties of the underlying deterministic system. This talk is based on a joint work with Deng Zhang and Chenwan Zhou.

• Amjad Tuffaha — Compressible Euler equations in an elastic domain

  Speaker: Amjad Tuffaha, American University of Sharjah
  

  Abstract:

  We address the compressible Euler equations in domain with a free elastic boundary, evolving according to a damped fourth order plate equation forced by the fluid pressure. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data in H^3. The main new device introduced is a variable coefficients space tangential-time differential operator Q of order 1, which is of transport type and allows the logarithm of the density g to evolve according to a wave-type equation Q^2 g − div_a(fg) = F with transmission boundary conditions,  where f is related to the speed of the waves and a is the change of variable. This operator captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics.

• Van Tien Nguyen — Multiple collapsing solutions to the 2D Keller-Segel system

  Speaker: Van Tien Nguyen, National Taiwan University
  

  Abstract:

  So far, except in some integrable PDEs where explicit formulas for multiple-soliton solutions are known, the question of the collision of solitons (soliton-soliton interaction) is much less understood. Most previous studies have focused on global-in-time solutions. For the first time, we develop a rigorous framework to construct such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons in a nonlinear parabolic problem. The talk presents a brand new mechanism of singularity formation formed by a collision of two sub-collapses, resulting in a finite-time blowup solution to the 2D Keller-Segel system with a 16-pi mass concentration at a single point. 

• Barbara Kaltenbacher — Forward and Inverse Problems in Nonlinear Acoustics

  Speaker: Barbara Kaltenbacher, Alpen-Adria-Universität Klagenfurt

  Abstract: 

  The importance of ultrasound is well established in the imaging of human tissue. In order to enhance image quality by exploiting nonlinear effects, recently, techniques such as harmonic imaging and nonlinearity parameter tomography have been put forward.

  As soon as the pressure amplitude exceeds a certain bound, the classical linear wave equation loses its validity and more general nonlinear versions have to be used.

  Another characteristic property of ultrasound propagating in human tissue is frequency power law attenuation leading to fractional derivative damping models in time domain.

  In this talk, we will first of all dwell on modeling of nonlinearity on one hand and of fractional damping on the other hand. Then we will give an idea on the challenges in the analysis of the resulting PDEs and discuss some parameter asymptotics.

  Finally, we address some relevant inverse problems in this context, in particular the above-mentioned task of  nonlinearity parameter imaging, which leads to a coefficient identification problem for a quasilinear wave equation.

   

• Hicham Kouhkouh — Long-time behavior of deterministic Mean Field Games with non-monotone interactions

  Speaker: Hicham Kouhkouh, University of Graz

  Abstract:

  We address the long-time behavior of the solutions to finite horizon deterministic MFG in the whole space with $H(x,p,m) = |p|^{2} - F(x,m)$. The cost functional $F$ is continuous with respect to the distribution of the agents and satisfies a gap condition at infinity. We do not assume the classical monotonicity condition on the cost functional (which implies a preference for less crowded areas and ensures uniqueness of the solution to the MFG system). Our main assumption instead is about the set of minima of the cost $F$. It allows the aggregation of the agents and the existence of multiple solutions. 

  Besides the long-time behavior, we will also prove its connection to the static MFG, and the ergodic MFG. 

  This is a joint work with Martino Bardi (Univ. of Padova) and can be found in: 

  M. Bardi, and H. Kouhkouh, Long-time behavior of deterministic Mean Field Games with nonmonotone interactions (SIAM J. Math. Analysis, 2024).

Friday, December 13, 2024

• Sameer Iyer — Uniform Inviscid Damping and Inviscid Limit of 2D Navier-Stokes with Navier Boundary Conditions

  Speaker: Sameer Iyer, University of California, Davis

  Abstract:

  We present a recent series of works, joint with J. Bedrossian, S. He, F. Wang, in which we prove nonlinear inviscid damping, enhanced dissipation, and inviscid limit for the 2D Navier-Stokes equations near Couette. The domain is the periodic channel, and Navier Boundary Conditions are prescribed vertically.

• Wei Wang — The existence of periodic-in-time solutions for nematic liquid crystal dynamics under given shear flow

  Speaker: Wei Wang, Zhe Jiang University

  Abstract:

  The periodic motions of nematic liquid crystals under shear flow have been predicted by experiments and numerical simulations for a long time. In this talk, we will demonstrate the existence of a specific type of periodic-in-time solution, referred to as "kayaking," under conditions of small velocity gradients.

• Malek Abid — The effect of vorticity on pre-breaking waves in shallow water

  Speaker: Malek Abid, Université Aix-Marseille

  Abstract:

  Two-dimensional nonlinear gravity waves traveling in shallow water on a vertically sheared current of constant vorticity are considered. Using Euler equations, in the shallow water approximation, hyperbolic equations for the surface elevation and the horizontal velocity are derived. Using Riemann invariants of these equations, that are obtained analytically, a closed-form nonlinear evolution equation for the surface elevation is derived. A dispersive term is added to this equation using the exact linear dispersion relation. With this new single first-order partial differential equation, vorticity effects on undular bores are studied. Within the framework of weakly nonlinear waves a Whitham equation with constant vorticity is derived from this new model. Within the framework of the new model and the Whitham equation a study of the effect of vorticity on the breaking time of dispersive waves and hyperbolic waves as well is carried out.

Saturday, December 14, 2024

• Haithem Taha — What Does Nature Minimize in Every Incompressible Flow?

  Speaker: Haithem Taha, University California, Irvine

  Abstract:

  Driven by an outdated problem in aerodynamics, we discovered a new principle in fluid physics. The Euler equation does not possess a unique solution for the flow over a multiply connected domain. This problem has serious repercussions in aerodynamics; it implies that the inviscid aero-hydrodynamic lift force over a two-dimensional object cannot be determined from first principles; a closure condition must be provided. The Kutta condition has been ubiquitously considered for such a closure in the literature, even in cases where it is not applicable.  In this talk, I will present a special variational principle that we revived from the history of analytical mechanics: Hertz’ principle of least curvature. Using this principle, we developed a novel variational formulation of Euler’s dynamics of ideal fluids that is fundamentally different from the previously developed variational formulations based on Hamilton’s principle of least action. Applying this new variational formulation to the century-old problem of the ideal flow over an airfoil, we developed a general (dynamical) closure condition that is, unlike the Kutta condition, derived from first principles. In contrast to the classical theory, the proposed variational theory is not confined to sharp-edged airfoils; i.e., it allows, for the first time, theoretical computation of lift over arbitrarily smooth shapes, thereby generalizing the century-old lift theory of Kutta and Zhukovsky. Moreover, the new variational condition reduces to the Kutta condition in the special case of a sharp-edged airfoil, which challenges the widely accepted wisdom about the viscous nature of the Kutta condition.

  We also generalized this variational principle to Navier-Stokes’ via Gauss’ principle of least constraint, thereby discovering the fundamental quantity that Nature minimizes in every incompressible flow. We proved that the magnitude of the pressure gradient over the field is minimum at every instant! We call it the Principle of Minimum Pressure Gradient (PMPG). We proved that the Navier-Stokes’ equation is the necessary condition for minimizing the pressure gradient cost subject to the continuity constraint. Hence, the PMPG turns a fluid mechanics problem into a minimization one where fluid mechanicians need not to apply Navier-Stokes’ equations, but minimize the pressure gradient cost. We close by posing two conjectures: one on nonlinear hydrodynamic stability and another on the mathematical problem of the inviscid limit of Navier-Stokes.

• David Ambrose — Some non-decaying, non-periodic existence theory for fluid equations

  Speaker: David Ambrose, Drexel University

  Abstract:

  We consider the irrotational Euler equations and the surface quasi-geostrophic equation in the case that the unknowns do not decay and are not spatially periodic.  In such settings, constitutive laws of convolution type (such as the Biot-Savart law) do not apply directly, as the convolution integral does not converge.  These can be replaced with identities of Serfati type, which separate the integrals into near-field and far-field pieces, with the far field contribution being able to be manipulated for better convergence properties.  We use these identities to find existence of solutions for the 2D Euler equations with bounded velocity and vorticity (generalizing a result of Serfati), for the 3D Euler equations in uniformly local Sobolev spaces, and for SQG in Holder spaces and in uniformly local Sobolev spaces. This includes joint work with Elaine Cozzi, Daniel Erickson, James Kelliher, Milton Lopes Filho, and Helena Nussenzveig Lopes.

• Mouhamed Fall — Uniqueness and non-degeneracy properties of the Generalized Benjamin-Ono equation

  Speaker: Mouhamed Fall, AIM Senegal

  Abstract:

  Solitary waves of the Benjamin-Ono equation are completely described by nontrivial solutions to the equation (-\Delta)^{\frac{1}{2}} u+u=u^2 on $\mathbb{\R}$. The uniqueness, up to translations, to this equation is a classical result by Amick and Toland  (Acta Math, 1991) and an alternative proof was obtained by Albert  (1992). A generalization to this equation is (-\Delta)^{s} u+u=|u|^p on $\mathbb{\R}$. A uniqueness, up to translation to this equation  for s not equal to 1/2 and p not equal to 2 has been a long open problem, besides uniqueness of ground-state solutions for  amore general equation has been proved by Frank and Lenzmann (Acta Math 2013). It still remains an open problem about the uniqueness of all nontrivial solutions, not only the ground-state for s not equal to 2. We will discuss the uniqueness and non degeneracy properties in the case  (-\Delta)^{s} u+u=u^2 on $\mathbb{\R}$, for the optimal values of the parameter s.

  Joint work with Tobias Weth. Ref https://arxiv.org/abs/2310.10577v5

• Riccardo Montalto — Nonlinear quasi-periodic oscillations in Fluid Mechanics

  Speaker: Riccardo Montalto, University of Milan

  Abstract:

  In this talk I shall discuss some recent results about the construction of quasi-periodic waves in Euler equations and other hydro-dynamical models in dimension greater or equal than two. I shall discuss quasi-peridic solutions and vanishing viscosity limit for forced Euler and Navier-Stokes equations and the problem of constructing quasi-periodic traveling waves bifurcating from Couette flow (and connections with inviscid damping). Time permitting, I also discuss some results concerning the construction of large amplitude quasi-periodic waves in MHD system and rotating fluids. The techniques are of several kinds: Nash-Moser iterations, micro-local analysis, analysis of resonances in higher dimension, normal form constructions and spectral theory.