Thomas Alazard (CNRS, Ecole Normale Supérieure Paris-Saclay)
Title: The water-wave equations in Eulerian coordinates
Abstract: These lectures are based on a point of view introduced by Walter Craig, Catherine Sulem and Vladimir Zakharov, which consists in studying the water-wave equations in Eulerian coordinates, using the Dirichlet to Neumann operator. I will present the equations and explain how to study the Dirichlet to Neumann operator using different techniques: either global with the method of multipliers, or microlocal using paradifferential operators. Applications will be given to different topics: study of the Cauchy problem, equipartition of energy, control theory and to other free boundary problems.
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Geoffrey Beck (INRIA, Rennes) and David Lannes (CNRS, université de Bordeaux)
Title: Wave-structure interactions
Abstract: This course is devoted to the derivation and mathematical analysis of several models describing the interaction of waves with partially immersed objects. A key point of the analysis is to understand initial boundary value problems for hyperbolic systems (e.g., the nonlinear shallow water equations) as well as for dispersive perturbations of such systems (e.g. the Boussinesq systems).
Chapter 1: A general method to model wave-structure interactions.
We describe here a strategy to obtain wave-structure interaction models that take the form of a coupled compressible-incompressible system. Even though the method works in any dimension, we consider in this course (except in Chapter 4) the case of horizontal dimension d=1.
1) Two examples of wave-structure interactions. This talk is focused on two situations exhibiting specific difficulties: a) a fixed object with non vertical sidewalls and b) an object with vertical sidewalls allowed to float freely vertically.
2) Choice of a wave-model. We present here two models commonly used to describe waves in shallow water: the nonlinear shallow water equations (hyperbolic) and a Boussinesq system (nonlinear dispersive).
3) Mixed constraints. We show that the pressure and the surface elevation satisfy different constraints under the object (interior region) and outside (exterior region).
4) Coupling conditions. We comment on the coupling conditions at the contact line.
5) Equation for the solid. The solid is driven by the hydrodynamic force in Newton's equations
6) Links with congested flows.
Chapter 2: Reduction to a transmission problem and mathematical analysis (nonlinear shallow water equations).
We show here in the case of the NSW equations in 1d, and in the case of an object with non vertical walls, that the model derived above can be reduced to atransmission problem on the two connected components of the exterior region. In this formulation, there is no need to compute the pressure exerted at the bottom of the object (it can be recovered once the transmission problem is solved). We then present some important steps in the analysis of (possibly free boundary) hyperbolic systems in 1D.
1) Resolution of the equations in the interior region.
2) Transmission problem in the exterior region
3) Comments in the case of an object with vertical walls
4) Mathematical analysis of 1d initial boundary value hyperbolic systems (Kreiss-Lopatinski condition, compatibility conditions, Kreiss symmetrizers)
5) The case of a free boundary
Chapter 3: Freely floating object with the Boussinesq equations.
The Boussinesq equations are a dispersive perturbationof the hyperbolic nonlinear shallow water equations. This dispersive perturbation induces drastic changes on the behavior and analysis of the associated initial boundary value problem.
1) Augmented formulation. We reformulate the equations as a system of
conservative laws with nonlocal flux and a localized source term; this
system is augmented with an equation on the trace of the surface elevation at the boundary.
2) Mathematical analysis. We insist here on the differences with the hyperbolic case, in particular for the regularity of the solution and the control of its trace at the boundary.
3) Numerical aspects. We propose a numerical scheme based on this augmented formulation.
4) The return to equilibrium case. In the linear regime we comment on the effect of dispersion on a particular setting.
Chapter 4: The two-dimensional case
If time permits, we will show how the handle the case of horizontal dimension d=2for the nonlinear shallow water equations.
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Ángel Castro (Instituto de Ciencias Matemáticas, Madrid)
Title: Traveling waves near shear flows
Abstract: In this talk we will consider the existence of traveling waves arbitrarily close to shear flows for the 2D incompressible Euler equations.
In particular we shall present some results concerning the existence of such solutions near the Couette, Taylor-Couette and the Poiseuille flows. In the first part of the talk we will introduce the problem and review some well known results on this topic. In the second one some of the ideas behind the construction of our traveling waves will be sketched.
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Guido Cavallaro (Sapienza Università di Roma)
Title: Concentrated vortex rings for Euler and Navier-Stokes equations
Abstract: We study the time evolution of an incompressible fluid (both inviscid and viscous) with axial symmetry without swirl, when the initial vorticity is very concentrated in N disjoint rings. We show that in a suitable limit, in which the thickness of the rings (and the viscosity, in the viscous case) tends to zero, the vorticity remains concentrated in N disjointed rings, each one of them performing a simple translation along the symmetry axis with constant speed.
Document: slides
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Maria Colombo (Ecole Polytechnique Fédérale de Lausanne)
Title: Instability and non-uniqueness for the Euler and Navier-Stokes equations
Abstract: We start introducing some basic facts about the Navier-Stokes equations, such as weak solutions, Leray global solutions, forward and backward self-similar solutions. We then explore recent developments in understanding the fundamental question of whether Leray-Hopf solutions to Navier-Stokes equations are unique. Following Jia-Sverak and Guillod-Sverak program, we describe how non-uniqueness can follow from instability in self-similarity variables. We then discuss a recent work in collaboration with Albritton and Brue’, where two distinct Leray solutions with zero initial velocity and identical body force are built. This nonuniqueness result builds in turn on Vishik's answer to another long-standing uniqueness problem about the 2D Euler equations.
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Michele Coti Zelati (Imperial College, London)
Title: Enhanced Dissipation Timescales and Mixing Rates
Abstract: Fluid mixing generically refers to a cascading mechanism that transfers information to smaller and smaller spatial scales, in a way that is time reversible and conservative for finite times but results in an irreversible loss of information at in infinite time. In simple terms, mixing is what is observed when stirring two liquids together, resulting in the creation of a homogenized mixture, but it can also be thought as a more complicated stabilizing mechanism for certain stationary structures, generating damping effects.
In this mini-course I will review several aspects linking Fourier analysis, spectral theory and fluid mixing, in the context of advection-diffusion equations driven by incompressible velocity fields, and also related to hydrodynamic stability problems in the two-dimensional Navier-Stokes equations.
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Noemi David (Université Claude Bernard Lyon 1)
Title: On the incompressible limit for porous medium models of tumor growth
Abstract:Both compressible and incompressible models of porous medium type have been used in the literature to describe the mechanical aspects of living tissues. Using a stiff pressure law, it is possible to bridge the gap between these two different representations. In the incompressible limit, density-based models generate free boundary problems of Hele-Shaw type where saturation holds in the moving domain. I will present some results for advection-porous medium equations motivated by tumor growth. The main novelty consists in establishing the limit pressure equation for which it is crucial to prove the strong compactness of the pressure gradient. Then, I will discuss the convergence rate of solutions of the compressible model to solutions of the Hele-Shaw problem.
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Michele Dolce (Ecole Polytechnique Fédérale de Lausanne)
Title: On maximally mixed equilibria of two-dimensional perfect fluid
Abstract: The motion of a two-dimensional perfect fluid can be described as an area preserving rearrangement of the initial vorticity that conserve the kinetic energy. In the infinite time limit, vorticity mixing is conjectured to occur for most initial conditions. A.I. Shnirelman in ’93 introduced the concept of maximally mixed states, by requiring that any further mixing of them would necessarily change their energy, and showed they are perfect fluid equilibria. We offer a new perspective on this theory by proving that many of them can be obtained as minimizers of a variational problem. We also show that maximally mixed states, in general, need not conform to the geometry of the domain. In particular, in a straight periodic channel, we find non stationary states which can be arbitrarily close to any shear flow in L^1 of vorticity but cannot converge back to a shear flow in the long-time limit. This is a joint work with T. D. Drivas.
Document: slides
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Tarek Elgindi (Duke University)
Title: Singularity formation in the incompressible Euler equation in finite and infinite time
Abstract: We will discuss various techniques that have been introduced to establish non-trivial dynamical properties of solutions to the 2d Euler equation, particularly infinite-time singularity formation.
The first lecture will be devoted to basic examples of Arnold's stability theorems as well as proofs of unbounded gradient growth in the 2d Euler equation.
The second lecture will be devoted to a different type of argument for unbounded gradient growth of the vorticity based on establishing unbounded gradient growth of the trajectory map along with a Baire-Category argument.
The third lecture will be devoted to a recent result on the stability of shearing and various applications.
Some papers that might be helpful:
1. S. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation
2. T. Drivas and T. Elgindi, Singularity formation in the incompressible Euler equation in finite and infinite time.
3. T. Drivas, T. Elgindi, and I. Jeong, Twisting in Hamiltonian Flows and Perfect Fluids
4. A. Kiselev, V. Sverak, Small scale creation for solutions of the incompressible two dimensional Euler equation
5. H. Koch, Transport and Instability for Perfect fluids
6. N. Nadirashvili, Wandering solutions of Euler's D-2 equation
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Marie Farge (CNRS-INSMI, LMD-ENS)
Title: How to analyze, model and compute turbulent flows using wavelets?
Document: slides, article.
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Emmanuel Grenier (Ecole Normale Supérieure Lyon) and Toan Nguyen (Penn State University)
Title: Instabilities in Boundary Layers
Abstract: Of great physical and mathematical interest is to establish the asymptotic behavior of solutions to the incompressible Navier-Stokes equations in the vanishing viscosity limit when fluid domains have a physical boundary. A complex structure of boundary layers with different small scales forms and causes great mathematical difficulties. The lectures are aimed to introduce novel analytical approaches, recent results, and conjectures in studying boundary layers.
Document: notes.
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Mei Ming (Yunnan University, China)
Title: Water waves with contact angles and surface tension
Abstract: We prove the local well-posedness of the water waves problem in a 2D domain with contact angles and surface tension. The geometric formulation by Shatah-Zeng is used and adapted here to our case. The contact angle is between $(0, \pi/2)$ and the singularities from related elliptic systems are considered. This talk is based on joint works with Chao Wang.
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Monica Musso (University of Bath, UK)
Title: Long time behavior for vortex dynamics in the 2 dimensional Euler equations
Abstract: The evolution of a two dimensional incompressible ideal fluid with smooth initial vorticity concentrated in small regions is well understood on finite intervals of time: it converges to a super position of Dirac deltas centered at collision-less solutions to the point vortex system, in the limit of vanishing regions. Even though for generic initial conditions the vortex point system has a global smooth solution, much less is known on the long time behavior of the fluid vorticity.
We consider the case of two vortex pairs traveling in opposite directions. Using gluing methods we describe the global dynamics of this configuration. This work is in collaboration with J Davila (U. of Bath), M. del Pino (U of Bath) and S. Parmeshwar (Imperial College London).
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Helena Nussenzveig Lopes (Federal University of Rio de Janeiro)
Title: An introduction to mathematical analysis of incompressible fluid flow
Abstract: These lectures concern the Euler and Navier-Stokes equations, which are models for incompressible fluid flow. The focus is the mathematical analysis of these partial differential equations.
We will discuss the main steps and ideas for local-in-time well-posedness of classical solutions, the problem of singularity formation and the Beale-Kato-Majda criterion and, finally, the issue of vortex stretching in three dimensions.
We then begin discussing weak solutions. We will explain the construction and proof of global-in-time existence of Leray-Hopf weak solutions of the 3D Navier-Stokes equations and the weak-strong uniqueness theorem due to Prodi-Serrin. Lastly we consider the special case of 2D flows and we will discuss results for weak solutions, in particular the Yudovich uniqueness theorem. In view of time constraints it will only be possible to give rough sketches of proofs but we will provide references where further details can be found. These lectures aim to serve as background material for the remaining lectures of the Summer School.
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Antoine Venaille (CNRS, Ecole Normale Supérieure Lyon)
Title: Introduction to Geophysical Flows
Abstract: Atmospheric winds and oceanic currents are strongly influenced by rotation, density stratification, and their small aspect ratio. We will present emblematic examples highlighting peculiar properties of these flows, with an emphasis on phenomena taking place at the equator. A useful general reference for the lectures is the book "Atmospheric and Oceanic Fluid Dynamics", 2nd Edition 2017, Cambridge University Press, by Geoffrey Vallis.
Lecture 1: Rotation
We will focus on a striking consequence of rotation : the trapping of planetary waves along the equator. The discovery of this waveguide in 1966 by Matsuno was the culmination of two centuries of endeavour on the computation of dynamical tides. It is nowadays a central piece in our understanding of climate phenomena such as El Nino.
Keywords: shallow water equations, Coriolis parameter, beta-plane approximation, Rossby waves, Inertial oscillations, Poincaré waves, Kelvin waves, Rossby number, Rossby radius, Laplace tidal equations.
References :
[1] Matsuno, Taroh. "Quasi-geostrophic motions in the equatorial area."Journal of the Meteorological Society of Japan. Ser. II 44, no. 1 (1966): 25-43.
[2] Gallagher, Isabelle, and Laure Saint-Raymond. "On the influence of the Earth’s rotation on geophysical flows." Handbook of mathematical fluid dynamics 4 (2007): 201-329.
Lecture 2: Curvature
We will provide a semi-classical interpretation to the equatorial wave spectrum, relating the existence of two unidirectional modes computed by Matsuno to peculiar topological properties of a dual wave problem introduced by Kelvin in 1880. It turns out that equatorial dynamics bears strong similarities with topological insulators, which are exotic materials that have attracted a lot of attention from physicists over the last decades.
Keywords: Berry curvature, Chern number, Bulk-interface correspondence, Wigner/Weyl transforms, Ray tracing.
References:
[1] Cheverry, Christophe, Isabelle Gallagher, Thierry Paul, and Laure Saint-Raymond. "Semiclassical and spectral analysis of oceanic waves." (2012): 845-892
[2] Faure, Frédéric. "Manifestation of the topological index formula in quantum waves and geophysical waves." arXiv preprint arXiv:1901.10592 (2019).
[3] Delplace, Pierre, J. B. Marston, and Antoine Venaille. "Topological origin of equatorial waves." Science 358, no. 6366 (2017): 1075-1077.
[4] Venaille, Antoine, Yohei Onuki, Nicolas Perez, and Armand Leclerc. "From ray tracing to waves of topological origin in continuous media." SciPost Physics 14, no. 4 (2023): 062.
Lecture 3: Stratification
Up in the stratosphere, equatorial winds change direction each 14 months. This quasi biennal oscillation is the archetype of a wave - mean flow interaction problem in geophysical fluid dynamics. It involves a two-way coupling between slowly varying winds and fast oscillating internal gravity waves that propagate in density stratified flows. We will present a dynamical system introduced in the seventies to account for this phenomenon.
Keywords: buoyancy, Boussinesq equations, internal gravity waves, streaming, doppler shift, quasilinear approach, Taylor-Goldstein equation, Lindzen-Holton Plumb model.
References:
[1] Plumb, R. A. "The interaction of two internal waves with the mean flow: Implications for the theory of the quasi-biennial oscillation." Journal of Atmospheric Sciences 34, no. 12 (1977): 1847-1858.
[2] Renaud, Antoine, and Antoine Venaille. "On the Holton–Lindzen–Plumb model for mean flow reversals in stratified fluids." Quarterly Journal of the Royal Meteorological Society 146, no. 732 (2020): 2981-2997.
