Initial boundary value problems for hyperbolic systems, and dispersive perturbations

D. Lannes, Lecture notes volume of the Winter School on Fluid Dynamics, Dispersive Equations and Quantum Fluids (Bressanone, december 2018), to appear. Download

The goal of these notes is to point out similarities and differences between two kinds of initial boundary value problems in dimension one. The first one concerns hyperbolic systems (such as the nonlinear shallow water equations) while the second one concerns dispersive perturbations of such systems (such as Boussinesq systems). In the absence of a boundary, that is, for the initial value problem, the link between both classes is quite obvious but in the presence of a boundary, the situation is more complex and dispersive boundary layers must be understood if one wants to understand the links between both classes of problems. After reviewing several types of initial boundary value problems (some of which being free boundary problems) arising in the study of waves in shallow water, we sketch the general theory for hyperbolic initial boundary value problems developed in [18] and that encompasses all of the above examples that involve hyperbolic systems. Such a general theory does not exist for dispersive perturbations of hyperbolic systems, but we treat two important examples involving Boussinesq systems. In the first one, we show that the nature of the initial boundary value problem shares little in common with the hyperbolic configuration. For instance, the problem has the structure of an ODE and no higher order compatibility conditions on the data are required to have solutions of high regularity. These differences naturally raise the questions of the control of the time of existence and of the dispersionless limit; they are addressed in a second example motivated by a wave-structure interaction problem. We explain the approach developed in [13] to treat this problem, pointing out the role played by dispersive boundary layers.

Offre de postdoc: Modélisation des vagues extrêmes en milieu littoral

Encadrement: Philippe Bonneton (EPOC) et David Lannes (IMB)

Le postdoc est pour une durée de 1 an et commencerait en septembre/octobre 2022. Il s’agit d’un projet interdisciplinaire financé par l’Institut des Mathématiques pour la Planète Terre et consacré à l’étude de la formation des vagues extrêmes en milieu côtier. Ce sujet comporte de l’analyse mathématique (EDP, proba-stat), des simulations numériques et de la modélisation physique. Une expertise sur aux moins deux de ces aspects est souhaitée. La description détaillée du sujet est disponible ici.

Relation between orbital velocities, pressure, and surface elevation in nonlinear nearshore water waves.

K. Martins, P. Bonneton, D. Lannes, and H. Michallet, Journal of Physical Oceanography 51 (2021), 3539–3556. Download

The inability of the linear wave dispersion relation to characterize the dispersive properties of nonlinear shoaling and breaking waves in the nearshore has long been recognized. Yet, it remains widely used with linear wave theory to convert between subsurface pressure, wave orbital velocities, and the free surface elevation associated with nonlinear nearshore waves. Here, we present a nonlinear fully dispersive method for reconstructing the free surface elevation from subsurface hydrodynamic measurements. This reconstruction requires knowledge of the dispersive properties of the wave field through the dominant wavenumbers magnitude k, representative in an energy-averaged sense of a mixed sea state composed of both free and forced components. The present approach is effective starting from intermediate water depths—where nonlinear interactions between triads intensify—up to the surf zone, where most wave components are forced and travel approximately at the speed of nondispersive shallow-water waves. In laboratory conditions, where measurements of k are available, the nonlinear fully dispersive method successfully reconstructs sea surface energy levels at high frequencies in diverse nonlinear and dispersive conditions. In the field, we investigate the potential of a reconstruction that uses a Boussinesq approximation of k, since such measurements are generally lacking. Overall, the proposed approach offers great potential for collecting more accurate measurements under storm conditions, both in terms of sea surface energy levels at high frequencies and wave-by-wave statistics (e.g., wave extrema). Through its control on the efficiency of nonlinear energy transfers between triads, the spectral bandwidth is shown to greatly influence nonlinear effects in the transfer functions between subsurface hydrodynamics and the sea surface elevation.

Ability of the nonlinear method to reconstruct an extreme wave (below in green, plot of the nonlinear correction)

Freely floating objects on a fluid governed by the Boussinesq equations

G. Beck, D. Lannes, Ann. IHP/Analyse non linéaire, to appear Download

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis.

Waves interacting with an object moving vertically

Influence of dispersion on the oscillations of an object released from an out of equilibrium position

Conférence Singflows

12-14 Avril, Institut de Mathématiques de Bordeaux
(salle de conférence)


Mardi 12 avril

11h00-13h00 Christophe Lacave : Synthèse des travaux récents dans la thématique « vortices »


14h00-15h00 Charlotte Perrin: Well-posedness of partially congested Navier-Stokes equation

15h00-16h00 Florent Noisette: Symbolic symmetriser and estimates for entering energy

16h00-16h15 Pause

16h15-17h15 Geoffrey Beck : Weak turbulence for water waves

Mercredi 13 avril

9h15-10h15 Richard Höfer: On the derivation of viscoelastic models for rod-like suspensions

10h15-10h30 Pause

10h30-12h30 David Lannes: Synthèse des travaux récents dans la thématique « objets flottants »

14h00-15h00 Martina Magliocca: Bifurcation results for a coupled incompressible Darcy’s free boundary problem with surface tension

15h00-16h00 Matthieu Menard: Étude d’un modèle de spray gyrocinétique

16h00-16h15 Pause

16h15-17h15 Ludovic Godard-Cadillac: Hölder regularity for collapses of point-vortices

Congress dinner

Jeudi 14 avril

9h15-10h15 Roberta Bianchini: Inviscid damping for the two-dimensional stably stratified Couette flow

10h15-10h30 Pause

10h30-12h30 Charles Collot: Singularity formation for the Burgers equation and unsteady separation for the Prandtl system

Résumés des exposés de synthèse

Charles Collot : Singularity formation for the Burgers equation and unsteady separation for the Prandtl system. We will study the first time at which a shock forms for the Burgers and Prandtl equations, triggering separation of the boundary layer for the latter ones. The first part will illustrate the main issues of singularity formation for the toy model of the Burgers equation, emphasising the role played by backward self-similar solutions. For this equation everything is explicit and computable by hand, a fact that is seldom noticed! The second part will focus on the inviscid unsteady Prandtl system ; where a complete – truly 2-dimensional – description of singularities can still be carried on. Finally, the third part will deal with the full Prandtl’s system, where we are able to handle viscosity effects on an axis, and show that analytic solutions remain analytic around this axis up to blow-up time with a universal bound for their analyticity radius. These are joint works with T.E. Ghoul, S. Ibrahim, and N. Masmoudi.

Christophe Lacave (Grenoble): Synthèse des travaux récents dans la thématique « vortices ». L’objectif de cet exposé est de faire un tour d’horizon de ce qui a été réalisé dans les 3 dernières années concernant la partie « vortices » de notre projet ANR et de faire un bilan des questions qui restent ouvertes. Je commencerai par les diverses contributions concernant la justification des équations du flot par courbure binormale, puis je parlerai des études récentes sur ce modèle. Je finirai mon panorama par une analogie classique entre les équations axi-symétriques sans swirl et les équations des lacs.

David Lannes (Bordeaux): Synthèse des travaux récents dans la thématiques « objets flottants ». On présentera dans cet exposé les principales avancées menées dans le cadre de cette ANR, et également par d’autres équipes, pour décrire l’interaction des vagues avec des objets flottants. On insistera sur plusieurs questions mathématiques dont l’intérêt dépasse ce cadre applicatif, comme les problèmes mixtes pour perturbations dispersives de systèmes hyperboliques. On essaiera de faire le lien avec d’autres problématiques faisant intervenir des écoulements congestionnés

Résumés des exposés

Geoffrey Beck (ENS Paris): Wave turbulence for the water waves equations.
In water-waves equations, the dispersion tends to force the solution to spread and non-linearity creates interactions between some oscillating modes. There are too many interactions to try to describe each Fourier mode evolution only by deterministic features and one rapidly needs to consider a statistical average between them. Wave-turbulence manages to give a quantitative description of the effective balance of the mean energy input from a source at low wave-numbers (such as wind for ocean water-waves), transfer of energy through reversible non-linearities to higher and higher wave-numbers. A heuristic derivation of Wave-Kinetic Equation (WKE) was proposed in the context of deep gravity water-waves by Hasselmann and Zakharov. The collision operator of (WKE) represents, in that context, the interactions between four resonant waves. Wave turbulence shares with traditional hydrodynamic turbulence three characteristic concepts : randomness, scalings and cascades. The « irreversibility » of the migration from low to high wave-number is a consequence of the quasi-resonance mechanism. Fortunately, recent math papers deal with rigorous derivation of the wave-kinetic equation for the cubic NLS equation. Recently Deng-Hani reached the expected kinetic time scale with one scaling law. The non-linearity for water-waves is really much more complicated than cubic NLS. Indeed, the non-linearities are given in particular by a Dirrichlet-to-Neuman operator which maps the trace of the velocity potential on a wave surface to the speed of deformation of the wave-surface. An asymptotic expansion leads to dyadic, cubic, quadratic, quintic… non-linearities at different scales. What is a good regime for water-waves turbulence

Roberta Bianchini: Inviscid damping for the two-dimensional stably stratified Couette flow. In this talk, we discuss the asymptotic stability of the two-dimensional inviscid Boussinesq equationsnear a stably stratified Couette flow, for small initial perturbations in a suitable Gevrey class. Under the classical Miles-Howard stability condition on the Richardson number, we prove that the system experiences a shear-buoyancy instability: the density variation and velocity undergo an inviscid damping of decay rate of order $t^{-1/2}$, while the vorticity and density gradient grow as $t^{1/2}$, within a certain time-scale dictated by the size of the initial fluctuations.This is a joint work with Jacob Bedrossian, Michele Coti Zelati and Michele Dolce.

Ludovic Godard-Cadillac : Hölder regularity for collapses of point-vortices. The first part of this talk studies the collapses of point-vortices for the Euler equation in the plane and for surface quasi-geostrophic equations in the general setting of $\alpha$ models. This consists in a Biot-Savart law with a kernel being a power function of exponent $-\alpha$. It is proved that, under a standard non-degeneracy hypothesis, the trajectories of the vorticies have a regularity Hölder on $[0,T]$ with $T$ the time of collapse. The Hölder exponent obtained is $1/(\alpha+1)$ and this exponent is proved to be optimal for all $\alpha$ by exhibiting an example of a $3$-vortex collapse.
The same question is then addressed for the Euler point-vortex system in smooth bounded connected domains.
It is proved that if a given vortex has an adherence point in the interior of the domain, then it converges toward this point and is Hölder continuous.
This is joint work with Martin Donati.

Richard Höfer (IMJ): On the derivation of viscoelastic models for rod-like suspensions. We consider effective properties of suspensions of inertialess, rigid, anisotropic, Brownian particles in Stokes flows. Recent years have seen tremendous progress regarding the rigorous justification of effective fluid equations for non-Brownian suspensions, where the complex fluid can be described in terms of an effective viscosity. In contrast to this (quasi-)Newtonian behavior, anisotropic Brownian particles cause an additional elastic stress on the fluid. A rigorous derivation of such visco-elastic systems starting from particle models is completely missing so far. In this talk I will present first results in this direction starting from simplified microscopic models where the particles evolve only due to rotational Brownian motion and cause a Brownian torque on the fluid. In the limit of infinitely many small particles with vanishing particle volume fraction, we rigorously obtain the elastic stress on the fluid in terms of the particle density that is given as the solution to an (in-)stationary Fokker-Planck equation.
Joint work with Marta Leocata (LUISS Rome) and Amina Mecherbet (Université de Paris)

Martina Magliocca (ENS Paris Saclay): Bifurcation results for a coupled incompressible Darcy’s free boundary problem with surface tension .In this talk, we will focus on traveling wave bifurcation results for an incompressible Darcy free boundary problem that describes cell motility. We will also compare two different techniques to prove the existence of bifurcation points: the Crandall-Rabinowitz argument and the Leray-Schauder degree theory.
This is a joint work with Thomas Alazard (Centre Borelli ENS Paris-Saclay) and Nicolas Meunier (University of Évry).

Matthieu Ménard (Grenoble): Étude d’un modèle de spray gyrocinétique. On présentera l’étude d’un modèle représentant une phase diffuse de solides plongée dans un fluide plan incompressible non visqueux. On justifiera tout d’abord son caractère localement bien posé en temps puis on montrera qu’il peut s’obtenir comme limite de champ moyen d’un système composé d’un nombre finis de solides plongés dans un fluide.

Florent Noisette (Bordeaux): Symbolic symetriser and estimates for entering energy. The 2D euler equation is well studied and understood. The evolution equation for the vorticity beeing a coupling between a transport problem and an elliptic problem make it simpler to tackle. Yudovich was the one to take on the case of a bounded domain with open boundary -meaning there is fluid entering and exiting the domain-, using sharp elliptic estimates. However to do so he needed a stronger regularity that what is normally needed in the case of an impermeable boundary. We discuss a technique first introduce by Papin and Weigant to enhance this result through the help of a smart test function.

Charlotte Perrin (Marseille): Well-posedness of partially congested Navier-Stokes equations. This talk addresses the mathematical analysis of 1D Navier-Stokes equations including a maximum packing constraint, that is a maximal constraint on the density. These equations arise naturally in the modeling of mixtures like suspensions or in the modeling of collective motion. The main feature of the model is the co-existence of two different phases. In the congested phase, the pressure is free and the dynamics is incompressible, whereas in the non-congested phase, the fluid obeys a pressureless compressible dynamics. I will discuss the Cauchy problem for initial data which are small perturbations in the non-congested zone of travelling wave profiles. This is a joint work with Anne-Laure Dalibard.

Wave-Structure Interactions and Wave Energies

Siam News Volume 54 | Number 04 | May 2021 Online article

Some issues on the mathematical analysis of wave-structure interaction are presented, and their relevance for the development of wave energies are reviewed.

Creation of the institute « Mathematics of the Planet Earth »

ICIAM Dianoia, Volume 9, Issue 2, April 2021 Online article

Interview with Laure Saint-Raymond and Arnaud Guillin, directors of the Institute of Mathematics of the Planet Earth

Interview: Building interdisciplinarity around Ocean sciences

ICIAM Dianoia, Volume 9, Issue 1, January 2021 Online article

The UN general assembly has proclaimed the Decade of Ocean Science for Sustainable Development (2021-30). The French National Centre for Scientific Research (CNRS) is among the world’s leading research institutions. Its scientists explore the living world, matter, mathematics, the universe, and the functioning of human societies in order to meet the major challenges of today and tomorrow. How does this institution plan to address the specific issue of the ocean, and what role may mathematics play?

Anne Corval, advisor to the scientific director of the CNRS, coordinated a Task Force Ocean to prepare the institution for this Decade, and Fabrizio D’Ortenzio, oceanographer at the CNRS, was one of the animators of the meetings organized to gather contributions from all sciences. In this interview, they address this question.

Modeling shallow water waves

D. Lannes, Nonlinearity 33 (2020), R1 Download

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the ‘turbulent’ and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe– Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d = 1; among them are the KdV, BBM, Camassa–Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

B. Desjardins, D. Lannes, J.-C. Saut, Water Waves (2020), 1-40 Download

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approxima-tion (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems.