Pierre Germain

Courant Institute of Mathematical Sciences

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Nader Masmoudi

New York University

Jalal Shatah

Courant Institute of Mathematical Sciences

Zaher Hani

Georgia Institute of Technology

Jacob Bedrossian

University of California, Los Angeles

Laurent Thomann

Université de Lorraine

Frédéric Bernicot

Laboratoire de Mathématiques Jean Leray

Tristan Buckmaster

Princeton University

Zaher Hani

University of Michigan–Ann Arbor

Frédéric Rousset

Centre National de la Recherche Scientifique

Fabio Pusateri

University of Toronto

Cooperative Institutions

Centre National de la Recherche Scientifique

Commissariat à l'Énergie Atomique et aux Énergies Alternatives

Institut National de Recherche pour l'Agriculture, l'Alimentation et l'Environnement

New York University

Institut de Radioprotection et de Sûreté Nucléaire

Courant Institute of Mathematical Sciences

Académie de Paris

Institut National de la Recherche Agronomique

Sorbonne Université

Université Paris Cité

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On the stability threshold for the 3D Couette flow in Sobolev regularity

Jacob Bedrossian Pierre Germain Nader Masmoudi

We study Sobolev regularity disturbances to the periodic, plane Couette flow in the 3D incompressible Navier-Stokes equations at high Reynolds number $\textbf{Re}$. Our goal is to estimate how the stability threshold scales in $\textbf{Re}$: the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In this work we prove that initial data which satisfies $\| u_{in} \|_{H^\sigma} \leq \delta\textbf{Re}^{-3/2}$ for any $\sigma > 9/2$ and some $\delta = \delta(\sigma) > 0$ depending only on $\sigma$, is global in time, remains within $O(\textbf{Re}^{-1/2})$ of the Couette flow in $L^2$ for all time, and converges to the class of 2.5 dimensional streamwise-independent solutions referred to as streaks for times $t \gtrsim \textbf{Re}^{1/3}$. Numerical experiments performed by Reddy et. al. with rough initial data estimated a threshold of $\sim \textbf{Re}^{-31/20}$, which shows very close agreement with our estimate.

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Dynamics Near the Subcritical Transition of the 3D Couette Flow II: Above Threshold Case

Memoirs of the American Mathematical Society (2022)

Jacob Bedrossian Pierre Germain Nader Masmoudi

This is the second in a pair of works which study small disturbances to the plane, periodic 3D Couette flow in the incompressible Navier-Stokes equations at high Reynolds number Re. In this work, we show that there is constant 0 > c 0 ≪ 1 0 > c_0 \ll 1 , independent of R e \mathbf {Re} , such that sufficiently regular disturbances of size ϵ ≲ R e − 2 / 3 − δ \epsilon \lesssim \mathbf {Re}^{-2/3-\delta } for any δ > 0 \delta > 0 exist at least until t = c 0 ϵ − 1 t = c_0\epsilon ^{-1} and in general evolve to be O ( c 0 ) O(c_0) due to the lift-up effect. Further, after times t ≳ R e 1 / 3 t \gtrsim \mathbf {Re}^{1/3} , the streamwise dependence of the solution is rapidly diminished by a mixing-enhanced dissipation effect and the solution is attracted back to the class of “2.5 dimensional” streamwise-independent solutions (sometimes referred to as “streaks”). The largest of these streaks are expected to eventually undergo a secondary instability at t ≈ ϵ − 1 t \approx \epsilon ^{-1} . Hence, our work strongly suggests, for all (sufficiently regular) initial data, the genericity of the “lift-up effect ⇒ \Rightarrow streak growth ⇒ \Rightarrow streak breakdown” scenario for turbulent transition of the 3D Couette flow near the threshold of stability forwarded in the applied mathematics and physics literature.

10.1090/memo/1377

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Citations (23)

Bilinear dispersive estimates via space-time resonances. Part I : the one dimensional case

arXiv (Cornell University) (2011)

Frédéric Bernicot Pierre Germain

We prove new bilinear dispersive estimates. They are obtained and described via a bilinear time-frequency analysis following the space-time resonances method, introduced by Masmoudi, Shatah, and the second author. They allow us to understand the large time behavior of solutions of quadratic dispersive equations.

10.48550/arxiv.1110.4794

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Les sols des oasis du Sahara algérien, source d’actinomycètes, rares producteurs d’antibiotiques

Science et changements planétaires / Sécheresse (1998)

Nasserdine Sabaou Hadjira Boudjella Achour Bennadji Abdellah Mostefaoui Abdelghani Zitouni

Les sols des oasis du Sahara algerien, bien que soumis a un climat aride, se sont reveles relativement riches en actinomycetes parfois reputes rares de par le monde et de surcroit producteurs d’antibiotiques. On les trouve non seulement dans les horizons de surface mais aussi a plus de 2 m de profondeur et en quantite appreciable. Cependant, si les horizons de surface contiennent des especes tres diversifiees, en revanche, chaque horizon profond est caracterise par la dominance nette d’une espece donnee. Au total, 727 souches ont ete isolees et rattachees a pres de 25 genres et 132 especes, des plus connus aux plus rares. Certaines souches semblent meme representer de nouvelles especes. Quelques-unes secretent des antibiotiques actifs contre plusieurs germes pathogenes. La possibilite de molecules actives originales n’est pas a ecarter, comme l’ont montre a cet effet les etudes preliminaires. Ceci montre donc a quel point les sols des oasis sahariennes restent encore a exploiter, particulierement dans le domaine des antibiotiques, et ceci surtout au vu de la recrudescence des maladies bacteriennes et fongiques observees ces dernieres annees.

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Backward self-similar solutions for compressible Navier-Stokes equations

arXiv (Cornell University) (2019)

Pierre Germain Tsukasa Iwabuchi Tristan Léger

This article is devoted to backward self-similar blow up solutions of the compressible Navier-Stokes equations with radial symmetry. We show that such solutions cannot exist if they either satisfy an appropriate smallness condition, or have finite energy. Furthermore, numerical simulations lead us to the conjecture that such solutions do not exist.

Symmetry in biology

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On the derivation of the homogeneous kinetic wave equation

arXiv (Cornell University) (2019)

Charles Collot Pierre Germain

The nonlinear Schr\"odinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localisation of the data and on the strength of the nonlinearity, on which the expectation of the squares of moduli of Fourier modes evolve according to an effective equation: the so-called kinetic wave equation. When the kinetic time for our setup is $1$, we prove this conjecture up to an arbitrarily small polynomial loss. When the kinetic time is larger than $1$, we obtain its validity on a more restricted time scale. The key idea of the proof is the use of Feynman interaction diagrams both in the construction of an approximate solution and in the study of its nonlinear stability. We perform a truncated series expansion in the initial data, and obtain bounds in average in various function spaces for its elements. The linearised dynamics then involves a linear Schr\"odinger equation with a corresponding random potential. We bound the expectation of the operator norm in Bourgain spaces using diagrams and random matrix tools. This gives a new approach for the analysis of nonlinear wave equations out of equilibrium, and gives hope that refinements of the method could help settle the conjecture.

10.48550/arxiv.1912.10368

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The weakly nonlinear large box limit of the 2D cubic nonlinear Schr\"odinger equation

arXiv (Cornell University) (2013)

Erwan Faou Pierre Germain Zaher Hani

We consider the cubic nonlinear Schr\odinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\R^2$ that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.

Hamiltonian (control theory)

Periodic boundary conditions

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On the Cubic Lowest Landau Level Equation

Archive for Rational Mechanics and Analysis (2018)

Patrick Gérard Pierre Germain Laurent Thomann

Stationary state

Complex system

Landau quantization

10.1007/s00205-018-1295-4

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Standing and Traveling Waves in a Nonlinearly Dispersive Lattice Model

arXiv (Cornell University) (2023)

Ross Parker Pierre Germain Jesús Cuevas–Maraver Alejandro B. Aceves P. G. Kevrekidis

In the work of Colliander et al. (2010), a minimal lattice model was constructed describing the transfer of energy to high frequencies in the defocusing nonlinear Schr\"odinger equation. In the present work, we present a systematic study of the coherent structures, both standing and traveling, that arise in the context of this model. We find that the nonlinearly dispersive nature of the model is responsible for standing waves in the form of discrete compactons. On the other hand, analysis of the dynamical features of the simplest nontrivial variant of the model, namely the dimer case, yields both solutions where the intensity is trapped in a single site and solutions where the intensity moves between the two sites, which suggests the possibility of moving excitations in larger lattices. Such excitations are also suggested by the dynamical evolution associated with modulational instability. Our numerical computations confirm this expectation, and we systematically construct such traveling states as exact solutions in lattices of varying size, as well as explore their stability. A remarkable feature of these traveling lattice waves is that they are of "antidark" type, i.e., they are mounted on top of a non-vanishing background. These studies shed light on the existence, stability and dynamics of such standing and traveling states in $1+1$ dimensions, and pave the way for exploration of corresponding configurations in higher dimensions.

Lattice (music)

Standing wave

Modulational Instability

10.48550/arxiv.2309.11649

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Global solutions for the gravity water waves equation in dimension 3

Annals of Mathematics (2012)

Pierre Germain Nader Masmoudi Jilal Shatah

We show existence of global solutions for the gravity water waves equation in dimension 3, in the case of small data.The proof combines energy estimates, which yield control of L 2 related norms, with dispersive estimates, which give decay in L ∞ .To obtain these dispersive estimates, we use an analysis in Fourier space; the study of space and time resonances is then the crucial point. • Since] and the mean curvature is given by κ

10.4007/annals.2012.175.2.6

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Citations (324)