Annales de la Faculté des Sciences de Toulouse

Thierry Gallay

Infinite energy solutions of the two-dimensional Navier–Stokes equations

Stéphane Druel

Regular foliations on weak Fano manifolds

Pietro Caputo

Julien Sohier

Convergence to equilibrium for a directed $(1+d)-$dimensional polymer

Jean Dolbeault

Maria J. Esteban

Michael Loss

Interpolation inequalities on the sphere: linear <i>vs.</i> nonlinear flows

Guillaume Deschamps

Espace des twisteurs d’une variété quaternionique Kähler généralisée

Dominique Bakry

Serge Cohen

Martin Hairer

Jean-Michel Roquejoffre

ANNALES
DE LA FACULTÉ
DES SCIENCES
Mathématiques
DOMINIQUE BAKRY, SERGE COHEN,
MARTIN HAIRER AND JEAN-MICHEL ROQUEJOFFRE
Preface
Tome XXVI, no 4 (2017), p. i-ii.
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Preface
This volume assembles a collection of expository articles based on some
of the courses delivered in Toulouse from February to May 2014, as part of
a program entitled Trimestre thématique EDP et Probabilités(1) and con-
centrated on the interactions between probability and partial differential
equations. These courses do not cover the full scope of the program, but
nevertheless present a quite large overview of the topics involved.
Interplay between probability theory and partial differential equations
has a long standing history, perhaps going back to Einstein’s 1905 paper
on Brownian motion. On the one hand, the law of many natural random
objects (Markov diffusion processes for example) are given through solutions
of partial differential equations (mostly of parabolic nature), and on the other
hand, many natural equations, arising for example from fluid mechanics or
statistical physics, have natural probabilistic representations.
At the present time, these topics are becoming more and more involved.
First, one is led to study specific partial differential equations, linear or
not, in which randomness appears either in the initial data, or in the evo-
lution process such as random forces. As such, stochastic partial differential
equations (such as KPZ, Φ4d fields, to name but a few) are under rapid de-
velopment, and require sophisticated tools such as regularity structures.
Randomness may also appear at another level, looking at partial differen-
tial equations from a geometric measure theory point of view. Indeed, even
first order differential equations (evolutions of particles, integration of vector
fields) may be studied in this context, where random vector fields or random
loops are in general not smooth and require for their study various tools of
probability theory. BV vector fields, rough path approaches, point particle
approximations, are some of the many tools which are used in this context
and exposed in this volume.
Another important aspect is data assimilation, where one has to couple
evolution equations (arising from fluid mechanics or geosciences) with the
observed data, in particular for the purpose of weather forecast. This may
(1) This thematic program took place within the activities of the LabEx CIMI (Centre
International de Mathématiques et d’Informatique de Toulouse) and was partially sup-
ported by ANR-11-LABX-0040-CIMI within the program ANR-11-IDEX-0002-02. The
organizers thank the CIMI staff for their invaluable help during this event.
– i –
be sought as part of nonlinear filtering theory, where the random unknown
state of a fluid has to be conditioned with the help of the observation. This
requires an important algorithmic effort, since those laws, when discretized,
are high dimensional and may not be handled by usual methods.
Finally, convergence to equilibrium and long time behavior are often con-
trolled through precise functional inequalities, which in turn may be studied
through evolution equations.
The choice of the talks presented here aims at providing an overview on
these different aspects.
D. Bakry, S. Cohen, M. Hairer, J.-M. Roquejoffre
– ii –


Preface

Michał Kowalczyk

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

Bernard Malgrange

Sur le problème d’équivalence de Cartan

Stepan Yu. Orevkov

Automorphism group of the commutator subgroup of the braid group

Mohammad Daher

Interpolation des opérateurs de Radon–Nikodym et des espaces $L_\Lambda ^p$, $\mathbf{h}_\Lambda ^p$

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Infinite energy solutions of the two-dimensional Navier–Stokes equations

Regular foliations on weak Fano manifolds

Convergence to equilibrium for a directed $(1+d)-$ dimensional polymer

Interpolation inequalities on the sphere: linear <i>vs.</i> nonlinear flows

Espace des twisteurs d’une variété quaternionique Kähler généralisée

Preface

Uniqueness and rigidity in nonlinear elliptic equations, interpolation inequalities, and spectral estimates

Sur le problème d’équivalence de Cartan

Automorphism group of the commutator subgroup of the braid group

Interpolation des opérateurs de Radon–Nikodym et des espaces $L_\Lambda ^p$ , $\mathbf{h}_\Lambda ^p$

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