On some properties of the curl operator and their consequences for the Navier-Stokes system
with François Vigneron,
file on HAL,
file on arXiv,
We investigate some geometric properties of the curl operator, based on its diagonalization
and its expression as a non-local symmetry of the pseudo-derivative (-Δ)1/2 among divergence-free vector fields
with finite energy. In this context, we introduce the notion of spin-definite fields, i.e. eigenvectors of
The two spin-definite components of a general 3D incompressible flow untangle the right-handed motion from the left-handed one.
Having observed that the non-linearity of Navier-Stokes has the structure of a cross-product
and its weak (distributional) form is a determinant that involves the vorticity, the velocity and a test function,
we revisit the conservation of energy and the balance of helicity in a geometrical fashion. We show that in the case
of a finite-time blow-up, both spin-definite components of the flow will explose simultaneously and with equal rates,
i.e. singularities in 3D are the result of a conflict of spin, which is impossible in the poorer geometry of 2D flows.
We investigate the role of the local and non-local determinants
∫ ℝ3 det(curl u, u, (-Δ)θ u) dx dt
and their spin-definite counterparts, which drive the enstrophy and, more generally, are responsible for
the regularity of the flow and the emergence of singularities or quasi-singularities.
As such, they are at the core of turbulence phenomena.
Integrating the Wigner Distribution on subsets of the phase space, a Survey,
file on arXiv, February 2022.
We review several properties of integrals of the Wigner distribution on subsets of the phase space. Along our way, we provide a theoretical proof of the invalidity of Flandrin's conjecture, a fact already proven via numerical arguments in our joint paper [MR4054880]
with B. Delourme and T. Duyckaerts. We use also the J.G. Wood & A.J. Bracken paper [MR2131219],
for which we offer a mathematical perspective.
We review thoroughly the case of subsets of the plane whose boundary is a conic curve and show that Mehler's formula can be helpful in the analysis of these cases, including for the higher dimensional case investigated in the paper
E. Lieb and Y. Ostrover.
Using the Feichtinger algebra, we show that, generically in the Baire sense,
the Wigner distribution of a pulse in
does not belong to
providing as a byproduct a large class of examples of subsets of the phase space
ℝ2n on which the integral of the Wigner distribution is infinite.
We study as well the case of convex polygons of the plane, with a rather weak estimate depending on the number of vertices, but independent of the area of the polygon.
An energy method for averaging lemmas,
with Diogo Arsénio,
Pure and Applied Analysis,
3 (2021), no. 2, 319-362,
file on arXiv, June 2020.
This work introduces a new approach to velocity averaging lemmas in kinetic theory. This approach---based upon the classical energy method---provides a powerful duality principle in kinetic transport equations which allows for a natural extension of classical averaging lemmas to previously unknown cases where the density and the source term belong to dual spaces. More generally, this kinetic duality principle produces regularity results where one can trade a loss of regularity or integrability somewhere in the kinetic transport equation for a suitable opposite gain elsewhere. Also, it looks simpler and more robust to rely on proving inequalities instead of constructing exact parametrices. The results in this article are introduced from a functional analytic point of view and they are motivated by the abstract regularity theory of kinetic transport equations.
Carleman Inequalities: an Introduction and More,
a new book, published in 2019
in the Springer-Verlag Series
Grundlehren der Mathematischen Wissenschaften
This is a 576-page book, dealing with various aspects of Carleman inequalities.
link, you will find a short description of the contents of the book as well as some previews opportunities.
On the anti-Wick symbol as a Gelfand-Shilov generalized function,
with Laurent Amour and Jean Nourrigat,
file on arXiv, May 2019,
published in the
Proceedings of the American Mathematical Society
148 (2020), 7, 2909-2914.
The purpose of this article is to prove that the anti-Wick symbol of an operator mapping
which is generally not a tempered distribution, can still be defined as a Gelfand-Shilov generalized function.
This result relies on test function spaces embeddings involving the Schwartz and Gelfand-Shilov spaces. An
additional embedding concerning Schwartz and Gevrey spaces is also given.
On integrals over a convex set of the Wigner distribution,
with Bérangère Delourme and Thomas Duyckaerts,
Journal of Fourier Analysis and Applications,
volume 26, February 2020.
We provide an example of a normalized
function u such
that its Wigner distribution W(u,u) has an integral >1 on the
for a suitable choice of a. This provides a
negative answer to a question raised by P. Flandrin in 1988. Our arguments are
based upon the study of the Weyl quantization of the indicatrix of
along with a precise numerical analysis of its
Mehler's formula and functional calculus,
Science China Mathematics, 62 (2019), no. 6, 1143-1166.
We show that Mehler's formula can be used to handle several formulas involving the quantization of singular Hamiltonians. In particular, we diagonalize in the Hermite basis the Weyl quantization of the characteristic function of several domains of the phase space.
Unique continuation through transversal characteristic hypersurfaces,
Journal d'Analyse Mathématique
,138, (2019), no.1, 135-156.
We prove a unique continuation result for an ill-posed characteristic problem.
A model problem of this type occurs in A.D. Ionescu & S. Klainerman article
(Theorem 1.1 in [MR2470908]) and we extend their model-result using only geometric assumptions.
The main tools are Carleman estimates and Hörmander's pseudo-convexity conditions.