Recent papers
On some properties of the curl operator and their consequences for the Navier-Stokes system
with François Vigneron,
Communications in Mathematical Research,
38 (2022), no.4, 449-497,
file on HAL,
file on arXiv,
March 2022.
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
(-Δ)-1/2 curl.
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
∫ [0,T]
∫ 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.
An energy method for averaging lemmas,
with Diogo Arsénio,
Pure and Applied Analysis,
3 (2021), no. 2, 319-362,
file on arXiv,
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.
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
S(ℝn) into
S'(ℝn),
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
L2(ℝ)
function u such
that its Wigner distribution W(u,u) has an integral >1 on the
square [0,a]×[0,a]
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
ℝ+×ℝ+
along with a precise numerical analysis of its
discretization.
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.
Sur deux contributions de Y. V. Egorov (1938-2018),
Annales de la Faculté des Sciences de Toulouse, (6) 28 (2019), no. 1, 1-9.
The Mathematician Yuri Egorov died on October 6, 2018 in Toulouse. This text outlines two fundamental aspects of his work, the quantification of canonical transformations and the study of subelliptic operators.
Some natural subspaces and quotient spaces of L1,
with Gilles Godefroy,
Advances in Operator Theory,
3 (2018), no.1, 73-86,
paper online.
Unique continuation through transversal characteristic hypersurfaces,
file on arXiv,
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.
The onset of instability in first-order systems,
with Toan T. Nguyen and Benjamin Texier,
Journal of the European Mathematical Society,
20, 6, 1303-1373, (2018),
paper on the JEMS website.
We study in this paper the Cauchy problem for first-order quasi-linear systems of partial differential equations. When the spectrum of the initial principal symbol is not included in the real line, i.e., when hyperbolicity is violated at initial time, then the Cauchy problem is strongly unstable, in the sense of Hadamard. This phenomenon, which extends the linear Lax-Mizohata theorem, was explained by G. Métivier in
Remarks on the well-posedness of the nonlinear Cauchy problem,
(Contemp. Math. 2005). In this article,
we are interested in the transition from hyperbolicity to non-hyperbolicity, that is the limiting case where hyperbolicity holds at initial time, but is violated at positive times: under such an hypothesis, we generalize a recent work by
N. Lerner, Y. Morimoto and C.-J. Xu,
Instability of the Cauchy-Kovalevskaya solution for a class of non-linear systems,
(American J. Math. 2010), on complex scalar systems, as we prove that even a weak defect of hyperbolicity implies a strong Hadamard instability. Our examples include Burgers systems, Van der Waals gas dynamics, and Klein-Gordon-Zakharov systems. Our analysis relies on an approximation result for pseudo-differential flows, introduced by B. Texier in
Approximations of pseudo-differential flows,
file on arXiv.
Energy decay for a locally undamped wave equation,
with Matthieu Léautaud,
Annales de la Faculté des Sciences de Toulouse,
Sér. 6, 26 no. 1 (2017), p. 157-205,
paper online.
We study the decay rate for the energy of solutions of a damped wave equation in a situation where the
Geometric Control Condition is violated. We assume that the set of undamped trajectories is a flat torus of positive codimension and that the metric is locally flat around this set. We further assume that the damping function enjoys locally a prescribed homogeneity near the undamped set in traversal directions.
We prove a sharp decay estimate at a polynomial rate that depends on the homogeneity of the damping function.
Our method relies on a refined microlocal analysis linked to a second microlocalization procedure
to cut the phase space into tiny regions respecting the uncertainty principle
but way too small to enter a standard semi-classical analysis localization.
Using a multiplier method, we obtain the energy estimates in each region
and we then patch the microlocal estimates together.
Gelfand-Shilov and Gevrey smoothing effect for the spatially inhomogeneous non-cutoff Kac equation,
with Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu,
published by the
Journal of Functional Analysis,
Volume 269, Issue 2, 15 July 2015, pages 459-535,
link to the file.
We consider the spatially inhomogeneous non-cutoff Kac's model of the Boltzmann equation. We prove that the Cauchy problem for the fluctuation around the Maxwellian distribution enjoys Gelfand-Shilov regularizing properties with respect to the velocity variable and Gevrey regularizing properties with respect to the position variable.
Gelfand-Shilov smoothing properties of the radially symmetric spatially homogeneous Boltzmann equation without angular cutoff,
with Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu,
file on arXiv,
Journal of Differential Equations,
256 (2014), no. 2, 797-831.
We prove that the Cauchy problem associated to the radially symmetric spatially homogeneous non-cutoff Boltzmann equation with Maxwellian molecules enjoys the same Gelfand-Shilov regularizing effect as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
Phase space analysis and functional calculus for the linearized Landau and Boltzmann operators,
with
Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu,
file on arXiv,
Kinetic and Related Models, 6 (2013), no. 3, 625-648.
We study the non-cutoff Boltzmann collision operator and the Landau collision operator linearized around a normalized Maxwellian distribution. For Maxwellian molecules, we prove that the linearized non-cutoff Boltzmann operator is equal to a fractional power of the linearized Landau operator. Furthermore, we provide the exact anisotropic phase space structure of these two linearized operators and display the sharp anisotropic coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.
Spectral and phase space analysis of the linearized non-cutoff Kac collision operator,
with
Yoshinori Morimoto, Karel Pravda-Starov and Chao-Jiang Xu,
file on arXiv,
Journal de mathématiques pures et appliquées, 100, (2013), no. 6, 832-867.
The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmetric Boltzmann operator are also proven.
Carleman estimates for anisotropic elliptic operators with jumps at an interface,
file on arxiv,
with Jérôme Le Rousseau,
Analysis & PDE
6-7 (2013), 1601--1648.
We consider a second-order selfadjoint elliptic operator with an
anisotropic diffusion matrix having a jump across a smooth
hypersurface. We prove the existence of a weight-function such that
a Carleman estimate holds true. We moreover prove that the
conditions imposed on the weight function are necessary.
Sharp hypoelliptic estimates for some kinetic equations,
with Karel Pravda-Starov,
January 2011,
Proceedings of the International Conference
the 26th Matsuyama Camp, Ryukoku University, Kyoto,
pdf.
We provide a simple overview of some hypoellipticity results with sharp
indices for a class of kinetic equations and we outline a general strategy based
on some geometrical properties.
Hypoelliptic estimates for a linear model of the Boltzmann equation
without angular cutoff
(file on
arxiv),
with Yoshinori Morimoto and Karel Pravda-Starov,
December 2010,
Communications in PDE,
volume 37, Issue 2, 2012, pages 234-284.
In this paper, we establish optimal hypoelliptic estimates for a class of
kinetic equations, which are linear models for the spatially inhomogeneous Boltzmann
equation without angular cutoff.
Instability of the Cauchy-Kovalevskaya solution
for a class of non-linear systems,
with Yoshinori Morimoto and Chao-Jiang Xu,
American Journal of Mathematics,
Vol. 132, 1, February 2010, pp. 99-123.
We prove that in any C-infinity neighborhood of an analytic Cauchy datum, there exists a smooth function such that the corresponding
initial value problem does not have any classical solution
for a class of first-order non-linear systems.
We use a method initiated by G. Métivier
for elliptic systems based on the representation of solutions and on the FBI transform; in our case the system can be hyperbolic at initial time,
but the characteristic roots leave the real line at positive times.
Fast rotating condensates in an asymmetric trap,
with Amandine Aftalion and Xavier Blanc,
article in press,
Journal of Functional Analysis,
Volume 257, Issue 3, 1 August 2009, Pages 753-806.
We investigate the effect of the anisotropy of a harmonic trap on the behaviour of a fast rotating Bose - Einstein condensate. This is done in the framework of the 2D Gross - Pitaevskii equation and requires a symplectic reduction of the quadratic form defining the energy. This reduction allows us to simplify the energy on a Bargmann space and study the asymptotics of large rotational velocity. We characterize two regimes of velocity and anisotropy; in the first one where the behaviour is similar to the isotropic case, we construct an upper bound: a hexagonal Abrikosov lattice of vortices, with an inverted parabola profile. The second regime deals with very large velocities, a case in which we prove that the ground state does not display vortices in the bulk, with a 1D limiting problem. In that case, we show that the coarse grained atomic density behaves like an inverted parabola with large radius in the deconfined direction but keeps a fixed profile given by a Gaussian in the other direction. The features of this second regime appear as new phenomena.
Fast rotating condensates in an asymmetric harmonic trap,
with Amandine Aftalion and Xavier Blanc,
Physical Review A,
Volume 79, Issue 1, Phys. Rev. A 79, 011603(R) (2009).
A note on the Oseen kernels,
an article in
Advances in Phase Space Analysis of Partial Differential Equations,
PNLDE, vol. 78, Birkhäuser,
2009.
We give an explicit expression for the kernels of the Oseen operators,
Δ-1
∂xj
∂xk
etΔ.
These Fourier multipliers involve the incomplete gamma function and
the confluent hypergeometric functions of the first kind. This explicit expression
provides directly the classical decay estimates with sharp bounds.
Semi-classical estimates for non-selfadjoint operators,
The Asian Journal of Mathematics, vol.11, 2, 217-250, (2007).
pdf.
This is a survey paper
on the topic of proving or disproving a priori L2 estimates
for non-selfadjoint operators.
Our framework will be limited to the case of scalar semi-classical pseudodifferential
operators of principal type.
We start with recalling the simple conditions following from the sign of the first bracket
of the real and imaginary part of the principal symbol.
Then we introduce the geometric condition (ψ)
and show the necessity of that condition for obtaining a weak L2 estimate.
Considering that condition satisfied, we investigate the finite-type case, where one iterated bracket
of the real and imaginary part does not vanish, a model of subelliptic operators.
The last section is devoted partly to rather recent results, although we begin
with a version of the 1973 theorem of R.Beals and C.Fefferman on solvability
with loss of one derivative under condition (P); next, we present a 1994 counterexample
by N.L.
establishing that condition (ψ) does not ensure an estimate with loss of one derivative
for P*.
Finally, we show that
condition (ψ)
implies an estimate with loss of 3/2 derivatives, following the recent papers by N.Dencker and N.L.
On the Fefferman-Phong inequality
and a Wiener-type
algebra of pseudodifferential operators,
with Yoshinori Morimoto,
Publications of the Research Institute for Mathematical Sciences
(Kyoto University)
43, 329-371, (2007),
pdf.
We provide an extension of the Fefferman-Phong inequality
to nonnegative symbols
whose fourth derivative belongs
to a Wiener-type algebra
of pseudodifferential operators
introduced by J.Sjöstrand. As a byproduct, we obtain that the number of derivatives needed
to get the classical Fefferman-Phong inequality in D dimensions is bounded above by
2D+4+ε.
Our method relies on some refinements of the Wick calculus,
which is closely linked to Gabor wavelets.
Also we use a decomposition of C3,1
nonnegative functions
as a sum of squares of C1,1 functions
with sharp estimates.
In particular, we prove that a C3,1 nonnegative function
can be written as a finite sum Σ bj2,
where each bj is C1,1, but also where each function
bj2 is
C3,1.
A key point in our proof is to give some bounds on
(bj'bj'')'
and on
(bjbj'')''.
Cutting the loss of derivatives
for solvability under condition
(Ψ),
Bulletin de la Société Mathématique de France,
vol.134, 4, 559-631, (2006).
For a principal type pseudodifferential operator,
we prove that condition (Ψ) implies local solvability
with a loss of 3/2 derivatives. We use many elements of
Dencker's paper on the proof of the Nirenberg-Treves conjecture
and we provide some improvements of the key energy estimates
which allows us to cut the loss of derivatives from 2 (Dencker's result)
to 3/2 (the present paper). It is already known that condition (Ψ)
does not imply local solvability with a loss of 1 derivative,
so we have to content ourselves with a loss >1.
Since this paper is quite technical, it could be a good idea to begin
with the
transparencies
of my talk at the Bourbaki seminar
in March 2006.
A more detailed presentation
pdf
appeared in the proceedings of that seminar
(Astérisque,
vol.311, exposé 960, (2007)).
Transport equations with partially BV velocities,
Annali della Scuola Normale Superiore di Pisa,
Classe di Scienze,
Serie V, Vol. III, fasc.4 (2004),
pdf,
dvi.
In this article, we prove the uniqueness of weak solutions
for a class of transport equations whose velocities are
partially with bounded variation. Our result deals
with the vector field
X = a1(x1)
.∂x1
+
a2(x1,x2)
.∂x2
where
a1(x1) is a BV function
and
a2(x1,x2)
is only L1 with respect to
x1
and
BV
with respect to
x2,
with a boundedness condition on the divergence of each
vector field
a1, a2. This model was studied
in a recent paper by P.-L.Lions and C.Le Bris
with a W1,1 regularity assumption replacing our
BV hypothesis. This settles partly a question
raised in a forthcoming paper by L.Ambrosio.
We examine the details of the argument of that article
and we combine some consequences of the Alberti rank-one
structure theorem for BV vector fields
with a regularization procedure. Our regularization
kernel is not restricted to be a convolution
and is introduced as an unknown function.
Our method amounts to commute a pseudo-differential
operator with a BV function.
Équations de transport
dont les vitesses sont partiellement
BV, texte de
l'exposé du 20 janvier 2004
au
séminaire X-EDP
, pdf.
Essentially a french version
of the above article.
However, we also go back to vector fields X as above with
W1,1
regularity: in that case, our boundedness
condition on the divergence
is only on the whole div X
and not on each divergence of a1,
a2.
Uniqueness of
L∞
solutions for
a class of conormal BV vector fields,
with Ferruccio Colombini,
pdf,
dvi, an article in
Geometric Analysis of PDE and Several
Complex Variables,(editors
S. Chanillo, P. Cordaro,
N. Hanges, J. Hounie, and A. Meziani)
Contemporary Mathematics #368. In this paper,
we prove the uniqueness of
bounded measurable solutions
for a class of vector fields with bounded variation.
Our class contains
the piecewise W1,1 class.
We use some arguments of geometric measure theory
to get rid of
sets
whose d-1 Hausdorff measure is 0.
Also we need an anisotropic regularization argument.
Une procédure de Calderón-Zygmund
pour le problème de la racine k-ième,
avec Ferruccio Colombini,
Annali di Matematica Pura ed Applicata,
volume 182, 231-246, 2003,
pdf.
The Wick calculus
of pseudo-differential operators
and some of its applications,
in the Chilean journal CUBO,
volume 5, (1), 2003.
pdf.
Solving pseudo-differential equations,
pdf,
dvi,
my article published in the
Proceedings of the ICM 2002 in Beijing,
Higher Education Press,
Volume II, pages 711-720.
Uniqueness of continuous solutions for BV vector fields,
with Ferruccio Colombini,
Duke Mathematical Journal,
volume 111, No.2, pages 357-384, 2002.
On the existence and uniqueness
of solutions to stochastic equations in infinite
dimension
with integral-Lipschitz coefficients,
with Ying Hu,
Journal of Mathematics of Kyoto University,
volume 42, (3), pages 579 - 598, 2002,
pdf.
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April 5, 2024