Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge-Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

Optimal Transport: Old and New

Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).

Gradient Flows: In Metric Spaces and in the Space of Probability Measures

日本物理学会誌及びJournal of the Physical Society of Japanの月刊について

We show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural. In order to convince the reader that it is mathematically natural, we show that the time asymptotic behavior can be easily understood in this framework. We use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.

/pdf/the-geometry-of-dissipative-evolution-equations-the-porous-4be4l1zzoc.pdf

The geometry of dissipative evolution equations: the porous medium equation

Using the notion of time-dependent rescalings introduced in [DR], we prove explicit dispersion
results. The main tool is a Lyapunov functional which is given by the energy after rescaling and the entropy
dissipation. The relation with asymptotically self-similar solutions is investigated. The method applies to the
solutions of the Boltzmann, Landau and BGK equations. In the case of the Vlasov-Fokker-Planck equation,
the diﬀerence with the self-similar solutions has a faster decay, which is estimated by a classical method for
parabolic equations and interpolation estimates.

Time-Dependent Rescalings and Dispersion for the Boltzmann Equation

The simplest version of the parabolic-elliptic Patlak-Keller-Segel system in the two-dimensional Euclidean space has an 8π critical mass which corresponds to the exact threshold between finite-time blow-up and self-similar diffusion towards zero. Among functions with mass 8π, we find a neighborhood of a radial function such that any solution with initial condition in this neighborhood is globally defined and blows-up in infinite time with an explicit scaling involving the square root of the logarithm of the time.

https://hal.archives-ouvertes.fr/hal-02394787/document

Existence and stability of infinite time blow-up in the Keller-Segel system

This paper is devoted to the study of the two-dimensional Dirac-Coulomb operator in presence of an Aharonov-Bohm external magnetic potential. We characterize the highest intensity of the magnetic field for which a two-dimensional magnetic Hardy inequality holds. Up to this critical magnetic field, the operator admits a distinguished self-adjoint extension and there is a notion of ground state energy, defined as the lowest eigenvalue in the gap of the continuous spectrum.

/pdf/critical-magnetic-field-for-2d-magnetic-dirac-coulomb-df3y3y4lov.pdf

Critical magnetic field for 2d magnetic Dirac-Coulomb operators and Hardy inequalities

The purpose of this work is to establish a quantitative and constructive stability result for a class of subcritical Gagliardo-Nirenberg-Sobolev inequalities which interpolates between the logarithmic Sobolev inequality and the standard Sobolev inequality (in dimension larger than three), or Onofri's inequality in dimension two. We develop a new strategy, in which the flow of the fast diffusion equation is used as a tool: a stability result in the inequality is equivalent to an improved rate of convergence to equilibrium for the flow. The regularity properties of the parabolic flow allow us to connect an improved entropy - entropy production inequality during an initial time layer to spectral properties of a suitable linearized problem which is relevant for the asymptotic time layer. Altogether, the stability in the inequalities is measured by a deficit which controls in strong norms (a Fisher information which can be interpreted as a generalized Heisenberg uncertainty principle) the distance to the manifold of optimal functions. The method is constructive and, for the first time, quantitative estimates of the stability constant are obtained, including in the critical case of Sobolev's inequality. To build the estimates, we establish a quantitative global Harnack principle and perform a detailed analysis of large time asymptotics by entropy methods.

https://hal.archives-ouvertes.fr/hal-02887010/document

Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

This note is devoted to the proof of convex Sobolev (or generalized Poincare) inequalities which interpolate between spectral gap (or Poincare) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux (11) and Carlen and Loss (10) for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case. Resume Cette note est consacreela preuve d'inegalites de Sobolev convexes (ou inegalites de Poincare generali- sees) qui interpolent entre des inegalites de trou spectral (ou de Poincare) et des inegalites de Sobolev logarith- miques. Nous ´ ` la famille des inegalites de Sobolev convexes toute entiere des resultats qui ontete obtenus recemment par Cattiaux (11) et Carlen et Loss (10) pour des inegalites de Sobolev logarithmiques. Sous des conditions locales sur la densite de la mesure par rapportune mesure de reference, nous demontrons que les inegalites de trou spectral entraˆonent toutes les inegalites de Sobolev convexes avec des constantes qui sont bornees uniformement dans la limite qui approche les inegalites de Sobolev logarithmiques. Nous retrouvons le cas des inegalites de Sobolev logarithmiques comme un cas particulier.

https://hal.archives-ouvertes.fr/docs/00/02/95/70/PDF/CSISG9.pdf

Jean Dolbeault

Papers

Time-Dependent Rescalings and Dispersion for the Boltzmann Equation

Existence and stability of infinite time blow-up in the Keller-Segel system

Critical magnetic field for 2d magnetic Dirac-Coulomb operators and Hardy inequalities

Stability in Gagliardo-Nirenberg-Sobolev inequalities: flows, regularity and the entropy method

Inegalites de Sobolev convexes et trou spectral