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

We consider radial solutions of an elliptic equation involving the p-Laplace operator and prove by a shooting method the existence of compactly supported solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane corresponding to an asymptotic Hamiltonian system and provides qualitative properties of the solutions.

https://hal.archives-ouvertes.fr/hal-00727573/document

Qualitative Properties and Existence of Sign Changing Solutions with Compact Support for an Equation with a p-Laplace Operator

Let $B$ denote the unit ball in $\mathbb R^2$. We consider the slightly
super-critical Gelfand problem for the $p$-Laplacian operator
$\Delta_p u =$ div $(|\nabla u|^{p-2}\nabla u)$, 
$
-\Delta_{2-\varepsilon} u=\lambdae^u$ in $B\quad
u =0 $ on $ \partial B,$
 
for small $\varepsilon>0$. We show that if $k\ge 1$ is given and $\lambda>0$
is fixed and small, then there is a family of radial solutions
exhibiting multiple blow-up as $\varepsilon\to 0$ in the form of a
superposition of $k$ bubbles of different blow-up orders and
shapes. Similar phenomena is found for the same problem involving
the operator $\Delta_{N-\varepsilon}$ in $\mathbb R^N$, $N\ge 3$.

https://www.ceremade.dauphine.fr/~dolbeaul/NumericalGallery/Fichiers_nb/0407_nb.pdf

Multiple bubbling for the exponential nonlinearity in the slightly supercritical case

The purpose of kinetic equations is the description of dilute particle
gases at an intermediate scale between the microscopic scale and the
hydrodynamical scale. By dilute gases, one has to understand a system with
a large number of particles, for which a description of the position and of the
velocity of each particle is irrelevant, but for which the decription cannot be
reduced to the computation of an average velocity at any time 
$t\in \mathbb R$ and any
position $x\in \mathbb R^d:$ one wants to take into account more than one possible velocity
at each point, and the description has therefore to be done at the level of
the phase space – at a statistical level – by a distribution function $f(t, x, v)$. 

This course is intended to make an introductory review of the literature on
kinetic equations. Only the most important ideas of the proofs will be given.
The two main examples we shall use are the Vlasov-Poisson system and the
Boltzmann equation in the whole space.

/pdf/an-introduction-to-kinetic-equations-the-vlasov-poisson-3pp7awvivh.pdf

An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation

This paper is devoted to Fokker-Planck and linear kinetic equations with very weak confinement corresponding to a potential with an at most logarithmic growth and no integrable stationary state. Our goal is to understand how to measure the decay rates when the diffusion wins over the confinement although the potential diverges at infinity.

https://www.aimsciences.org/article/exportPdf?id=2578ad96-e4b6-465e-93e2-f2822200733b

Diffusion and kinetic transport with very weak confinement

This paper is dealing with two L2 hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard L2 norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein-Taylor model in one-dimension.

Jean Dolbeault

Papers

Qualitative Properties and Existence of Sign Changing Solutions with Compact Support for an Equation with a p-Laplace Operator

Multiple bubbling for the exponential nonlinearity in the slightly supercritical case

An introduction to kinetic equations: the Vlasov-Poisson system and the Boltzmann equation

Diffusion and kinetic transport with very weak confinement

Sharpening of decay rates in Fourier based hypocoercivity methods