Variational methods for the kinetic Fokker-Planck equation

67 pagesInternational audienceWe develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical $H^1$ theory of uniformly elliptic equations. In particular, we identify a function space analogous to $H^1$ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the~$C^\infty$ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit

Analysis of Participant Descriptions in Single-Case Design Experimental Research

This project examines the extent to which common demographic descriptors of race/ethnicity, gender, age, grade level, and disability identifiers are present in published special education research using single-case experimental design

Analysis of Participant Descriptions in Single-Case Design Experimental Research

This project examines the extent to which common demographic descriptors of race/ethnicity, gender, age, grade level, and disability identifiers are present in published special education research using single-case experimental design

Construction of vicinal 4°/3°-carbons via reductive Cope rearrangement

Herein reported is a strategy for constructing vicinal 4°/3° carbons via reductive Cope rearrangement. Substrates have been designed which exhibit Cope rearrangement kinetic barriers in the range of ~23 kcal/mol with isothermic favorability (G ~0). These fluxional/shape-shifting molecules can be driven forward by chemoselective reduction to useful polyfunctionalized building blocks.</jats:p

Construction of vicinal 4°/3°-carbons via reductive Cope rearrangement

Herein reported is a strategy for constructing vicinal 4°/3° carbons via reductive Cope rearrangement. Substrates have been designed which exhibit Cope rearrangement kinetic barriers in the range of ~23 kcal/mol with isothermic favorability (G ~0). These fluxional/shape-shifting molecules can be driven forward by chemoselective reduction to useful polyfunctionalized building blocks

Construction of vicinal 4°/3°-carbons<i>via</i>reductive Cope rearrangement

Herein reported is a strategy for constructing vicinal 4°/3° carbonsviareductive Cope rearrangement.</jats:p

Selective ring-rearrangement or ring-closing metathesis of bicyclo[3.2.1]octenes

Explored was the competitive ring-closing metathesis vs. ring-rearrangement metathesis of bicyclo[3.2.1]octenes prepared by a simple and convergent synthesis from bicyclic alkylidenemalononitriles and allylic electrophiles.</p