PDE CDT Mini-Courses

A Comparison of Classical & Modern Methods for ODEs

Glimpses of Lipschitz Truncations & Regularity

Prof Bianca Stroffolini (Naples) - October 2015

Oxford 1&2.pdf


oxford3.pdf


Oxford 4.pdf

Some Degenerate PDEs

Prof Hua Chen (Wuhan) - November 2015

Degenerate PDEs Mini-Course Nov 2015.pdf
 

Some Degenerate PDEs Lecture Notes Nov 2015.pdf
 

Carleman inequalities and applications to thin obstacles

An invitation to microlocal analysis of PDEs

Prof Jared Wunsch (Northwestern) 4.5 hour mini-course - May 2016
 
Microlocal analysis is the study of PDEs in phase space.  I will give an introduction to what these tools are and why you might care to use them, focusing on an axiomatic "black-box" approach to pseudodifferential operators and how to use them rather than giving a rigorous construction.  Applications to wave and SchrÖdinger equations will be discussed.

Functions of Bounded Variation

Dr Panu Lahti (Aalto and Oxford) 6 hour mini-course - June 2016
 
The aim of the course is to study the theory of functions of bounded variation, or BV functions, in the Euclidean setting. We begin by considering possible definitions of the BV class - BV functions are usually defined as integrable functions whose distributional derivatives are measures - and notions of convergence of BV functions. Then we show that the BV class satisfies a compactness property that is not available for the Sobolev class W^{1,1}, making BV functions a natural class to work with in many variational problems.

A central topic of the course will be proving De Giorgi's structure theorem for sets of finite perimeter, that is, sets whose characteristic functions are in the BV class. By combining this with the BV version of the coarea formula, which states that the variation measure of a BV function can be expressed in terms of the perimeters of its super-level sets, one can then derive various fine properties of BV functions, including a decomposition of the variation measure into the so-called absolutely continuous, Cantor, and jump parts.

Time permitting, we consider Alberti's rank one theorem for the Cantor and jump parts of the variation measure, as well as some generalizations of the BV theory to the setting of metric measure spaces. 

BV_minicourse_Slides_Version3.pdf
 

Turbulent Flow

Prof James Glimm (Stony Brook University) 27 July 2016

An Introduction to Theories of Turbulence.pdf
 

We present some of the most basic and most widely useful concepts related to the modeling of turbulence, starting with the notion of multiscale science, to illustrate the nature of the fundamental problems. Large eddy simulations, with the use of dynamic subgrid models are developed. Kolmogorov’s 1941 scaling law, and the renormalization group are explained. We conclude with a discussion of classical and exotic notions of mathematical convergence for solutions of the Navier Stokes equation.

Statisical Models of Turbulent Flow.pdf
 

The fundamental scaling law of Kolmogorov 1941 has the form

                      <v’v’> ~ epsilon2/3 k-5/3

Here we focus on models for epsilon. Epsilon is the energy dissipation rate and a fundamental measure of turbulent intensity. Many advanced theories of turbulence depend on modeling of epsilon. According to Kolmogorov 1962 and also Obukhov, epsilon has a log normal distribution.

Here we propose a random field stochastic model for epsilon. The random field is log normal and thus parameterized by a mean and a covariance. For application to large eddy simulations, the mean and s time scale describe the statistics of the unresolved scales, whose influence is removed to obtain a universal law for the subgrid statistics. We propose a multiscale, but otherwise simple parameterization of the covariance, in terms of a new scaling law.

Applications to clustering of particles in particle laden flow and to the fractal structure of turbulence are discussed.

Image Processing and Related PDEs

Dr Yves Van Gennip (Unversity of Nottigham) - 6 & 7 September 2016

Image_Processing_Oxford1.pdf

Image_Processing_Oxford2.pdf

Image_Processing_Oxford3.pdf

Image_Processing_Oxford4.pdf
 

Isometric embedding - continuum mechanics duality

Prof Marshall Slemrod (University of Wisconsin - Madison, Emeritus) will be giving a short lecture course on Isometric embedding - continuum mechanics duality in Trinity term and has provided the lecture notes.
 
Elementary Concepts
Compensated Compactness
Nash-Kuiper
Fluids, Elasticity, Geometry, Wrinkled Solutions
 
The lectures are scheduled for 9am Tuesday 2 & 9 May and midday Wednesday 3 & 10 May 2017 in L6.

Measure-Theoretical  Analysis and Applications to Nonlinear PDEs

Professor Monica Torres (Purdue University) will be giving a short lecture course at 11am on Tuesday 6 and Wednesday 7 June in L6, Mathematical Institute:
Measure-Theoretical  Analysis and Applications to Nonlinear PDEs
Part I: Divergence-Measure Fields; The Solvability of divF = T and the Dual BV
Part II: Divergence-Measure Fields and Nonlinear Conservation Laws

Analysis of Nonlinear PDEs

Professor Lawrence Craig Evans (UC Berkeley) will be giving two lectures at 2.30pm Thursday 22 June at St Luke's Chapel, Radcliffe Observatory Quarter:
Analysis of Nonlinear PDEs
Part I: Weak Convergence and Adiabatic Invariants
Part II: Riccati Equation Methods for Weak KAM Theory

Mathematical Models of the Heart: The electrical patterns driving the muscle contractions

Professor James Glimm (Distinguished Professor, Stony Brook University) will give two lextures on Mathematical Models of the Heart: The electrical patterns driving the muscle contractions at 2pm, Tuesday 1 August 2017.

Poster

Gaussian Measures in Analysis and Probability

From the central limit theorem to Brownian motion, Gaussian measures are a central object in probability theory. 
It also plays an important role in Fourier analysis and partial differential equations. In these lectures we will discuss several
topics related to Gaussian measures. We will prove some well known and not well know but equally interesting properties
of Gaussian measures. We will also show how Gaussian measures can be used to study many problems in probability and
analysis. The lectures are accessible to advanced undergraduate students and beginning postgraduate students. 

Prof Elton Hsu (Northwestern) will give a 6-hour mini-course on Friday 27 April, Tuesday 1 May and Wednesday 2 May 2018.

The Monge-Ampere Equation

We will discuss the regularity problem for the Monge-Ampere equation. Topics will include weak solutions, the classical Dirichlet problem, singular solutions, Caffarelli's interior C^{1,alpha}, C^{2, alpha} and W^{2, p} estimates, partial regularity, and applications. 

Dr Connor Mooney (ETH Zurich) will give a 4-hour mini-course on Tuesday15 and Wednesday 16 May 2018.

The Analysis of the Development of Shocks in Compressible Fluids

Professor Demetrios Christodoulou (ETH Zurich)
C3, 11am-1pm, Tuesday 9, Wednesday 10 October and 10am-midday Thursday 11 October.

Divergence-free positive tensors and applications to gas dynamics

Professor Denis Serre (ENS Lyon)
L5, 11am to 1pm, Tuesday and Wednesday 13 and 14 November 2018

A lot of physical processes are modelled by conservation laws (mass, momentum, energy, charge, ...) Because of natural symmetries, these conservation laws express often that some symmetric tensor is divergence-free, in the space-time variables. We extract from this structure a non-trivial information, whenever the tensor takes positive semi-definite values. The qualitative part is called Compensated Integrability, while the quantitative part is a generalized Gagliardo inequality.

In the first part, we shall present the theoretical analysis. The proofs of various versions involve deep results from the optimal transportation theory. Then we shall deduce new fundamental estimates for gases (Euler system, Boltzmann equation, Vlaov-Poisson equation).

One of the theorems will have been used before, during the Monday seminar (PDE Seminar 4pm Monday 12 November)