Summer School 2011
Analytic Methods in PDEs
- Monday 12 September 2011, 09:00 to Friday 16 September 2011, 12:00
- Department of Mathematics, University of Surrey
- Open to:
- Staff, Students
This summer school is aimed primarily at PhD students but is also open to postdoctoral fellows and others with interests in partial differential equations. The course will aim to give the participants a background on various techniques commonly employed in the study of partial differential equations arising in applied problems.
Dates and times: Arrival Monday morning, 12th September. Lectures Monday afternoon, all day on Tuesday, Wednesday, Thursday and Friday morning. The program is now available.
Speakers and topics
Alexej Ilyin (Moscow, Russia), “Spectral inequalities and applications to the Navier-Stokes equations”
Spectral inequalities play an important role in the theory of the Navier—Stokes equations and other parabolic equations, especially in the estimates of the dimension of their global attractors. We prove lower bounds of the Berezin-Li-Yau-type for sums of eigenvalues of elliptic operators and systems with constant coefficients (including the Stokes system). We also discuss recent results on the Lieb-Thirring inequalities in the Euclidean space and prove Lieb-Thirring inequalities with improved constants on some manifolds (periodic case, 2-D and 3-D torus and sphere). Notes
Armen Shirikyan (Sergy et Pontoise, France), “Ergodic theory of random dynamical systems and applications”
The aim of this short course is to give a quick introduction to the ergodic theory of randomly perturbed dynamical systems and to show how to apply it to some problems arising in mathematical fluid dynamics. We begin with a general study of a class of Markov random dynamical systems (RDS) in a phase space X. It will be shown, in particular, that such an RDS defines a semigroup in the space of probability measures on X. We then introduce the concept of a stationary distribution and describe the Bogolyubov-Krylov argument for its construction. The core of the course is a result on uniqueness and mixing properties of stationary distribution. Its proof if based on a coupling argument which enables one to show that the above-mentioned semigroup is a contraction in the space of measures endowed with the Kantorovich-Wassertein metric. In conclusion, we prove that the results obtained for Markov RDS apply to 2D Navier-Stokes equations perturbed by an external random force. Notes
Luigi Berselli (Pisa, Italy), “Some Selected Topics in Incompressible Fluid Mechanics”
The aim of these lectures is to present the basic equations of incompressible inviscid and viscous fluids, explaining also possible connections with wide classes of non-newtonian fluids. We will point out the very specific issues that make their analytical an numerical study peculiar in the wider field of dissipative partial differential equations. The role played by the pressure in the regularity theory and in the construction of numerical approximations will be emphasized. Finally, we will consider the behavior of solutions with respect to the viscosity and the role of the dissipation in three main problems: growth estimates for smooth solutions, singular vanishing viscosity limits, and generation of small scales as in the Kolmogorov K41 theory. Notes
Jan Kristensen (Oxford), “Aspects of regularity theory for minimizers of multiple Integrals”
There are by now several examples of regular variational problems that admit singular minimizers (non-differentiable Sobolev maps, and worse) in the multi-dimensional vectorial case. This is in line with what one can prove: minimizersare partially regular, meaning they are smooth outside a small relatively closed subset of their domain. We refer to this set of singularities as the singular set. General measure theory easily gives that the singular set has zero Lebesgue measure, but over the years much better estimates of their size, in particular in terms of Hausdorff measures, have been derived. This is often done by proving that minimizers admit higher order weak derivatives. Still the gap between the examples provided by singular minimizers and the theoretical bounds for the size of the singular sets remains a major challenge. In these lectures we start by brief reviewing the known examples of singular minimizers and some related background results. We then discuss recent progress on the problem of higher differentiability of minimizers in both the convex and nonconvex cases. Notes
Edriss Titi (Irvine, USA), “Determining modes, nodes for nonlinear dissipative PDEs with applications”
Participants need to pay their own accommodation, travel and meals. If you are an EPSRC funded PhD student your university may be able to refund these costs. Please ask your supervisor or Head of Department.
There is no registration fee.
No university accommodation is available, but it may be possible to accommodate you at the Guildford YMCA, at a rate of £51.50 per night including breakfast, Travelodge from £55.75 per night, alternatively check out the website Places to Stay. If you require accommodation, please request it through the organisers.
To register, please contact Sergey Zelik (S.Zelik@surrey.ac.uk) or Michele Bartuccelli (M.Bartuccelli@surrey.ac.uk) with an indication of whether you will require accommodation. We will reply to indicate if there are still places on the course. The registration deadline is Friday 26th August 2011.