South Wales Analysis and Probability Seminar (SWAP)

Cardiff University and Swansea University

About

The South Wales Analysis and Probability Seminar (SWAP) is a regional seminar bringing together mathematicians based in South Wales, with interests in analysis, probability, PDEs, SDEs, dynamical systems, and other related fields.

 

Next Meeting (poster)

The next meeting is scheduled for 27 September 2018 at Cardiff University. The schedule is:

  9:30-10:00 Coffee & Registration
10:00-10:45 Marco Marletta (Cardiff) Inverse problems for Maxwell systems with partial boundary data
10:45-11:30 Jiang-Lun Wu (Swansea) BMO and Morrey-Campanato estimates for stochastic singular integral operators and their applications to parabolic SPDEs
11:30-11:50 Coffee
11:50-12:35 Federica Dragoni (Cardiff) Geometrical properties for solutions of subelliptic nonlinear PDEs
12:35-14:05 Lunch
Colloquium:
14:05-15:05 Peter Hintz (Berkeley) Stability of black holes
15:05-15:30 Coffee
15:30-16:15 Dmitri Finkelshtein (Swansea) Long-time behavior for monostable (doubly) nonlocal reaction-diffusion equations

 

Abstracts

Marco Marletta (Cardiff) - Inverse problems for Maxwell systems with partial boundary data

We prove that the coefficients in the time-harmonic Maxwell system are uniquely determined by measurements of the tangential components of the electric and magnetic fields on any (arbitrarily small) open subset of the boundary. The proof is based on a generalisation of the Alessandrini identity, due to Lassas, Salo and Uhlmann, and on a Runge principle for the Maxwell system and the related Schrodinger operator with matrix-valued potential. This is joint work with Malcolm Brown and Juan Manuel Reyes Gonzales, funded by EPSRC grant EP/K024078/1.

 

Jiang-Lun Wu (Swansea) - BMO and Morrey-Campanato estimates for stochastic singular integral operators and their applications to parabolic SPDEs

Stochastic integral operators defined in the stochastic integral of convolution manner appeared naturally in the mild formulation of SPDEs. In this talk, we derive BMO and Morrey-Campanato estimates for stochastic singular integral operators, We then apply our results to discuss various estimates (including the Schauder estimates) of the solutions of SPDEs with additive noises. Joint work with Guangying Lv, Hongjun Gao and Jinlong Wei.

 

Federica Dragoni (Cardiff) - Geometrical properties for solutions of subelliptic nonlinear PDEs

Geometrical properties for solutions to elliptic and parabolic PDEs is a classic mathematical problem. In particular we will focus on starshapedness, a geomet- rical notion deeply connected to convexity but weaker than that. For elliptic PDEs it is know that the capacitary potential defined on a starshaped ring has starshaped level sets. We generalise this results in the setting of subelliptic PDEs, in particular to PDEs associated to Carnot groups. Carnot groups are non-commutative nilpotent Lie groups which are not isomorphic to R^N at any scale. Going into these more degenerate geometries some unexpected phenom- ena appear: e.g. there are different possible notions of starshapedness and they may not be all equivalent. We use a notion associated to the the natural scaling in Carnot groups (dilations) and show that this geometrical property is inher- ited by the level sets of a large class of nonlinear PDEs. Joint work with Nicola Garofalo and Paolo Salani.

 

Peter Hintz (Berkeley) - Stability of black holes

More than a hundred years ago, Schwarzschild first wrote down the mathematical description of a black hole; on a technical level, black holes are certain types of solutions of Einstein's equations of general relativity. While they have since become part of popular culture, many fundamental questions about them remain unanswered: for example, it is not yet known mathematically if they are stable! I will explain what that means and outline a recent proof of full nonlinear stability (obtained in joint work with A. Vasy) in the case that the cosmological constant is positive, a condition consistent with current cosmological models of the universe. The talk is intended as a non-technical introduction to the subject, with a focus on the central role played by modern microlocal and spectral theoretical techniques.

 

Dmitri Finkelshtein (Swansea) - Long-time behavior for monostable (doubly) nonlocal reaction-diffusion equations

We consider a class of monostable equations in R^d with nonlocal diffusion and local or nonlocal reaction, which satisfy the linear determinacy principle. For the case where either the diffusion kernel or the initial condition have appropriately regular heavy tails, we find sharp estimates to describe accelerated front propagation. For the one-dimensional case we show also that the propagation to the right direction is fullydetermined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone-like initial conditions which may yield different rates of the acceleration. For the case where (anisotropic multidimensional) kernel is decaying at least exponentially fast in a direction and if the initial condition has the similar property, we show that the solution propagates in this direction at most linearly in time. For a particular equation arising in population ecology, we study then traveling waves and describe the front of propagation. This is a joint work with Pasha Tkachov (GSSI, L’Aquila).


Organisers

Jonathan Ben-Artzi (Cardiff)
Dmitri Finkelshtein (Swansea)