Jean-Marc Bouclet (Toulouse)

13 May 2019

Sharp time decay and resolvent estimates for dispersive equations

I will present recent results obtained with Nicolas Burq about local energy decay (and the related resolvent estimates) for the wave, Klein-Gordon and Schrödinger equations on an asymptotically Euclidean background. I'll make most of the talk accessible to a large audience and only touch the technical aspects if time permits.

Mahir Hadžić (KCL)

10 May 2019

Room M/0.40

Stability and instability of self-gravitating relativistic matter distributions

We prove that the compactly supported steady states of the Einstein-Vlasov system (representing galaxies) and Einstein-Euler system (representing stars) become linearly unstable when their central redshift is sufficiently large. Our work confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich and Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, Wheeler (for the Einstein-Euler system). This is a joint work with Gerhard Rein and Zhiwu Lin.

Mohammed Lemou (Rennes)

7 May 2019

Room E/2.20

Nonlinear stability and instability results for gravitational kinetic models

I will start by giving a short overview around stability and instability issues in gravitational systems driven by kinetic equations. A well-known conjecture in both astrophysics and mathematics communities was that  "all steady states of the gravitational Vlasov-Poisson system which are decreasing functions of the energy, are stable up to space translations".  We will  explain why the traditional variational approaches are not sufficient to answer this conjecture and  briefly present an alternative approach (that we introduced in 2012), which, although nonquantitative, solves this conjecture in the case of the Vlasov-Poisson system.  More recently, we proved a quantitative version of this nonlinear stability. This has been achieved by using an extended rearrangement notion of functions and a refined functional inequality on this rearrangement, together with a Poincaré-like  inequality. Another advantage of this new approach is its applicability to other systems like the so-called Hamiltonian Mean Field (HMF), where the space domain is bounded and where the decreasing property of the steady states is no more sufficient to guarantee their stability. In fact, an additional explicit criteria is needed for HMF, under which the non-linear stability is proved. This criteria is sharp since a nonlinear instability result  can be proved for HMF if it is not satisfied.

Alberto Debernardi (Bar-Ilan University)

15 April 2019

Weighted norm inequalities for integral transforms with splitting kernels

Given an integral transform on the positive real line, we say that its kernel is
splitting if it satisfies upper pointwise estimates given by products of two func-
tions, each of them taken in a different variable. We discuss necessary and/or
sufficient conditions weighted norm inequalities involving these transforms to
hold. Sharpness of the results is directly related to the sharpness of the upper
estimates for the kernels that we find.

Anne Nouri (Marseille)

8 April 2019

On some quantum kinetic equations

The study of gases at very low temperatures is an important field in contemporary physics. We present different models like a Boltzmann type equation for anyons and the Boltzmann-Nordheim quantum equation for bosons. A study  of the well-posedness of their related Cauchy problem is presented.

Kangwei Li (BCAM)

1 April 2019

Extrapolation for multilinear Muckenhoupt class of weights and applications

In this talk, I will introduce our recent progress on extrapolation theory. In the linear case, the extrapolation theory is well understood. However, in the multilinear case, the extrapolation was only known for product $A_p$ weights. The multilinear $A_{\vec P}$ weight, which was introduced in 2009, no extrapolation theory was known before. In this talk, I will give a full solution to this problem. As applications, we can improve the weighted estimates for the bilinear Hilbert transform, the multilinear Marcinkiewicz-Zygmund inequality etc. Endpoint cases are also provided. This talk is based on joint work with José María Martell and Sheldy Ombrosi, and joint work with Henri Martikainen, José María Martell, Sheldy Ombrosi and Emil Vuorinen.

Filip Rindler (Warwick)

25 March 2019

Theme & variations on div \mu = \sigma

The PDE div \mu = \sigma for (vector) measures \mu and \sigma appears -- sometimes in a slightly hidden way -- in many different problems of geometric measure theory and the calculus of variations, for instance in the structure theory of normal currents, Lipschitz functions, varifolds, and liftings for BV maps. In this talk I will survey a number of recent results about this equation and other related PDEs. As applications, I will discuss the structure of singularities of solutions, dimensional estimates, integral functionals on BV, and several converses of Rademacher's theorem (both in Euclidean and non-Euclidean settings).

Choi-Hong Lai (Greenwich)

18 March 2019

A generalised defect correction method and its applications in option pricing

The Black-Scholes partial differential equation is well known to be an option pricing tool for an ideal market. This requires the use of constant volatility over the period of time from zero up to the mature time. For realistic markets the volatility may become nonlinear, and the partial differential equation becomes nonlinear. A typical example is Frey and Patie nonlinear volatility model which contains a very small parameter in it. In addition to the nonlinearity a correction term may be added to the Black-Scholes partial differential equation to facilitate discrete hedging. The technique of defect correction method is extended to handle this type of nonlinear partial differential equations.

Several examples of the defect correction method are used to motivate the use of defect correction method and the adaptation of it to handle asymptotic expansion and nonlinear correction term is to be discussed. Numerical examples of a European option pricing will be used to demonstrate the algorithm.

Jessica Guerand (Cambridge)

11 March 2019

Quantitative parabolic regularity à la De Giorgi

De Giorgi method is a way to prove interior Hölder regularity of solutions of parabolic equations. While in the elliptic case the proof is completely quantitative, in the parabolic case it seems to remain a non-quantitative step: the intermediate value lemma. The purpose of this talk is to present a quantitative version of this step after introducing how it is useful to get Hölder regularity.

Jan Lang (Ohio State)

3:10 - 4:10

4 March 2019

Approximation of non-compact Sobolev embeddings

For limiting noncompact Sobolev embeddings into continuous functions, we study the behavior of approximation, Gelfand, Kolmogorov, Bernstein and isomorphism s-numbers. In the one-dimensional case, the exact values of the above-mentioned strict s-numbers are obtained, and in the higher dimensions sharp estimates for asymptotic behavior of the strict s-numbers are established. As all known results for s-numbers of Sobolev type embeddings are studied mainly under the compactness assumption, our work is an extension of existing results and reveal an interesting behavior of s-numbers in the limiting case when some of them (approximation, Gelfand, and Kolmogorov) have positive lower bound and others (Bernstein and isomorphism) are decreasing to zero. It also follows from our results that such limiting noncompact Sobolev embeddings are finitely strictly singular maps. Based on joint work with Vit Musil.

Davoud Cheraghi (Imperial)

2:10 - 3:10

4 March 2019

Quasi-periodic dynamics in complex dimension one

Quasi-periodic dynamics in one complex variable reveals fascinating interplays between complex analysis and Diophantine approximations. The question of whether a nonlinear perturbation of a linear rotation is conjugate to a linear rotation (linearisation) dates back to more than a century ago, with remarkable contributions by C. Siegel, A. Brjuno, and J.-C. Yoccoz. The behaviour of non-linearisable maps is very complicated. Indeed, there is not a single example of a non-linearisable map whose local behaviour is completely understood. There is major recent advances on this problem which has lead to a complete description of the topological behaviour of typical orbits. This is an introductory talk to demonstrate some of these results.

Gianne Derks (Surrey)

25 February 2019

Existence and stability of fronts in inhomogeneous wave equations

Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down.
In this talk, wave equations with finite length inhomogeneities will be considered, assuming that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions and the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. Criteria for the existence of and stability of stationary solutions will be discussed. The results will be illustrated  with an example related to a Josephson junction system with a finite length inhomogeneity associated with variations in the Josephson tunnelling critical current and an application to DNA-RNAP interactions.

Stephen Pankavich (Colorado School of Mines)

18 February 2019

Classical solutions of the Vlasov-Maxwell-Fokker-Planck system

The Vlasov-Maxwell system is a fundamental kinetic model of plasma dynamics. When one includes effects due to collisions with a fixed background of particles, the result is the Vlasov-Maxwell-Fokker-Planck system. The first Lorentz-invariant model, which considers relativistic velocities, was derived by Calogero in 2010. Here, we shall discuss well-posedness results for global-in-time classical solutions of this system and its non-relativistic analogue, each posed in a variety of dimensional settings. Our methods utilize a gain in regularity stemming from the (momentum) diffusion operator to arrive at smooth solutions arising from initial data which lack even weak differentiability properties.

Yuzhao Wang (Birmingham)

11 February 2019

Further renormalization and unconditionally global well-posedness of the cubic
nonlinear Schrödinger equation (NLS).

We first develop a normal form approach to study NLS in Fourier-Lebesgue spaces. By applying an infinite iteration of normal form reductions, we derive a normal form equation which is
equivalent to the renormalized cubic NLS for regular solutions. For rough functions, the normal form equation behaves better than the renormalized cubic NLS, thus providing a further renormalization of the cubic NLS. Then we establish a global-in-time bounds in Fourier-Lebesgue spaces for the NLS by exploiting its complete integrability and the Galilean symmetry. As a consequence, we obtain the global well-posedness of the cubic NLS.

This is a joint work with Tadahiro Oh at the University of Edinburgh.

Monica Musso (Bath)

4 February 2019

Gluing methods for vortex dynamics in Euler flows

A classical problem for the two-dimensional Euler  flow for an incompressible fluid confined to a smooth domain is that of finding regular solutions with highly concentrated vorticities around N moving vortices. The formal dynamic law for such objects was first derived in the 19th century by Kirkhoff and Routh. We devise a gluing approach for the construction of smooth N-vortex solutions. We capture in high precision the core of each vortex as a scaled finite mass solution of Liouville's equation plus small, more regular terms.
This work is in collaboration with J. D avila, M. del Pino, J. Wei.

Ilya Molchanov (Bern)

28 January 2019

The semigroup of metric measure spaces and stable random spaces

The family of Polish spaces equipped with probability measures can be turned into a semigroup using the Cartesian product as the semigroup operation. The main result is the Fundamental Theorem of Arithmetics for this semigroup, which establishes an analogue of the prime numbers decompositions for such spaces. Further, random metric measure spaces are considered in view of their infinite divisibility and stability properties with respect to the semigroup operation. A characterisation of stable metric measure spaces is also provided.
(joint work with Steve Evans)

Sabine Bögli (Imperial)

21 Januaury 2019

The essential numerical range for unbounded linear operators

If a linear operator T is approximated by projection or domain truncation methods, eigenvalues may accumulate at a point that does not belong to the spectrum of T. The occurrence of such a spurious eigenvalue is commonly known as spectral pollution. A useful tool to describe the set of spectral pollution is the notion of essential numerical range W_e(T) which was introduced in the late 1960s for bounded T. We discuss the generalisation of this notion to unbounded operators, including equivalent characterisations and perturbation results.
This is based on joint work with M.Marletta and C.Tretter.

Michela Ottobre (Herriot-Watt Uni)

10 December 2018

On a class of SDEs with multiple invariant measures

In 1968 Hoermander introduced a sufficient condition to ensure hypoellipticity of second order partial differential operators. As is well known, this seminal work of Hormander had deep repercussions both in the analysis of PDEs and in probability theory and a large strand of literature has been devoted to studying ergodic properties of processes which do satisfy the Hoermander condition  (HC). While such literature has mostly been concerned with study of convergence to equilibrium for dynamics which admit a unique invariant measure, it is a known fact that Hoermander-type diffusions need not be ergodic, i.e. they need not admit a unique invariant measure. In this talk we will present the UFG condition, which is weaker than the Hormander condition. Such a condition was introduced by Kusuoka and Strook with probabilistic motivations, and, independently, by Sussman, Hermann and Lobry, this time in the field of control theory. We will present new results on the geometry and long time behaviour of diffusion semigroups that do not satisfy the Hoermander condition. We will highlight how, loosely speaking, UFG diffusions constitute a large class of SDEs which exhibit multiple equilibria (invariant measures) and such that it is possible to determine in a systematic way the basin of attraction of each  equilibrium state.

Christian Maes (KU Leuven)

3:10 - 4:10

3 December 2018

Time-symmetric aspects of stationary flow

We start by recalling the relation between detailed balance and gradient flow. We generalize it to the case of GENERIC.  We discuss how time-symmetric aspects matter crucially when nonequilibrium driving is added.

David Lafontaine (Bath)

2:10 - 3:10

3 December 2018

About wave and Schrödinger equations in the exterior of many strictly convex obstacles

In order to study the non-linear Schrödinger and wave equations, it is crucial to understand the decay of solutions of the associated linear equations. When a trapped trajectory exists, a loss is unavoidable for a first family of a-priori estimates of the linear flow: the so-called smoothing estimates. In contrast, we will show that in the exterior of many strictly convex obstacles, the estimates of space-time norms of solutions, known as Strichartz estimates, hold with no loss with respect to the flat case, as soon as the dynamic of the trapped trajectories is sufficiently unstable. Finally, if time permits, we will say a word about the associated non-linear equations: if the geometry does not induce too much concentration of energy, we expect that the solutions behave linearly in large times.

Benjamin Gess (Leipzig)

3:10 - 4:10

26 November 2018

Random dynamical systems for stochastic PDE with nonlinear noise.

In this talk we will revisit the problem of generation of random dynamical systems by solutions to stochastic PDE. Despite being at the heart of a dynamical system approach to stochastic dynamics in infinite dimensions, most known results are restricted to stochastic PDE driven by affine linear noise, which can be treated via transformation arguments. In contrast, in this talk we will address instances of stochastic PDE with nonlinear noise, with particular emphasis on porous media equations driven by conservative noise. This class of stochastic PDE arises in particular in the analysis of stochastic mean curvature motion, mean field games with common noise and is linked to fluctuations in non-equilibrium statistical mechanics.

Kirill Cherednichenko (Bath)

2:10 - 3:10

26 November 2018

Periodic PDEs with critical contrast: unified approach to homogenisation and links to time-dispersive media.

I shall discuss a novel approach to the homogenisation of high-contrast periodic PDEs, which yields an explicitly construction of their norm-resolvent asymptotics. A practically relevant outcome of this result is that it interprets  composite media with micro-resonators as a class of time-dispersive media. This is joint work with Yulia Ershova and Alexander Kiselev.

Random dynamical systems for stochastic PDE with nonlinear noise.

In this talk we will revisit the problem of generation of random dynamical systems by solutions to stochastic PDE. Despite being at the heart of a dynamical system approach to stochastic dynamics in infinite dimensions, most known results are restricted to stochastic PDE driven by affine linear noise, which can be treated via transformation arguments. In contrast, in this talk we will address instances of stochastic PDE with nonlinear noise, with particular emphasis on porous media equations driven by conservative noise. This class of stochastic PDE arises in particular in the analysis of stochastic mean curvature motion, mean field games with common noise and is linked to fluctuations in non-equilibrium statistical mechanics.

Mahir Hadžić (KCL)

19 November 2018

Cancelled

Petr Siegl (Belfast)

12 November 2018

Spectral instabilities of Schrödinger operators with complex potentials.

We present an overview of recent results on pseudospectra and basis proper-
ties of eigensystems of one-dimensional Schrödinger operators with unbounded
complex potentials. In particular, we address the problem of localizing the
transition between spectral (Riesz basis of eigenvectors and “normal” behavior
of the resolvent norm) and pseudospectral (vast regions in the complex plane
where the resolvent norm explodes) character of these operators depending on
the size of real and imaginary parts of the potential.
The talk is based on:
[1] B. Mityagin and P. Siegl: Local form-subordination condition and Riesz
basisness of root systems, Journal d’Analyse Math´ ematique, to appear,
arXiv:1608.00224
[2] D. Krejˇ ciˇ r´ık and P. Siegl: Pseudomodes for Schrödinger operators with
complex potentials, Journal of Functional Analysis, to appear, arXiv:1705.01894.

Nicolas Dirr (Cardiff)

Elaine Crooks (Swansea)

Tristan Pryer (Reading)

Carlo Mercuri (Swansea)

Gui-Qiang Chen (Oxford)

11:00 - 5:30

9 November 2018

LMS South-West Network on Generalised Solutions for Nonlinear PDEs

11.30-12.00 Welcome coffee.
12.00-12.35  Nicolas Dirr (Cardiff)
12.40-13.40 LUNCH
13.40-14.15  Elaine Crooks (Swansea)
14.20-14.55 Tristan Pryer (Reading)
15.00-15.15 Coffee break
15.15-15.50 Carlo Mercuri (Swansea)
15.55- 16.30  Gui-Qiang Chen (Oxford)
16.30-17.30 Discussions on EPSRC Network Application

David Beltran (BCAM)

5 November 2018

Local smoothing estimates for Fourier Integral Operators and wave equations

The sharp fixed-time Sobolev estimates for Fourier Integral Operators (and therefore solutions to wave equations in Euclidean space or compact manifolds) were established by Seeger, Sogge and Stein in the early 90s. Shortly after, Sogge observed that a local average in time leads to a regularity improvement with respect to the sharp fixed-time estimates. Establishing variable-coefficient counterparts of the Bourgain—Demeter decoupling inequalities, we improved the previous known local smoothing estimates for FIOs, and we show, in particular, that our results are sharp in both the Lebesgue and regularity exponent (up to the endpoint) in odd dimensions. This is joint work with Jonathan Hickman and Christopher D. Sogge.

Maria Carmen Reguerra (Birmingham)

22 October 2018

Sparse bounds for Bochner-Riesz operators

Sparse operators are positive dyadic operators that have very nice boundedness properties. The L^p bounds and weighted L^p bounds with sharp constant are easy to obtain for these operators. In the recent years, it has been proven that singular integrals (cancellative operators) can be pointwise controlled by sparse operators. This has made the sharp weighted theory of singular integrals quite straightforward. The current efforts focus in understanding the use of sparse operators to bound rougher operators, such a oscillatory integrals. Following this direction, our goal in this talk is to describe the control of Bochner-Riesz operators by sparse operators.

Sanju Velani (York)

15 October 2018

Inhomogeneous Diophantine Approximation on M_0 -sets with restricted denominators

Let F ⊆ [0,1] be a set that supports a probability measure µ with the property that | b µ(t)| ≪ (log|t|) −A for some constant A > 0. Let A = (q n ) n∈ N be a sequence of natural numbers. If A is lacunary and A > 2, we establish a quantitative inhomogeneous Khintchine-type theorem in which (i) the points of interest are restricted to F and (ii) the denominators of the ‘shifted’ rationals are restricted to A. The theorem can be viewed as a natural strengthening of the fact that sequence (q n x mod1) n∈ N is uniformly distributed for µ almost all x ∈ F. Beyond lacunary, our main theorem implies the analogous quantitative result for sequences A for which the prime divisors are restricted to a finite set of k primes and A > 2k.

Frank Rösler (Cardiff)

8 October 2018

Norm-resolvent convergence in perforated domains.

For several different types of boundary conditions (Dirichlet, Neumann and Robin), we prove norm-resolvent convergence for the operator −∆ in the perforated domain Ω without balls of small radius, to the limit operator −∆+µ on L 2 (Ω), where µ ∈ C is a constant depending on the choice of boundary conditions.
This is an improvement of previous results [Cioranescu & Murat. A Strange Term Coming From Nowhere, Progress in Nonlinear Differential Equations and Their Applications, 31, (1997)], [S. Kaizu. The Robin Problems on Domains with Many Tiny Holes. Proc. Japan Acad., 61, Ser. A (1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.

Anton Savostianov (Durham)

1 October 2018

Homogenisation with error estimates of attractors for damped semi-linear anisotropic wave equations

It is well known that long time behaviour of  a dissipative dynamical system generated by an evolutionary PDE can be described in terms of attractor, an attracting set which is essentially thinner than a ball of the corresponding phase space of the system. In this talk we compare long time behaviour of damped anisotropic wave equations with the corresponding homogenised limit in terms of their attractors. First we will formulate order sharp estimates between the trajectories of the corresponding systems and will see that the hyperbolic nature of the problem results in extra correction comparing with parabolic equations. Then, after brief review on previous results on homogenisation of attractors, we will discuss new results. It appears that the Hausdorff distance between the anisotropic attractors and their homogenised counter-parts, in suitable norms, can be estimated via operator norm of the difference of the resolvents of the corresponding elliptic operators. Furthermore, we show that the homogenised attractor admits first-order correction suggested by the natural asymptotic expansion. The corrected homogenised attractors, as expected, are close to the anisotropic attractors already in the strong energy norm. The corresponding quantitative estimates on the Hausdorff distance between the corrected homogenised attractors and anisotropic ones, with respect to the strong energy norm, are also obtained. Our results are applied to Dirchlet, Neumann and periodic boundary conditions. This is joint work with Shane Cooper.

Marco Marletta (Cardiff University)

Jiang-Lun Wu (Swansea)

Federica Dragoni (Cardiff University)

Peter Hintz (Berkeley)

Dmitri Finkelshtein (Swansea)

27 September 2018

South Wales Analysis and Probability Seminar

9:30-10:00 Coffee and registration
10:00-10:45 Marco Marletta (Cardiff)
10:45-11:30 Jiang-Lun Wu (Swansea)
11:30-11:50 Coffee
11:50-12:35 Federica Dragoni (Cardiff)
12:35-14:05 Lunch
14:05-15:05 Colloquium: Peter Hintz (Berkeley)
15:05-15:30 Coffee
15:30-16:15 Dmitri Finkelshtein (Swansea)