# LMS Hyperbolic Network

## UK Network on Hyperbolic Equations and Related Topics, 2017-2018

Supported by the LMS (Scheme 3 Grant Ref 31703) and by the Edinburgh Mathematical Society

Departments

## Organisers

Linear and nonlinear hyperbolic partial differential equations (PDEs) arise in basically all sciences (physics, chemistry, medicine, engineering, astronomy, etc.). In physics, they model several important phenomena, from propagation of waves in a medium (for instance propagation of seismic waves during an earthquake) to refraction in crystals and gas-dynamics. The purpose of this UK network on hyperbolic equations and related topics is to bring together the expertise on hyperbolic equations of three different mathematics department (Edinburgh, Imperial, Loughborough), to strengthen the existing research collaborations and to create new ones. Three 1-day workshops per year are planned focused on different approaches to hyperbolic equations and related topics (inverse problems, kinetic theory, imaging, microlocal analysis, general relativity, etc.).

Meeting 1
1 December 2017 (Friday), Imperial College London, Huxley 410
Planned program

11:00-12:00: Camil Muscalu (Cornell): The helicoidal method
12:00-12:30: Aparajita Dasgupta (Imperial College London): Eigenfunction expansions of ultradifferentiable functions and ultradistributions
LUNCH
13:30-14:30: Pieter Blue (Edinburgh): Hidden symmetries and decay of fields outside black holes
14:30-15:00: Wagner Augusto Almeida de Moraes (Curitiba/ICL): Global hypoellipticity on compact Lie groups
COFFEE BREAK
15:30-16:00: Chiara Taranto (Imperial College London): Well-posedness of the Rockland wave equation on graded groups and sub-Laplacian Gevrey spaces
16:10-16:40: Christian Jäh (Loughborough): Recent progress in hyperbolic systems with variable multiplicities

All are welcome to attend.

Meeting 2
7 March 2018 (Wednesday), Loughborough University
Schofield Building on Loughborough Campus Map

Planned program: TBA

Meeting 3
TBA, University of Edinburgh

### Abstracts

ABSTRACTS for MEETING 1 at Imperial College London:

Camil Muscalu (Cornell): The helicoidal method
Not  long  ago  we  discovered  a  new method  of  proving  vector  valued  inequalities  in  Harmonic Analysis. With the help of it, we have been able to give  complete positive answers to a number of questions that have been circulating for some time. The plan of the talk is to describe (some of) these,  and  to  also  explain  how  this  method  implies  sparse domination  results  for  various  multi-linear operators and their (multiple) vector valued extensions. Joint work with Cristina BENEA.

Wagner Augusto Almeida de Moraes (Curitiba/ICL): Global hypoellipticity on compact Lie groups
The objective of this talk is to analyse the global hypoellipticity and the global solvability of constant vector fields on product of compact Lie groups. For this we study the behavior of the Fourier coefficients in which naturally appear conditions that associate the constants of the operator with the structure of the Lie groups that we are working. In addition, it is possible to modify these conditions to obtain what we will call global komatsu hypoellipticity and solvability of Roumieu type and Beurling type. Finally, we will analyse the relations that these properties have with each other.

Pieter Blue (Edinburgh): Hidden symmetries and decay of fields outside black holes
I will discuss energy and Morawetz (or integrated local decay) estimates for fields outside black holes, in particular the Vlasov equation. This builds on earlier work for the wave and Maxwell equation. Much of the work on these problems in the last decade has used the vector-field method and its generalisations. One generalisation has focused on using symmetries, differential operators that take solutions of a PDE to solutions. In this context, a hidden symmetry is a symmetry that does not decompose into first-order symmetries coming from a smooth family of isometries of the underlying manifold. In this talk, I will build on applications of the vector-field method to the Vlasov equation to prove an integrated energy decay for the Vlasov equation outside a very slowly rotating Kerr black hole, and I will discuss some new features of the symmetry algebra for the Vlasov equation, which illustrate the difficulties in passing to pointwise-decay estimates for the Vlasov equation in this context. This is joint work with L. Andersson and J. Joudioux.

Aparajita Dasgupta (Imperial College London): Eigenfunction expansions of ultradifferentiable functions and ultradistributions
In this talk a global characterisation of classes of ultradiﬀerentiable functions and corresponding ultradistributions will be given in terms of the eigenfunction expansion of an elliptic operator on a compact manifold. This extends the result for analytic functions on compact manifolds obtained by Seeley in 1969, and the characterisation of Gevrey functions and Gevrey ultradistributions on compact Lie groups and homogeneous spaces by the authors (2014). Joint work with Michael Ruzhansky.

Chiara Taranto (Imperial College London): Well-posedness of the Rockland wave equation on graded groups and sub-Laplacian Gevrey spaces
In a recent work [2], C. Garetto and M. Ruzhansky investigate the Cauchy problem for the time-dependent wave equation for sums of squares of vector ﬁelds on compact Lie groups. In particular, they establish the well-posedness in spaces that compare to the Gevrey spaces. We generalise their result to graded groups and to more general operators, [3]. Furthermore, modelled on the spaces of Gevrey-type appearing in [2], we deﬁne the sub-Laplacian Gevrey spaces on manifolds and partially characterise these spaces. A full characterisation for the sub-Laplacian Gevrey spaces is achieved for certain groups whose symbolic calculus is well-known, such as the Heisenberg group. In this talk, I will ﬁrst introduce some preliminary facts about the Fourier analysis and the quantisation on Lie groups, focusing on the case of the Heisenberg group. Subsequently, I will present the results mentioned above. This is a joint work with my supervisor Prof. Ruzhansky.

References
[1] D. Dasgupta, M. Ruzhansky, Gevrey functions and ultradistributions on compact Lie groups and homogeneous spaces, Bulletin des Sciences Math´ematiques, 138 (2014), 756-782.
[2] C. Garetto, M. Ruzhansky, Wave equation for sum of squares on compact Lie groups, J. Diﬀerential Equations, 258 (2015), 4324-4347.
[3] M. Ruzhansky, C. Taranto, Time-dependent wave equations on graded groups, arXiv:1705.03047.

Christian Jäh (Loughborough): Recent progress in hyperbolic systems with variable multiplicities

Hyperbolic systems with variable multiplicities are notoriously difficult since many well developed techniques for the treatment of systems are not available. Though there are results available, the field is relatively bleak in comparison to hyperbolic systems with constant multiplicities.
In this talk we discuss some recent progress in the $C^\infty$ well-posedness of such systems with coefficients depending only on $t$ and some new results on Sobolev well-posedness for systems depending on $t$ and $x$. In the first case [1], some restrictions on the roots of a general m x m system are imposed and some results form linear algebra are used. In the second case, we discuss some work in progress which deals with systems in upper triangular form.
References
[1] C. Garetto and Ch. Jäh, Well-posedness of hyperbolic systems with multiplicities and smooth coefficients, Mathematische Annalen, 369(1-2), pp. 441-485.
[2] C. Garetto, Ch. Jäh, and M. Ruzhansk,y Cauchy problem for a class of m by m hyperbolic systems with non-diagonalisable principal part and variable multiplicity, in preparation.