# 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
21 February 2018 (Wednesday), Loughborough University, Room SCH 1.01
Schofield Building on Loughborough Campus Map

Planned program: Room SCH 1.01

10:30-11:30: Karima Khusnutdinova (Loughborough): Ring waves in stratified fluids over parallel currents
11:30-12:00: Matt Tranter (Loughborough): Scattering of long solitary waves in delaminated bars
12:00-13:00: Baptiste Morisse (Cardiff): Well-posedness of weakly hyperbolic systems of PDEs in Gevrey regularity
LUNCH
14:00-15:00: Julio Delgado (Imperial College London):  Schatten-von Neumann classes of integral operators
15:00-15:30: Daulti Verma (Miranda House, University of Delhi/Imperial): Hardy Inequalities on Metric Measure Spaces
COFFEE
16:00-16:30: Marianna Chatzakou (Imperial): Pseudo-differential operators on the Engel group

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.

ABSTRACTS for MEETING 2 at Loughborough University

Karima Khusnutdinova (Loughborough): Ring waves in stratified fluids over parallel currents

We consider long ring waves in a stratified fluid in the presence of a depth-dependent parallel current (e.g., oceanic internal waves generated in narrow straits) within the scope of the full set of Euler equations with the boundary conditions appropriate for oceanographic applications. We show that despite the clashing geometries of the waves and the current, there exists a new linear modal decomposition (separation of variables) in the set of Euler equations describing the waves. We use it to describe distortion of the wavefronts of surface and internal waves, and to derive a new 2+1D weakly-nonlinear model for the amplitudes of the waves.  The wavefronts are described by the singular solution (envelope of the general solution) of a nonlinear first-order differential equation. We find the singular solution for the case of a two-layer fluid with the piecewise-constant current and discuss some properties of the waves. The wavefronts of surface and interfacial waves propagating over the same current look strikingly different. This is joint work with Xizheng Zhang.

Matt Tranter (Loughborough): Scattering of long solitary waves in delaminated bars

In this talk we will discuss the scattering of long longitudinal bulk strain solitary waves in delaminated elastic bars within the scope of Boussinesq-type equations. We consider two cases: pure solitary waves for a perfectly bonded symmetric waveguide and radiating solitary waves for a layered waveguide with a soft bonding between the layers. We develop direct and semi-analytical numerical approaches to the problem, using asymptotic multiple-scale expansions and averaging with respect to the fast variables. Results are compared to theoretical estimates using the Inverse Scattering Transform. Our results indicate that solitary waves may be able to help detect delamination. This is a joint work with Karima Khusnutdinova.

Baptiste Morisse (Cardiff): Well-posedness of weakly hyperbolic systems of PDEs in Gevrey regularity

I consider systems of first-order PDEs, which are weakly hyperbolic: the spectrum of the principal symbol is real but eigenvalues may cross. Close to one of those crossing eigenvalues,  lower order linear terms may induce a typical Gevrey growth in frequency. I will present an energy estimate in Gevrey regularity, using an approximate symmetrizer of the principal symbol. The symbol of such an approximate symmetrizer is in a special class of symbols, related to a specific metric in phase space. For such symbols, composition of associated operators lead to error terms that only can be handle thanks to the Gevrey energy.

Julio Delgado (Imperial College London):  Schatten-von Neumann classes of integral operators

In this talk we present sharp kernel conditions ensuring that the corresponding integral operators belong to Schatten-von Neumann classes. The conditions are given in terms of the spectral properties of operators acting on the kernel. As applications we establish several criteria in terms of different types of differential operators and their spectral asymptotics in different settings. Joint work with Michael Ruzhansky.

Daulti Verma (Miranda House, University of Delhi/Imperial): Hardy Inequalities on Metric Measure Spaces

In this talk, we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no dierentiable structure on such spaces, the inequalities will be given in the integral form in the spirit of Hardy’s original inequality. We give examples obtaining new weighted Hardy inequalities on Rn, on homogeneous groups, on hyperbolic spaces, and on Cartan-Hadamard manifolds. This is a joint work with Prof. Ruzhansky.

Marianna Chatzakou (Imperial): Pseudo-differential operators on the Engel group

We will give the concrete formulas of the Fourier transform and of the Plancherel measure of the Engel group (a particular case of a Carnot group), as well as, of the innitesimal representation of its Lie algebra. Following this, we apply the group Fourier transform on some left invariant operators, such as the sub-Laplacian, which is in fact a particular case of the anharmonic oscillator of order two. Finally we will present some results on the spectral properties of this operator and an upper bound of the partial derivatives of the symbol of its inverse.