**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**

School of Mathematics, The University of Edinburgh, Maxwell Institute for Mathematical Sciences

Department of Mathematics, Imperial College London

Department of Mathematical Sciences, Loughborough University

## Organisers

Pieter Blue (Edinburgh)

Claudia Garetto (Loughborough)

Michael Ruzhansky (Imperial College London)

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**

Add to Google Calendar

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 **

Add to Google Calendar

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

**27 April 2018 (Friday), University of Edinburgh, ICMS, 15 South College Street**

Add to Google Calendar

Planned program

11.00 Oana Pocovnicu (Heriot Watt University): *Long time regularity of the 2D Euler-Poisson system for electrons with vorticity*

12.00 Zoe Wyatt (University of Edinburgh): *The zero-mode stability of Kaluza Klein spacetimes*

12.30 LUNCH

13.30 Martin Taylor (Imperial College London): *Global nonlinear stability of Minkowski space for the massive and massless Einstein–Vlasov systems*

14.30 Qian Wang (Oxford University): *Sharp local well-posedness for general quasilinear wave equations*

15.30 COFFEE

16.00 Leonardo Tolomeo (University of Edinburgh): *Global well-posedness of the two-dimensional cubic stochastic nonlinear wave equation*

16.30 Michael Ruzhansky (Imperial College London): *Very weak solutions to wave equations*

17.30 CLOSE

All are welcome to attend.

### 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*

*Well-posedness of hyperbolic systems with multiplicities and smooth coefficients*, Mathematische Annalen, 369(1-2), pp. 441-485.

**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.

**Abstracts for MEETING 3 at the University of Edinburgh**

Oana Pocovnicu (Heriot Watt University)

Title: Long time regularity of the 2D Euler-Poisson system for electrons with vorticity

Abstract: The Euler-Poisson system for electrons is one of the simplest two-fluid models used to describe the dynamics of a plasma. From the point of view of analysis, it can be re-written as a quasilinear hyperbolic PDE. In this talk, we will discuss the long time existence for the two-dimensional Euler-Poisson system, with a particular attention to the dependence of the time of existence on the size of the vorticity. This talk is based on joint work with A. Ionescu (Princeton).

Zoe Wyatt (University of Edinburgh)

Title: The zero-mode stability of Kaluza Klein spacetimes

Abstract: In string theory, our most developed theory of quantum gravity to date, one is interested in spacetimes of the form $R^{1+3} \times K$ where $K$ is some $n-$dimensional compact Ricci-flat manifold. In the first and simplest case considered by Kaluza and later Klein, $K$ is the $n-$torus with the flat metric. An interesting question to ask is whether this solution to the Einstein equations, viewed as an initial value problem, is stable to small perturbations of the initial data. Motivated by this problem, I will outline the proof of stability in a restricted class of perturbations, and discuss the physical justification behind this restriction. Furthermore the resulting quasilinear hyperbolic PDE exhibits the weak-null condition, and I will discuss how it can be treated by generalising the proof of the non-linear stability of Minkowski spacetime given by Lindblad and Rodnianski.

Martin Taylor (Imperial College London)

Title: Global nonlinear stability of Minkowski space for the massive and massless Einstein–Vlasov systems

Abstract: The Einstein—Vlasov system describes an ensemble of collisionless particles interacting via gravity, as modelled by general relativity. Under the assumption that all particles have equal mass there are two qualitatively different cases according to whether this mass is zero or nonzero. I will present two theorems concerning the global dispersive properties of small data solutions in both cases. The massive case is joint work with Hans Lindblad.

Qian Wang (Oxford University)

Title: Sharp local well-posedness for general quasilinear wave equations

Abstract: The commuting vector fields approach, devised for Strichartz estimates by Klainerman, was employed for proving the local well-posedness in the Sobolev spaces $H^s$ with $s>2+\frac{2-\sqrt{3}}{2}$ for general quasi-linear wave equation in ${\mathbb R}^{1+3}$ by him and Rodnianski. Via this approach they obtained the local well-posedness in $H^s$ with $s>2$ for $(1+3)$ vacuum Einstein equations, by taking advantage of the vanishing Ricci curvature. The sharp, $H^{2+\epsilon}$, local well-posedness result for general quasilinear wave equation was achieved by Smith and Tataru by constructing a parametrix using wave packets. Using the vector fields approach, one has to face the major hurdle caused by the Ricci tensor of the metric for the quasi-linear wave equations. This posed a question that if the geometric approach can provide the sharp result for the non-geometric equations. I will present my work, which proves the sharp local well-posedness of general quasilinear wave equation in ${\Bbb R}^{1+3}$ by a vector field approach, based on geometric normalization and new observations on the mass aspect functions.

Leonardo Tolomeo (University of Edinburgh)

Title: Global well-posedness of the two-dimensional cubic stochastic nonlinear wave equation

Abstract: In this talk, we consider the Cauchy problem for the defocusing cubic stochastic nonlinear wave equation (SNLW) with additive space-time white noise forcing, posed in two spatial dimensions, both on the torus and on the Euclidean space. Because of the roughness of the forcing, we will use a modified version of the I-method to get global well posedness for this equation on the torus. Furthermore, we will discuss how to combine this argument with finite speed of propagation to get the same result on R2.

Michael Ruzhansky (Imperial College London)

Title: Very weak solutions to wave equations

Abstract: In this talk we will discuss the recently introduced notion of very weak solutions allowing one to deal with equations with very singular (distributional) coefficients. We will also give some examples and numerical experiments.