Hyperbolic systems and sub-elliptic operators: to extend the results obtained on compact Lie groups to a wider class of hyperbolic equations

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Project Detail:
Linear and nonlinear partial differential equations (PDEs) arise in basically all sciences (physics, chemistry, medicine, engineering, astronomy, etc.). In particular, 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. From a mathematical point of view we say that a linear PDE is hyperbolic if its characteristic roots are all real and analogously a system is hyperbolic if its eigenvalues are all real. In the last fifty years numerous mathematicians have devoted their studies to hyperbolic equations and systems and realised quickly the main role played by multiplicities. Hyperbolic equations and systems with multiplicities arise in the analysis of sub-elliptic operators. Typical example is the wave equation (or a Schrödinger equation) associated to Hörmander’s sum of squares of vector fields on a group or a manifold. Such equations can be transformed into first order hyperbolic systems. The well-posedness of the Cauchy problem for wave equations associated to sum of squares has been recently investigated in some specific cases (t-dependent coefficients only and Lie groups) but the theory is far to be complete. It is aim of this project to extend the results obtained on compact Lie groups to a wider class of hyperbolic equations (not only t-dependent) and to a more general context (non compact Lie groups and manifolds).

Find out more:
http://www.lboro.ac.uk/departments/maths/staff/academic/claudia-garetto/
http://homepages.lboro.ac.uk/~mamm4/
http://wwwf.imperial.ac.uk/~ruzh/
http://www.sciencedirect.com/science/article/pii/S0022039615000467

Entry requirements:
Applicants should have, or expect to achieve, at least a 2:1 Honours degree (or equivalent) in mathematics or a related subject. A relevant Master’s degree and/or experience in the following will be an advantage: analysis.

Funding information:
The 3 year studentship provides a tax free stipend of £14,553 per annum (in line with the standard research council rates) for the duration of the studentship plus tuition fees at the UK/EU rate. International (non EU) students may apply however the total value of the studentship will be used towards the cost of the International tuition fee in the first instance.

Application details:
Reference number: MA/CG-2/2017
Start date: 1st October 2017
Closing date: 25th July 2017

Supervisors:
Primary supervisor: Claudia Garetto
Secondary supervisor: Marta Mazzocco
External supervisor: Michael Ruzhansky

Contact details:
Name: Claudia Garetto
Email address: [Email Address Removed]
Telephone number: +44 (0) 1509222870

How to apply:
All applications should be made online at http://www.lboro.ac.uk/study/apply/research/. Under programme name, select ‘Mathematics’.

Please quote reference number: MA/CG-2/2017