Details of Grant 

EPSRC Reference:	GR/R67583/01
Title:	Regularity of solutions and spectral asymptotics for systems of partial differential equations
Principal Investigator:	Ruzhansky, Professor M
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	Imperial College London
Scheme:	Fast Stream
Starts:	04 November 2002	Ends:	03 June 2004	Value (£):	62,032
EPSRC Research Topic Classifications:	Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 07 Mar 2001 MARCH 2001 Mathematics Responsive Mode Deferred
Summary on Grant Application Form
We are going to apply recent developments in the regularity theory of Fourier integral operators with complex phases to several problems. First, we are going to analyze systems with multiplicity 2, when expressions of parametrices in terms of Fourier integrals are known. Then, we will study systems with finite order of intersection of bicharacteristics, where parametrices can still be understood asymptotically as Fourier integrals. We will determine the loss of regularity for solutions of the Cauchy problem for such systems. Also, we will study spectral asymptotics for elliptic systems with the same properties. As the methods here are often similar, we should be able to relate geometric properties of parametrices (in terms of Fourier integrals) for hyperbolic systems to the spectral asymptotics of elliptic systems with multiplicity. Finally, we will use the global existence of complex phases to analyze global (in time) properties of solution to hyperbolic PDEs. There are strong consequences of this, for example one can attempt to create a general theory of global well-posedness of semilinear hyperbolic PDEs. Finally, the use of complex phases will allow us to extend ail the results to equations (systems) with complex characteristics with constant signs of imaginary parts.
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Organisation Website:	http://www.imperial.ac.uk