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 CoInvestigators: 

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: 

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 wellposedness 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 