Mathematical Sciences: Global Properties of Nonlinear       Hyperbolic Equations Arising in Mathematical Physics
{'Code': '03040102', 'Directorate': {'Abbreviation': 'MPS', 'LongName': 'Direct For Mathematical & Physical Scien'}, 'Division': {'Abbreviation': 'DMS', 'LongName': 'Division Of Mathematical Sciences'}}
{'SignBlockName': 'John V. Ryff', 'PO_EMAI': None, 'PO_PHON': None}
This project continues work of the principal                               investigator on problems related to nonlinear partial                           differential equations arising in mathematical physics.  The                    equations are typically of hyperbolic type.  Objectives of                      this work include the development of methods for proving                        global existence, stability and asymptotic behavior results                     for classical field equations.  Among the problems to be                        considered are the global nonlinear stability of Minkowski,                     Schwartzchild and Kerr solutions of the Einstein vacuum                         equations.  Also the asymptotic behavior of the Yang-Mills                      equations for general initial conditions will be studied.                       Work will also be done on establishing global existence and                     asymptotic results for the Maxwell-Dirac equations subject to                   small initial conditions.                                                            A second line investigation involves analysis of the                       formation of singularities for quasilinear wave equations in                    three space dimensions.  It has been known for some time that                   solutions to certain classes of equations have developed                        singularities in finite time .  Very little is known about                      the nature of these singularities near the onset of blow up                     and the three dimensional case is the least understood of                       all.  One of the difficulties to be overcome is conceiving a                    means for visualizing solutions whose singularities only                        occur after exponentially long periods of time have elapsed.                         Work will continue toward developing new methods for the                   study of global smooth solutions for semilinear wave                            equations.  In wave equations where the nonlinear term is a                     power of the unknown function, it is known that solutions                       remain smooth indefinitely if the power is less than five.                      For higher powers almost nothing is known, although it is                       believed that some version of singular local energy estimates                   will provide the information necessary for the resolution of                    this problem.                                                                        In addition to natural applications to mathematical                        physics, this work will provide methods which will be                           applicable to the analysis of large classes of differential                     equations.
{'FirstName': 'Sergiu', 'LastName': 'Klainerman', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Sergiu Klainerman', 'EmailAddress': 'seri@math.princeton.edu', 'NSF_ID': '000161442', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Princeton University', 'CityName': 'PRINCETON', 'ZipCode': '085442001', 'PhoneNumber': '6092583090', 'StreetAddress': '1 NASSAU HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'New Jersey', 'StateCode': 'NJ', 'CONGRESSDISTRICT': '12', 'CONGRESS_DISTRICT_ORG': 'NJ12', 'ORG_UEI_NUM': 'NJ1YPQXQG7U5', 'ORG_LGL_BUS_NAME': 'THE TRUSTEES OF PRINCETON UNIVERSITY', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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