Mathematical Sciences: Geometric Methods and Nonlinear      Hyperbolic Equations
{'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}
New geometrical methods will be used in the analysis of two                problems concerning nonlinear hyperbolic equations.  One thrust                 will focus on the establishment of a new decay estimate for the                 linear nonhomogeneous Klein-Gordon equation in three space                      dimensions.  Decay estimates arise when one studies linear                      problems associated with nonlinear hyperbolic equations.  This                  approach has been suggested by successful applications by                       Klainerman and Hormander to quadratically nonlinear wave                        equations.  A similar program is planned for the Klein-Gordon                   equation where uniform decay estimates will be sought.  The                     solution is to be bounded by an appropriate norm of the forcing                 function - which is not assumed to have compact support.  When                  this program is carried out, it should be possible to give a                    direct proof of the existence of global small solutions of the                  equation.                                                                            A second line of investigation continues earlier work on                   obtaining lower bounds for the life-span of smooth three-                       dimensional compressible flows.  These ideal, compressible flows                are considered with small initial disturbances that are smooth                  and localized.  Near the front of the ensuing disturbance a                     singularity or discontinuity is eventually detected.  These are                 shocks in the acoustical waves and are predictable.  The thrust                 of this effort is to determine whether or not singularities can                 form at an earlier time and, if so, to characterize their nature.               The functional relationship between the solution and its initial                configuration is expected to be an exponential involving the size               of the initial support.                                                              This work will have natural application to studies of wave                 motion and fluid dynamics.
{'FirstName': 'Thomas', 'LastName': 'Sideris', 'PI_MID_INIT': 'C', 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Thomas C Sideris', 'EmailAddress': 'sideris@math.ucsb.edu', 'NSF_ID': '000210522', 'StartDate': '07/15/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'University of California-Santa Barbara', 'CityName': 'SANTA BARBARA', 'ZipCode': '931060001', 'PhoneNumber': '8058934188', 'StreetAddress': '3227 CHEADLE HALL', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'California', 'StateCode': 'CA', 'CONGRESSDISTRICT': '24', 'CONGRESS_DISTRICT_ORG': 'CA24', 'ORG_UEI_NUM': 'G9QBQDH39DF4', 'ORG_LGL_BUS_NAME': 'UNIVERSITY OF CALIFORNIA, SANTA BARBARA', 'ORG_PRNT_UEI_NUM': None}
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{'Code': '126100', 'Text': 'CLASSICAL ANALYSIS'}
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