Mathematical Sciences: Microlocal Analysis and Applications
{'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}
Two themes will be developed during the course of this                     project.  Each centers on fundamental questions related to                      partial differential equations.  In the first instance work will                be done investigating the propagation of singularities for                      solutions of nonlinear partial differential equations.  The basic               question is one of predicting singularities in positive time                    knowing the singularities which occurred in the past.  An                       analogous problem is the description of the singularities from                  knowledge of the Cauchy data at the initial time.  The                          propagation of singularities is a microlocal phenomenon, as                     pointed out by Huygens centuries ago.  The modern exposition of                 this principle is given by the wave front set and, depending on                 which Sobolev space contains a solution, the singularities travel               along bicharacteristics, as in the linear case, or else require                 intricate description - which represents one of the goals of this               work.                                                                                A second line of research will deal with the local                         solvability for pseudo-differential operators of principal type.                One is interested in solutions of the associated nonhomogeneous                 equation.  Earlier work of Nirenberg and Treves proved a                        sufficient condition for local solvability in the analytic case                 for differential operators.  Roughly the same condition gives a                 necessary condition for local solvability for pseudo-differential               operators.  Recently, sufficiency of the condition was shown for                the two-dimensional case.  Work will concentrate on lifting this                dimension restriction.
{'FirstName': 'Nicolas', 'LastName': 'Lerner', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Nicolas Lerner', 'EmailAddress': None, 'NSF_ID': '000290781', 'StartDate': '06/01/1988', 'EndDate': None, 'RoleCode': 'Principal Investigator'}
{'Name': 'Purdue Research Foundation', 'CityName': 'WEST LAFAYETTE', 'ZipCode': '479064353', 'PhoneNumber': '3174946200', 'StreetAddress': '1281 WIN HENTSCHEL BLVD STE 2500', 'StreetAddress2': None, 'CountryName': 'United States', 'StateName': 'Indiana', 'StateCode': 'IN', 'CONGRESSDISTRICT': '04', 'CONGRESS_DISTRICT_ORG': 'IN04', 'ORG_UEI_NUM': 'C1G1LGYJF5G5', 'ORG_LGL_BUS_NAME': 'PURDUE RESEARCH FOUNDATION', 'ORG_PRNT_UEI_NUM': 'YRXVL4JYCEF5'}
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{'Code': '1261', 'Text': 'CLASSICAL ANALYSIS'}
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