Mathematical Sciences: Harmonic Analysis and Operator-Theory
{'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 provides continued support of mathematical                    research into problems in partial differential equations and                    potential theory.  Particular emphasis will be placed on                        equations with coefficients which lack smoothness.                              Specifically, to obtain mean-square estimates of operators                      arising in the solution of equations with such coefficients.                    The form of the analysis is one of studying singular integral                   operators defined along curves.  The kernel is of Cauchy-type                   and the boundedness of integral transforms reflects the                         smoothness of the solution of the related differential                          equation.      A second line of investigation will focus on the                 non-linear phenomena of singularity formation.  For example,                    three-dimensional incompressible fluid flow with high Reynolds                  number exhibits singularities in finite time even with smooth                   initial data which vanishes outside bounded sets.                                    This project is specifically concerned with what is called                 Kato's conjecture.  For certain differential operators a square                 root can be defined.  The domain of this square root is thought                 to be one of the basic classes of functions used in the study                   of differential equations, Sobolev space.  This has been                        verified for one-dimensional operators by making the connection                 with boundedness of Cauchy-kernel transforms on certain curves.                  What remains to be done is the classification of the square                    roots' domain in higher dimensions.                                                  Work on singularity formation combines numerical                           experimentation with general theory.  It is known that                          singularities in the flow can occur only through a stretching                   of the vorticity, i.e. the curl of the velocity field becomes                   unbounded.  This makes numerical approximations of solutions                    which form singularities a subtle issue.  Emphasis in this                      research will be placed on a simple prototype model in which                    the transport term will be studied closely.  The object is to                   see whether the transport terms delay or accelerate breakdown                   of the solution.
{'FirstName': 'Jean-Lin', 'LastName': 'Journe', 'PI_MID_INIT': None, 'PI_SUFX_NAME': None, 'PI_FULL_NAME': 'Jean-Lin Journe', 'EmailAddress': None, 'NSF_ID': '000496274', 'StartDate': '03/15/1989', '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': '1261', 'Text': 'CLASSICAL ANALYSIS'}
{'url': 'https://www.nsf.gov/awardsearch/download?DownloadFileName=1988&All=true', 'xml': '8815651.xml'}