| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | May 18, 2015 |
| Latest Amendment Date: | June 9, 2017 |
| Award Number: | 1500424 |
| Award Instrument: | Continuing Grant |
| Program Manager: |
Marian Bocea
mbocea@nsf.gov (703)292-2595 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | July 1, 2015 |
| End Date: | June 30, 2019 (Estimated) |
| Total Intended Award Amount: | $169,691.00 |
| Total Awarded Amount to Date: | $169,691.00 |
| Funds Obligated to Date: |
FY 2017 = $57,463.00 |
| History of Investigator: |
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| Recipient Sponsored Research Office: |
3400 N CHARLES ST BALTIMORE MD US 21218-2608 (443)997-1898 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
3400 N. Charles Street Baltimore MD US 21218-2608 |
| Primary Place of
Performance Congressional District: |
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| Unique Entity Identifier (UEI): |
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| Parent UEI: |
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| NSF Program(s): | ANALYSIS PROGRAM |
| Primary Program Source: |
01001718DB NSF RESEARCH & RELATED ACTIVIT |
| Program Reference Code(s): | |
| Program Element Code(s): |
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| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This project will study dispersive partial differential equations. Such equations model many different phenomena, among them the propagation of various kinds of waves, such as water waves and laser light. These equations also model certain phenomena in particle physics. This project attempts to understand long-time behavior of such equations and related systems.
There are a number of problems that will be studied in the course of this endeavor. The study will mainly revolve around the three well-known dispersive partial differential equations: wave, Schrodinger, and Korteweg de Vries. A great deal is unknown for the focusing Schrodinger and Korteweg de Vries (KdV) problems, particularly for the mass-critical problem. The PI intends to study the focusing, mass-critical Schrodinger problem for mass above the mass of the ground state, as well as the focusing gKdV problem for large mass below the mass of the ground state. The PI also plans to extend recent work with the I-method to the wave and Klein-Gordon equations. Finally, the PI will study the ultra hyperbolic Schrodinger equation.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
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PROJECT OUTCOMES REPORT
Disclaimer
This Project Outcomes Report for the General Public is displayed verbatim as submitted by the Principal Investigator (PI) for this award. Any opinions, findings, and conclusions or recommendations expressed in this Report are those of the PI and do not necessarily reflect the views of the National Science Foundation; NSF has not approved or endorsed its content.
In this project, we study several equations which fall into the broad category of dispersive partial differential equations. Dispersive partial differential equations arise frequently in the natural sciences, modeling the behavior of waves, many particle systems, and other phenomena.
In this project we specifically studied the long time behavior of a number of different equations. One such equation was the nonlinear wave equation. The nonlinear wave equation possesses a scaling symmetry. There exists a unique Hilbert Sobolev space, called the critical space, for which the norm of the initial data is invariant under the scaling symmetry.
This symmetry completely determines the local theory of such an equation. That is, if the initial data lies in the critical space, then the local theory is well established for a broad class of wave equations. However, the local theory is known to fail for less regular data.
We prove that for a class of wave equations with radially symmetric initial data lying in the critical Sobolev space, the local theory may be extended to all time. Moreover, the solution scatters to a free solution, that is, exhibits the behavior of a solution to a linear wave equation as time tends to positive or negative infinity.
We prove that the same behavior also occurs for many other nonlinear dispersive equations. We do the same for the nonlinear Schrodinger equation, which models many particle dynamics in quantum physics.
It is worthwhile to study such equations for the insight that the equations give into the physical sciences. Such equations are also important because the technqiues developed in their study can be used to study other equations arising in the physical world. Simply put, many equations have an important scaling symmetry which determines the local behavior of the solution.
The pursuit of this project brought together a diverse group of researchers. Researchers from the level of graduate student, postdoc, and professor all participated in this project. The project also brought together contributions from many researchers both in the United States and abroad. This project also involved contributions of many underrepresented groups in mathematics.
Last Modified: 08/27/2019
Modified by: Benjamin Dodson
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