| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
|
| Initial Amendment Date: | August 4, 2016 |
| Latest Amendment Date: | August 4, 2016 |
| Award Number: | 1613471 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Victor Roytburd
DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2016 |
| End Date: | July 31, 2020 (Estimated) |
| Total Intended Award Amount: | $191,109.00 |
| Total Awarded Amount to Date: | $191,109.00 |
| Funds Obligated to Date: |
|
| History of Investigator: |
|
| Recipient Sponsored Research Office: |
202 HIMES HALL BATON ROUGE LA US 70803-0001 (225)578-2760 |
| Sponsor Congressional District: |
|
| Primary Place of Performance: |
LA US 70803-4918 |
| Primary Place of
Performance Congressional District: |
|
| Unique Entity Identifier (UEI): |
|
| Parent UEI: |
|
| NSF Program(s): | APPLIED MATHEMATICS |
| Primary Program Source: |
|
| Program Reference Code(s): |
|
| Program Element Code(s): |
|
| Award Agency Code: | 4900 |
| Fund Agency Code: | 4900 |
| Assistance Listing Number(s): | 47.049 |
ABSTRACT
This grant supports the research program of the Principal Investigator on mathematical models of superconductivity and the behavior of the solutions of these models. Superconductors are metals that at a sufficiently low temperature exhibit two important properties: they lose entirely their electrical resistivity, and the magnetic field is excluded from the superconducting area. Superconductors have great technological potential for applications ranging from magnetic sensors, through generators of large magnetic fields, to high power transmitters. Even for low temperatures, a sufficiently strong electric current would revert the superconductor to the state of a normal metal. The Principal Investigator will study the transition of superconducting materials from the normal state to the superconducting one and vice versa, with either increasing or decreasing current. Of particular interest is the disparity between experimental measurements of the critical current and theoretical predictions of the critical current in the absence of magnetic fields, and the maximal amount of current that can flow through a sample before the superconducting state loses its stability. The project will shed light on both the nucleation of superconductivity for decreasing currents and on the loss of superconducting properties when the electric current increases. The results are expected to have an effect on other areas of Applied Mathematics, such as hydrodynamic stability, magnetic resonance imaging, and more.
The research will address the Ginzburg-Landau model of superconductivity in the presence of electric currents. The Principal Investigator will study several fundamental theoretical problems related to this model. Most of the project will involve analytical work to be performed by the investigator and his collaborators. The following problems will be studied: (1) the existence and stability of fully superconducting solutions away from the boundary, which is adjacent in part to a normal material from which the current enters and exits the superconducting sample, and solutions will be studied both in the presence and in the absence of a magnetic field, (2) analysis of the behavior of solutions with decreasing current density below the critical current where the normal state loses its stability, (3) generalization of some recent results on the local and global stability of the normal state in the presence of strong electric currents, both in the presence and the absence of magnetic fields.
PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH
Note:
When clicking on a Digital Object Identifier (DOI) number, you will be taken to an external
site maintained by the publisher. Some full text articles may not yet be available without a
charge during the embargo (administrative interval).
Some links on this page may take you to non-federal websites. Their policies may differ from
this site.
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.
The project involved analysis of some Schrodinger operators and their application to various problems in Mathematical Physics. In particular, the results are pertinent to superconductivity (Ginzburg-Landau theory), Diffusion MRI (the Bloch-Torrey equations), and Hydrodynamic stability.
Superconductivity is characterized by the loss of electric resistivity below a certain critical temperature. Strong current will however destroy superconductivity and revert the sample to the normal state. If the current is decreased again the normal state would become unstable. We have obtained the critical current where the normal state loses its stability in a variety of settings. Additionally, the existence of a fully superconducting solution (such that current flows with very little potential drop) has been proved.
Diffusion-weighted magnetic resonance imaging is the use of MRI sequences that generates images from the resulting data using the diffusion of water molecules to achieve better contrast . A common model for it is the Bloch-Torrey equations. We have considered a three-dimensional applied magnetic field in this project. Previously, analysis focused on unidirectional magnetic fields (that depend on two spatial coordinates). We have obtained in this case some estimates on magnetization of the sample (or more precisely of the spectrum of the Bloch-Torrey operator)
Finally, we have managed to prove stability of laminar flows with monotone increasing velocity profile in a two-dimensional channel. Some progress was made on problems involving symmetric flows in a channel.
Last Modified: 10/15/2020
Modified by: Yaniv Almog
Please report errors in award information by writing to: awardsearch@nsf.gov.
