| NSF Org: |
DMS Division Of Mathematical Sciences |
| Recipient: |
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| Initial Amendment Date: | August 12, 2013 |
| Latest Amendment Date: | August 12, 2013 |
| Award Number: | 1309084 |
| Award Instrument: | Standard Grant |
| Program Manager: |
Tomek Bartoszynski
tbartosz@nsf.gov (703)292-4885 DMS Division Of Mathematical Sciences MPS Directorate for Mathematical and Physical Sciences |
| Start Date: | August 15, 2013 |
| End Date: | July 31, 2017 (Estimated) |
| Total Intended Award Amount: | $150,675.00 |
| Total Awarded Amount to Date: | $150,675.00 |
| Funds Obligated to Date: |
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| History of Investigator: |
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| Recipient Sponsored Research Office: |
3112 LEE BUILDING COLLEGE PARK MD US 20742-5100 (301)405-6269 |
| Sponsor Congressional District: |
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| Primary Place of Performance: |
Mathematics Building COLLEGE PARK MD US 20742-5121 |
| 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): | PROBABILITY |
| Primary Program Source: |
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| 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
The main goal of this project is to investigate a large class of problems in which asymptotic analysis allows for a detailed study of multi-scale phenomena. Among the problems to be studied are the asymptotic behavior of branching diffusion processes, transition between the averaging and homogenization regimes for transport by cellular flows, and asymptotics of solutions to nonlinear parabolic PDEs. Several problems concerning random perturbations of incompressible flows will also be studied. These include the transport properties of the Benard convection, large deviations for randomly perturbed incompressible flows, and mathematical analysis of Brownian motors.
The proposed research will provide a rigorous mathematical foundation for several phenomena that have been actively discussed in natural sciences. In particular, we will consider branching diffusion processes, which are central in the study of evolution of various populations such as bacteria, cancer cells, carriers of a particular gene, etc., where each member of a population may die or produce offspring independently of the rest. We plan to describe the long-time behavior of the population in different regions of space when, in addition to branching, the members of the population move in space and the branching mechanism depends on the location. Other models to be considered describe the movement of particles (e.g., molecules) due to a combination of a macroscopic motion and a small random diffusion. In particular, we'll examine mechanisms that create directed motion out of fluctuations of a random or periodic velocity field even in situations when the field itself has no preferred direction. Such mechanisms are very important in many applications and have been discussed in hundreds of physics and chemistry papers, although mostly at computational and experimental levels.
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.
Intellectual merit:
The main goal of the project was to investigate a large class of problems in which asymptotic analysis allows for a detailed study of multi-scale phenomena.
1) Branching diffusion processes are central in the study of evolution of various populations such as bacteria, cancer cells, carriers of a particular gene, etc., where each member of a population may die or produce offspring independently of the rest. We described the long-time behavior of the population in different regions of space when, in addition to branching, the members of the population move in space and the branching mechanism may depend on the location. In particular, the phenomenon of intermittency (strongly non-uniform distribution of population clusters) was explored for branching in homogeneous media.
2) We obtained rigorous mathematical results that explain the mechanism for creating directed motion out of fluctuations of a random or periodic velocity field even in situations when the field itself has no preferred direction. Such mechanisms (Brownian motors) are very important in many applications and had earlier been discussed in numerous physics and chemistry papers, although mostly at computational and experimental levels.
3) We described the transition between the averaging and homogenization regimes for transport by cellular flows.This work establishes interesting connections between different problems of asymptotic analysis.
4) We described the long time behavior of randomly perturbed flows with trapping regions. The limiting processis described in terms of a new class of elliptic problems with non-standard boundary conditions.
Eleven research papers were published (or accepted for publication) during the period of the award.
Broader impact:
The results that were obtained during the period of the award provide a rigorous mathematical foundation for several phenomena that are of importance in natural sciences (population dynamics, propagation of chemical reactions, intracellular transport, etc.).
Three graduate students were actively involved in the work on this project. Two of those students recently defended their PhD dissertations under the PI's supervision, and one is a current student who already produced excellent results.
Last Modified: 09/28/2017
Modified by: Leonid B Koralov
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