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Research Spending & Results

Award Detail

Doing Business As Name:Louisiana State University
  • Andrei Tarfulea
  • (219) 902-9175
Award Date:01/06/2020
Estimated Total Award Amount: $ 73,026
Funds Obligated to Date: $ 73,026
  • FY 2018=$73,026
Start Date:09/01/2019
End Date:06/30/2021
Transaction Type:Grant
Awarding Agency Code:4900
Funding Agency Code:4900
CFDA Number:47.049
Primary Program Source:040100 NSF RESEARCH & RELATED ACTIVIT
Award Title or Description:Bounds and Asymptotic Dynamics for Nonlinear Evolution Equations
Federal Award ID Number:2012333
DUNS ID:075050765
Parent DUNS ID:940050792
Program Officer:
  • Victor Roytburd
  • (703) 292-8584

Awardee Location

Street:202 Himes Hall
City:Baton Rouge
County:Baton Rouge
Awardee Cong. District:06

Primary Place of Performance

Organization Name:Louisiana State University & Agricultural and Mechanical College
Street:303 Lockett Hall
City:Baton Rouge
County:Baton Rouge
Cong. District:06

Abstract at Time of Award

Many physical, engineering, and biological phenomena are described by mathematical models with a large number of strongly-interacting particles. The range of these phenomena includes such diverse examples as complex and compressible fluids (combustion, aerospace engineering, and meteorology), nonlocal reaction-diffusion processes (nuclear physics, population biology, and genetics), and kinetic theory (plasma physics, swarm dynamics, and astrophysics). This project focuses on novel approaches to determining two fundamental characteristics of solutions to equations modeling large numbers of strongly interacting particles: their regularity and asymptotic behavior. The regularity of such problems establishes that the models are well-behaved, which often means the equations remain numerically tractable in computer simulations. The asymptotic theory seeks to find simplified limiting behavior for equations, in which many complex interactions average out and have a residual effect that governs the behavior of the system. Information about the limiting behavior is instrumental for applications such as medical imaging or materials science. For many important phenomena that demonstrate complex, nonlinear behavior, the application of known methods for analysis and control is greatly limited and not always possible. The aim of this project is to investigate three new techniques that partly overcome the difficulties caused by nonlinearity. The project will also provide training and research opportunities for both graduate and undergraduate students. The principal investigator will use techniques of nonlinear analysis, viscosity theory, and probability to establish bounds and asymptotic dynamics for the three major parts of the project. The first part focuses on exploring thermally enhanced dissipation for hydrodynamic equations where the viscosity grows with local temperature. From kinetic considerations and empirical observations, the kinematic viscosity of a compressible fluid flow increases with the local temperature and the local temperature is produced by friction. The intuition is that, in such models, regions of high turbulence self-regularize by producing hot spots which boost the viscosity exactly where it is needed to prevent the development of singularities. Prior work has identified this effect in two model problems (along with corresponding bounds). One of the main goals of the project is to push these types of estimates to physical models of compressible thermal fluids such as the Navier-Stokes-Fourier system, the equations of magneto-hydrodynamics, and the Poisson-Nernst-Planck-Fourier system for electrokinetic complex fluids. Enhanced thermal dissipation is a truly novel source of regularization compared to other known energy-based methods and lends itself naturally to dynamic weighted Sobolev estimates and entropy methods. The second part focuses on developing methods to extract asymptotic behavior from strongly nonlocal heterogeneous reaction-diffusion equations. There is a growing interest in extracting simpler macroscopic dynamics (often taking the form of geometric equations) from certain scaling limits of more complicated models. The nonlocal operators in these models present unique challenges in determining their residual impact on the (sometimes discontinuous) homogenized equation. The investigator plans to implement the techniques of viscosity theory to pursue homogenization phenomena for nonlocal periodic Fisher-KPP and bistable (Allen-Cahn) equations. The third part focuses on the regularity theory for kinetic equations (i.e., Landau and Boltzmann). Most regularity results for these equations rely on the assumption of having the lower bound on the density (as this often yields a minimum dissipation in the velocity variables). The investigator will explore the emergence of such lower bounds through probabilistic techniques, writing the kinetic equation as an approximate Fokker-Planck equation for a certain stochastic process. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

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