Tuesday, April 5, 2016

the most critical point of Jiahong Wu, along with Peter Constantin, Edriss Titi, Jerry Bona, Deborah Lockhart, michael Steuerwalt, Donghou Chae, Netra Khanal, Sharma Rajee,

ward Abstract #0907913
Two Partial Differential Equations Modeling Geophysical Fluids

NSF Org: DMS
Division Of Mathematical Sciences
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Initial Amendment Date: June 15, 2009
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Latest Amendment Date: June 15, 2009
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Award Number: 0907913
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Award Instrument: Standard Grant
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Program Manager: Michael H. Steuerwalt
DMS Division Of Mathematical Sciences
MPS Direct For Mathematical & Physical Scien
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Start Date: June 15, 2009
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End Date: May 31, 2013 (Estimated)
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Awarded Amount to Date: $175,941.00
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ARRA Amount: $175,941.00
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Investigator(s): Jiahong Wu jahong@math.okstate.edu (Principal Investigator)
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Sponsor: Oklahoma State University
101 WHITEHURST HALL
Stillwater, OK 74078-1011 (405)744-9995
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NSF Program(s): APPLIED MATHEMATICS
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Program Reference Code(s): 0000, 6890, 9150, OTHR
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Program Element Code(s): 1266
ABSTRACT

Wu

DMS-0907913

This award is funded under the American Recovery and

Reinvestment Act of 2009 (Public Law 111-5). The project focuses

on two well-known partial differential equations modeling

geophysical fluids: the surface quasi-geostrophic (SQG) equation

and the two-dimensional Boussinesq equations. The major

objective is to develop strategies and effective approaches for

solving the global regularity problem on the classical solutions

of these equations. The global regularity issue concerning these

equations has recently attracted substantial attention and much

important progress has been made. However, it remains open in

the cases of the inviscid SQG equation, the SQG equation with

supercritical dissipation, and the inviscid Boussinesq equations.

To deal with the inviscid or supercritical SQG equation, the

investigator combines extensive numerical computations with

analytic and geometric approaches. The immediate plan is to

study the curvature of the level curves in the spatial regions

where the gradients are comparable to the maximal gradient. The

boundedness of the curvature in these regions would rule out any

finite-time singularities. The strategy on the global regularity

issue for the two-dimensional Boussinesq equations is to

gradually reduce the dissipation and thermal diffusion. The

first aim is at the case when there is only vertical dissipation

or thermal diffusion. In contrast to the recently resolved case

with horizontal dissipation or thermal diffusion, the situation

now is more sophisticated due to the "mismatch" of derivatives.

To handle this case, new tools such as logarithmic type

inequalities involving Sobolev norms of derivatives in different

directions are developed.

The three-dimensional quasi-geostrophic equations derived by

J. G. Charney in the 1940s have been very successful in modeling

large-scale motions of atmosphere and oceans. The dynamics of

these three-dimensional equations with uniform potential

vorticity reduces to the SQG equation. The SQG equation has been

very useful in studying many weather phenomena such as

frontogenesis, the formation of sharp fronts between hot and cold

air. Mathematically, frontogenesis corresponds to the

fundamental issue of whether classical solutions of this equation

can develop finite-time singularities. This project helps

improve the understanding of many weather phenomena governed by

this equation. Boussinesq equations model many flows in nature

such as oceanic circulation, central heating and natural

ventilation. The study here of the potentially singular behavior

of solutions to the Boussinesq equations not only yields a

significant contribution to the mathematical issue of global

regularity but also has potential environmental applications. As

part of this project, several Ph.D. students of the investigator

are actively involved in the proposed research and develop

analytic and computational skills that enable them to become

capable scholars and highly skilled workforce.

PUBLICATIONS PRODUCED AS A RESULT OF THIS RESEARCH

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Xiao, YL; Xin, ZP; Wu, JH. "Vanishing viscosity limit for the 3D magnetohydrodynamic system with a slip boundary condition," JOURNAL OF FUNCTIONAL ANALYSIS, v.257, 2009, p. 3375. View record at Web of Science  doi:10.1016/j.jfa.2009.09.01

Yuan, JM; Wu, JH. "A dual Petrov-Galerkin method for integrable fifth-order Korteweg-de Vries type equations," Discrete and Continuous Dynamical Systems-Series A, v.26, 2010, p. 1525.

Cao, C; Wu, J. "Two regularity criteria for the 3D MHD equations," Journal of Differential Equations, v.248, 2010, p. 2263.

Khanal, N; Wu, J; Yuan, JM; Zhang, B. "Complex-valued Burgers and KdV-Burgers equations," J. Nonlinear Science, v.20, 2010, p. 341.

Cao, C; Wu, J. "Global regularity for the 2D magnetohydrodynamic equations with partial dissipation and magnetic diffusion," Adv. in Mathematics, v.226, 2011, p. 1803.

Devuyst, E; Garosi, J; Wu, J. "Firm behavior under illiquidity risk," Applied Mathematics Letters, v.24, 2011, p. 709.

Wu, J. "Global regularity for a class of generalized magnetohydrodynamic equations," J. Mathematical Fluid Mechanics, v.13, 2011, p. 295.

Adhikari, D; Cao, C; Wu, J. "The 2D Boussinesq equations with vertical viscosity and vertical diffusivity," Journal of Differential Equations, v.249, 2010, p. 1078.

Constantin, P; Lai, M.; Sharma, R; Tseng, Y; Wu, J. "New numerical results for the surface quasi-geostrophic equation," J. Scientific Computing, v.50, 2012, p. 1.

Chae, D; Constantin, P; Wu, J. "Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations," Archive for Rational Mechanics and Analysis, v.202, 2011, p. 35.


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BOOKS/ONE TIME PROCEEDING

Netra Khanal. "A study on the solutions of the Kawahara, Complex-valued Burgers and Kdv-Burgers equations", 06/15/2009-05/31/2010,  2009, "Oklahoma State University library".

Netra Khanal. "A study on the solutions of the Kawahara, Complex-valued
Burgers and Kdv-Burgers equations", 06/01/2010-05/31/2011,  2009, "Oklahoma State University library".

Sharma, R. "Global regularity or finite time
singularity of the surface
quasi-geostrophic equations", 06/01/2010-05/31/2011,  2010, "Oklahoma State University Library".

Netra Khanal. "A study on the solutions of the Kawahara, Complex-valued
Burgers and Kdv-Burgers equations", 06/01/2011-05/31/2012,  2009, "Oklahoma State University library".

Sharma, R. "Global regularity or finite time
singularity of the surface
quasi-geostrophic equations", 06/01/2011-05/31/2012,  2010, "Oklahoma State University Library".


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