Zoran Grujic
Associate Professor of Mathematics
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Office: Kerchof Hall 329 Phone: 434.243.7702 Fax: 434.982.3084
Email: zg7c@virginia.edu
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TEACHING
Fall 2011 – On Leave
Spring 2012
MATH 2310-001
MATH 8250-001
Office hours: W 1300-1500 and by appointment.
RESEARCH
Analyzing singularity formation (or the lack thereof) in nonlinear partial differential equations arising as models in fluid and plasma dynamics. Mathematical theory of turbulent cascades in physical scales of incompressible fluid and plasma flows.
Current research includes:
Local geometric measure-type
conditions on the regions of intense fluid activity preventing the formation of
singularities in solutions to the 3D Navier-Stokes equations.
• A geometric measure-type regularity criterion for solutions to the 3D Navier-Stokes equations arXiv (submitted)
Elucidating the role of coherent
vortex structures in the study of enstrophy cascade in physical scales of
3D viscous incompressible flows.
• Coherent vortex structures and 3D enstrophy cascade (with R. Dascaliuc) arXiv (CMP, to appear)
vs.
RECENT/UPCOMING MAJOR/PLENARY LECTURES
Conference celebrating the 60th birthday of Professor Peter Constantin (Xi'an, China)
Onsager Seminar (Kyoto, Japan)
CURRENT RESEARCH SUPPORT
The Research Council of Norway, Prosjekt 213474 – FRINATEK (Independent basic research projects in Mathematics, Physical Science and Technology), Nonlinear PDE in spaces of analytic functions, 1/1/2012-12/31/2015, NOK 7, 777, 000, PI; collaborative research with H. Kalisch, University of Bergen, PI (project manager) and S. Selberg, Norwegian University of Science and Technology, PI.
PUBLICATIONS
1. Space analyticity for the Navier-Stokes and related equations with initial data in L^p (with I. Kukavica), J. Funct. Anal. 152 (1998), 447-466 pdf.
2. The role of spatial analyticity in the local alignment of vorticity directions in 3D viscous fluids Nonlinearity, 12 (1999), 1239-124 pdf.
3. Space analyticity for the nonlinear heat equation in a bounded domain (with I. Kukavica), J. Differential Equations 152 (1999), 42-54 pdf.
4. On the smallness of the (possible) singular set in space for 3D Navier-Stokes equations, Electron. J. Differential Equations 1999 (1999), 1-8 pdf.
5. Spatial analyticity on the global attractor for the Kuramoto-Sivashinsky equation, J. Dynam. Differential Equations 12 (2000), 217-227 pdf.
6. Dynamics of complex singularities in 1D nonlinear parabolic PDE's , Studia Math. 147 (2001), 183-195 pdf.
7. The geometric structure of the super-level sets and regularity for 3D Navier-Stokes equations, Indiana Univ. Math. J. 50 (2001), 1309-1317 pdf.
8. Local well-posedness of the generalized KdV equation in spaces of analytic functions (with H. Kalisch), Differential Integral Equations 15 (2002), 1325-1334 pdf.
9. A remark on time-analyticity for the Kuramoto-Sivashinsky equation (with I. Kukavica), Nonlinear Anal. 52 (1) (2003), 69-78 pdf.
10. Spatial analyticity properties of nonlinear waves (with J.L. Bona), Math. Models & Methods in Appl. Sci. 13 (3) (2003), 345-360 pdf.
11. The derivative nonlinear Schrodinger equation in analytic classes (with H. Kalisch), J. Nonlinear Math. Physics 10 (2003), 1-10 pdf.
12. Interpolation between algebraic and geometric conditions for smoothness of the vorticity in the 3D NSE (with A. Ruzmaikina) , Indiana Univ. Math. J. 53 (2004), 1073-1080 pdf.
13. On depletion of the vortex-stretching term in the 3D Navier-Stokes equations (with A. Ruzmaikina), Comm. Math. Phys. 247 (2004), 601-611 pdf.
14. Algebraic lower bounds on the uniform radius of spatial analyticity for the generalized KdV equation (with J.L. Bona and H. Kalisch), Ann. Inst. Henri Poincare, Anal. Non Lineaire 22 (2005), 783-797 pdf.
15. Regularity of forward-in-time self-similar solutions to the 3D NSE, Discrete Contin. Dyn. Syst. 14 (2006), 837-843 pdf.
16. Global solutions of the derivative Schrodinger equation in a class of functions analytic in a strip (with J.L. Bona and H. Kalisch), J. Differential Equations 229 (2006), 186-203 pdf.
17. Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE (with Qi Zhang), Comm. Math. Phys. 262 (2006), 555-564 pdf.
18. Boundary control model of a nonlinear system of fluid-structure interactions (with V. Barbu, I. Lasiecka and A. Tuffaha), Proceedings of the 12th IEEE International Conference on Methods and Models in Automation and Robotics, 29th-31st August 2006, Miedzyzdroje, Poland pdf.
19. Existence of the energy-level weak solutions to a nonlinear fluid-structure interaction model (with V. Barbu, I. Lasiecka and A. Tuffaha), Contemporary Mathematics 440 (2007), 55-82 pdf.
20. Smoothness of solutions to a nonlinear fluid-structure interaction model (with V. Barbu, I. Lasiecka and A. Tuffaha), Indiana Univ. Math. J. 57 (2008), 1173-1207 pdf.
21. Gevrey regularity for a class of water wave models (with H. Kalisch), Nonlinear Anal. 71 (2009), 1160-1170 pdf.
22. A bound on oscillations in an unsteady undular bore (with H. Kalisch), Appl. Anal. 88 (2009), 1701-1712 pdf.
23. A Kdv-type Boussinesq system in a scale of Bourgain-type spaces: from the energy level to analytic spaces (with J.L. Bona and H. Kalisch), Discrete Contin. Dyn. Syst. 26 (2010), 1121-1139 pdf.
24. Localization and geometric depletion of vortex-stretching in the 3D NSE, Comm. Math. Phys. 290 (2009), 861-870 pdf.
25. Fluid-Structure
Interaction Model: Wellposedness, Regularity and Control (with V. Barbu,
26. A family of regularity classes for the 3D NSE approximating a critical class, Indiana Univ. Math. J. 59 (2010), 707-719 pdf.
27. Localization of analytic regularity criteria on the vorticity and balance between the vorticity magnitude and coherence of the vorticity direction in the 3D NSE (with R. Guberovic), Comm. Math. Phys. 298 (2010), 407-418 pdf.
28. A regularity criterion for the 3D NSE in a local version of the space of functions of bounded mean oscillations (with R. Guberovic), Ann. Inst. Henri Poincare, Anal. Non Lineaire 27 (2010), 773-778 pdf.
29. Energy cascades and flux locality in physical scales of the 3D NSE (with R. Dascaliuc), Comm. Math. Phys. 305 (2011), 199-220 pdf.
30. Anomalous dissipation and energy cascade in 3D inviscid flows (with R. Dascaliuc), Comm. Math. Phys. 309 (2012), 757-770 pdf.
31. Dissipation anomaly and energy cascade in 3D incompressible flows (with R. Dascaliuc), C. R. Math. Acad. Sci. Paris 350 (2012), 199-202 pdf.
Searching for clues behind a coral reef..
