Zoran Grujic

Professor of Mathematics

Department of Mathematics

Kerchof Hall

University of Virginia

Charlottesville, VA 22904

Reflections on mathematical fluid dynamics, March 15-17, 2019, UVA

**RESEARCH INTERESTS**

Analyzing possible singularity formation in nonlinear partial differential equations arising as models in 3D incompressible fluid and plasma dynamics. Mathematical theory of turbulent cascades and turbulent dissipation in 3D flows.

Of particular interest
has been the study of *formation*
*of*
*small scales*
and *turbulent
dissipation* in
3D Navier-Stokes equations within a mathematical framework based on
the suitably defined scale of `sparseness' of the near-max
super-level sets of the positive and negative parts of the vorticity
components [in the context of a blow-up-type argument].

Black Rain – an example of art-making using raw turbulence (solar wind) data.

A primer on the Navier-Stokes equations.

PUBLICATIONS

*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.

*The role of spatial
analyticity in the local alignment of vorticity directions in 3D
viscous fluids* Nonlinearity,
**12**
(1999), 1239-124 pdf.

*Space analyticity for
the nonlinear heat equation in a bounded domain*
(with I. Kukavica),
J. Differential Equations **152**
(1999), 42-54 pdf.

*On*
*the smallness of
the (possible) singular set in space for 3D Navier-Stokes equations*,
Electron. J. Differential Equations **1999**
(1999), 1-8 pdf.

*Spatial analyticity on
the global attractor for the Kuramoto-Sivashinsky equation,*
J. Dynam.
Differential Equations **12**
(2000), 217-227 pdf.

*Dynamics of complex
singularities in 1D nonlinear parabolic PDE's*
,
Studia
Math. **147**
(2001), 183-195 pdf.

*The geometric
structure of the super-level sets and regularity for 3D Navier-Stokes
equations*,
Indiana Univ. Math. J. **50
** (2001),
1309-1317 pdf.

*Local well-posedness
of the generalized KdV equation in spaces of analytic functions*
(with H. Kalisch),
Differential Integral Equations **15**
(2002), 1325-1334
pdf.

*A remark on
time-analyticity for the Kuramoto-Sivashinsky equation *(with
I. Kukavica), Nonlinear Anal. **52
**(1)
(2003), 69-78 pdf.

*Spatial analyticity
properties of nonlinear waves*
(with J.L. Bona),
Math. Models & Methods in Appl. Sci. **13**
(3) (2003), 345-360
pdf.

*The derivative
nonlinear Schrodinger equation in analytic classes *(with
H. Kalisch), J. Nonlinear Math. Physics **10**
(2003), 1-10 pdf.

*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.

*On depletion of the
vortex-stretching term in the 3D Navier-Stokes equations *(with
A. Ruzmaikina)*,
*Comm.
Math. Phys. **247
**(2004),
601-611 pdf.

*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.

*Regularity of
forward-in-time self-similar solutions to the 3D NSE, *Discrete
Contin. Dyn. Syst. **14
**(2006),
837-843 pdf.

*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.

*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.

*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.

*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.

*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.

*Gevrey*
*regularity for a
class of water wave models *(with
H. Kalisch), Nonlinear Anal. **71
**(2009),
1160-1170 pdf.

*A bound on
oscillations in an unsteady undular bore *(with
H. Kalisch), Appl. Anal. **88
**(2009),
1701-1712 pdf.

*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.

*Localization and
geometric depletion of vortex-stretching in the 3D NSE, *Comm.
Math. Phys. **290
**(2009),
861-870 pdf.

*Fluid-Structure
Interaction Model: Wellposedness, Regularity and Control *(with
V. Barbu, I. Lasiecka and A. Tuffaha)*,
*In:
Advances in Dynamics and Control: Theory, Methods and Applications.
Chapter 2, 21-32. Cambridge Scientific Publishers, 2010 pdf.

*A family of regularity
classes for the 3D NSE approximating a critical class, *Indiana
Univ.
Math. J. **59**
(2010), 707-719 pdf.

*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.

*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.

*Energy cascades and
flux locality in physical scales of the 3D NSE*
(with R. Dascaliuc),
Comm. Math. Phys. **305**
(2011), 199-220 pdf.

*Anomalous dissipation
and energy cascade in 3D inviscid flows*
(with R. Dascaliuc),
Comm. Math. Phys. **309
**(2012),
757-770 pdf.

*Dissipation anomaly
and energy cascade in 3D incompressible flows*
(with R. Dascaliuc),
C. R. Math. Acad. Sci. Paris **350
**(2012),
199-202 pdf.

*Vortex stretching and
criticality for the 3D NSE *(with
R. Dascaliuc), J. Math. Phys. **53,
**115613
(2012) pdf.

*Coherent vortex
structures and 3D enstrophy cascade *(with
R. Dascaliuc), Comm. Math. Phys. **317**
(2013), 547-561 pdf.

*A geometric
measure-type regularity criterion for solutions to the 3D
Navier-Stokes equations, *Nonlinearity
**26**
(2013), 289-296
pdf.

*On the transport and concentration of
enstrophy in 3D magnetohydrodynamic turbulence *(with
Z. Bradshaw), Nonlinearity **26**
(2013), 2373-2390
pdf.

*Energy
cascades in physical scales of 3D incompressible magnetohydrodynamic
turbulence *(with
Z. Bradshaw), J. Math. Phys. **54**,
093503 (2013) pdf.

*Blow-up scenarios for
3D NSE exhibiting sub-criticality with respect to the scaling of
one-dimensional local sparseness *(with
Z. Bradshaw), J. Math. Fluid Mech. **16
**(2014),
321-334 pdf.

*A spatially localized
L log L estimate on the vorticity in the 3D NSE *(with
Z. Bradshaw), Indiana Univ. Math. J. **64
**(2015),
433-440 pdf.

*A note on the surface
quasi-geostrophic temperature variance cascade *(with
Z. Bradshaw), Comm. Math. Sci. **13
**(2015),
557-564 pdf.

*Vortex stretching and
anisotropic diffusion in 3D Navier-Stokes equations*,
Contemporary Mathematics **666
**(a
special issue dedicated to Hugo Beirao da Veiga's 70^{th}
birthday)
pdf.

*Local analyticity
radii of solutions to the 3D Navier-Stokes equations with locally
analytic forcing *(with
Z. Bradshaw and I. Kukavica), J. Differential Equations **259**
(2015),
3955-3975 pdf.

*Effect of vorticity coherence on
energy-enstrophy bounds for the 3D Navier-Stokes Equations *(with
R. Dascaliuc and M.S. Jolly), J. Math. Fluid Mech. **17**
(2015),
393-410 pdf.

*On energy cascades in the forced
3D Navier-Stokes equations *(with
R.Dascaliuc), J. Nonlinear Sci. **26**
(2016),
683-715 pdf.

*Vorticity direction and
regularity of solutions to the Navier-Stokes equations *(with
H. Beirao Da Veiga and Y. Giga), a chapter in Handbook
of Mathematical Analysis in Mechanics of Viscous
Fluids. Springer, 2016.

*The
space B^{-1}_{\infty, \infty}, volumetric sparseness, and 3D NSE
*(with
A. Farhat and K. Leitmeyer), J. Math. Fluid Mech. **19
**(2017),
515-523 arXiv.

*Frequency
localized regularity criteria for the 3D Navier-Stokes equations
*(with
Z. Bradshaw), Arch. Rational Mech. Anal. **224**
(2017),
125-133 pdf.

*An algebraic
reduction of the `scaling gap' in the Navier-Stokes regularity
problem *(with
Z. Bradshaw and A. Farhat), Arch. Rational Mech. Anal. **231**
(2019),
1983-2005 arXiv.

*Local near-Beltrami
structure and depletion of the nonlinearity in the 3D Navier-Stokes
equations *(with
A. Farhat), J. Nonlinear Sci. (to appear) arXiv.

*Oscillations and
integrability of the vorticity in the 3D NS flows*
(with
Y. Do, A. Farhat and L. Xu), Indiana Univ. Math. J. (to appear)
arXiv.