Zoran Grujic


Professor of Mathematics 


Department of Mathematics

Kerchof Hall

University of Virginia

Charlottesville, VA 22904



Math fluids workshop at UVA May 12-13.



Analyzing singularity formation [or the lack thereof] 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].

a recent paper on arXiv

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

A primer on the Navier-Stokes equations.


Nonlinear Analysis: Real World Applications






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 70th 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), to appear in J. Math. Fluid Mech. arXiv.

Frequency localized regularity criteria for the 3D Navier-Stokes equations (with Z. Bradshaw), to appear in Arch. Rational Mech. Anal. pdf.

An algebraic reduction of the `scaling gap' in the Navier-Stokes regularity problem (with Z. Bradshaw and A. Farhat) arXiv.