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Scholarly Interest Report
         
Alexander Kiselev
Professor
Professor of Mathematics
 
e-mail:Alexander.Kiselev@rice.edu
 
  • Ph.D. Mathematics (1996) Caltech, Pasadena, California
  • B.S. Physics (1992) St Petersburg State University, St Petersburg, Russia
 
Primary Department
   Department of Mathematics
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Websites
 Rice home page
 
Research Areas
 I currently have three main research areas. The first is on classical equations of fluid dynamics, namely Euler and Navier-Stokes equations for incompressible fluid. Whether these equations have global smooth solution in 3D is a famous Clay Institute million dollar prize problem. I work on related models trying to get better insight into 3D fluid motion. Progress in these equations is important in many applications and for better understanding ubiquitous complex phenomena like turbulence. Second direction is related, but the focus is on mixing in fluids. I study which flows are most efficient mixers, and how they help speed up or quench chemical reactions. The third direction is in mathematical biology. I study mathematical models of chemotaxis, transport controlled by chemical sensing. It is crucial for many processes in nature, from monocytes fighting infections to morphology of organisms to survival of many ecological systems.
 

Properties of solutions to classical equations of fluid mechancis

 

The classical equations of fluid mechanics are some of the most studied partial differential equations of applied analysis. Euler equation for incompressible inviscid fluid has been discovered in 1755. Navier-Stokes equation adds fluid viscosity to the model and has been derived in 1820-1850s. These equations are one of the central ones in modern applied analysis because they underlie an amazing range of applications, from weather and climate forecasting to simulations necessary for airplane and car design. These equations can be used to model and describe complex and ubiquitous phenomena such as turbulence, better understanding of which are crucial in many applications. Despite much research carried out over much of the last century, the fluid mechanics equations are so complex that many of its fundamental properties are still poorly understood. One famous open problem is whether three dimensional Navier-Stokes equation has a unique global smooth solution. This question is one of the seven Clay Institute Millenium million dollar prize problems. The problem places a focus on better understanding of spontaneous intense fluid motions, such as those present in tornado or in a hurricane. If the solution to the equation can have a singularity, this will have important consequences in terms of better understanding of the limitations of the model, necessary adjustments in the numerical algorithms, and better insight into the mechanisms for onset of turbulence. My work focuses on studying the properties of solutions of Euler and Navier-Stokes equations as well as related models of fluid mechanics. A particular focus is on scenarios and mechanisms for spontaneous generation of intense fluid motion, as well as descirption of constraints under which such motion cannot develop. The primary technical tools I use in this research are partial and ordinary differential equations techniques and methods of Fourier analysis.   


The highlight of my 2015 research in this direction involves two papers on the fluid models motivated by geophysics. These are so-called alpha patach equations, which model evolution of temperature on the surface of a planet. We showed finite time singularity formation for a class of these models very close to two-dimensional Euler equation. The singularity formation happens on the boundary. To the best of our knowledge, this is the first proof of singularity formation in this class of fluid mechanics equations.  


In 2016, my main focus was on understanding a new scenario of singlualrity formation in solutions of Euler equation, discovered by Hou and Luo based on extensive numerical computations. Several simplified models of this scenario have been analyzed with the eventual goal of understnading the actual scenario.  

 

Mixing in fluid flows

 

A crucial role of fluids in mixing processes can be illustrated by a simple example many of us observe every day. When one adds creme to coffee, one can wait for a diffusion process to make the mixture uniform - but very few people engage into this meditative exercise. Instead one makes a single quick swirl with a spoon. Mixing by both laminar and turbulent flows is crucial for many processes in nature of engineering, from ozone distribution in the atmosphere to nuclear burning in stars to proper function of internal combustion engine. My work in the are is one studying what proerties of the fluid flow makes it an efficient mixer, and what are the bounds on mixing efficiency given some natural constraints such as energy budget. I also study the effect mixing can have on chemical reactions. Depeding on the conditions, sometimes fluid flow can lead to dramatic speed up and sometimes it can quench the flames. The main models I employ for these studies are passive scalar equation and reaction-diffusion equations. The latter go back to classical works of Fisher and Kolmogorov-Petrovski-Piskunov in 1930s.  

 

Biomixing with chemotaxis

 

Chemotaxis is a term describing transport directed by chemical sensing. Its role is crucial in many processes in bliology, ecology and medicine. For example, chemical sensing plays a key role in morphology of the organisms where cells use it to specialize and find their proper place in the structure. Monocytes, immune system killer cells, use chemical agents released by distressed tissues to find and fight the infection. Corlas and many other marine animals use chemotaxis in their reproduction processes, and this part of their life cycle may be vulnerable to ocean pollution. Mathematically, the first mean field partial differential equation model of chemotaxis was written down by Keller and Segel in 1970s. This model and its variations been much studied since, and provides qualitative insight into many properties of the chemotactic process. My research on chemotaxis focuses on two questions. The first is quantitative bounds on the effect of chemotaxis for biological reaction efficiency. Here the success of moncytes in fighting infection or fertilization rates in coral reproduction process can be modeled as a reaction. The goal of such estimates is to better understand to what extent chemotaxis is responsible for the success of these processes. The second direction is the development of more advanced models of chemotaxis. In this direction, I have initiated an interaction with biologists to see if it may be possible to better calibrate mathematical models of chemotaxis using available experimental data.    


During 2015, the main advance in this direction has been work joint with my graduate student Xiaoqian Xu on Keller-Segel equation with fluid advection. The solutions of Keller-Segel equation are known to have dramatic analytical properties called "collapse": the density can collapse into a point, forming a singularity. This corresponds to behavior observed for some types of mold and bacteria under stress. We show that the presence of fluid flow may prevent such singularity formation if the flow is sufficiently mixing.  


In 2016, I worked on a model of "flocking", a tendency of animals and other agents to align their speeds when traveling together. We discovered a completely new phenomenon, where solutions to a well known model become regular if one adds a nonlocal interaction modulated by species denisty. Such interaction is very natural, as alignment is likely to be stronger in regions of higher density. 

 
Teaching Areas
 Mathematics
 
Selected Publications
 Refereed articles
 

Alexander Kiselev and XIaoqian Xu "Suppression of chemotactic explosion by mixing." Archive for Rational Mechanics and Analysis, 222 (2016) : 1077-1112.

 
 

Alexander Kiselev snd Changhui Tan "Finite time blow up in the hyperbolic Boussinesq system." Submitted

 
 

Tam Do, Alexander Kiselev, Lenya Ryzhik, and Changhui Tan "Global regularity for the fractional Euler alignment system." Submitted

 
 

A. Kiselev and A. Zlatos "Blow up for the 2D Euler equation on some bounded domains." Journal of Differential Equations, 259 (2015) : 34903494.

 
 

A. Kiselev, L. Ryzhik, Y. Yao and A. Zlatos "Finite time singularity for the modified SQG patch equation." Annals of MathematicsSubmitted

 
 

A. Kiselev, Y. Yao and A. Zlatos "Local regularity for the modified SQG patch equation." Communications on Pure and Applied MathematicsSubmitted

 
 

K. Choi, A. Kiselev and Y. Yao "Finite time blow up for a 1D model of 2D Boussinesq system." Communications in Mathematical Physics, 334 (2015) : 16671679.

 
 

K. Choi, T. Hou, A. Kiselev, G. Luo, V. Sverak and Y. Yao "On the finite time blow up of a 1D model for the 3D axisymmetric Euler equation." Journal of the American Mathematical SocietySubmitted

 
 

A. Kiselev and V. Sverak "Small scale creation for solutions of the incompressible two dimensional Euler equation." Annals of Mathematics, 180 (2014) : 1205--1220.

 
 

G. Iyer, A. Kiselev and A. Zlatos "Lower bounds on the mix norm of passive scalars advected by incompressible enstrophy-constrained flows." Nonlinearity, 27 (2014) : 973--985.

 
 

M. Dabkowski, A. Kiselev, L. Silvestre and V. Vicol "Global well-posedness of slightly supercritical active scalar equations." Analysis and PDE, 7 (2014) : 43--72.

 
Presentations
 Invited Talks
 

"Caltech Colloquium." (March 2016)

 
 

"Conference in Harmonic Analysis in honor of Michael Christ." (May 2016)

 
 

"Operator Theory, Analysis and Mathematical Physics (OTAMP) Conference, Euler Institute, St. Petersburg, Russia." (August 2016)

 
 

"Penn State University, Colloquium." (April 2016)

 
 

"University of Central Florida, Colloquium." (March 2016)

 
 

Workshop ”Mathematical Aspects of Hydrodynamics”, Oberwolvach, Germany. Invited speaker.

 
 

Workshop ”Analysis and Computation in Kinetic Theory”, Stanford. Invited speaker. 

 
 

Workshop ”Water Waves and Related Fluid Models”, Oxford, United Kingdom. Invited speaker. 

 
 

Workshop on turbulent and coherent convection, UW-Madison. Invited speaker.

 
 

San Antonio AMS-MAA joint meeting, Special session on mathematical fluid mechanics. Invited talk. 

 
 

Colloquium, CAAM, Rice University

 
 

Colloquium, Mathematics, Rice University

 
 Keynote Speaker
 

"International Conference on Reaction-Diffusion Equations and their Applications to the Life, Social and Physical Sciences, Renmin University,
Beijing, China." (May 2016)

 
 

"Summer school on Partial Differential Equations, Hebrew University, Jerusalem, three 75 minute talks." (September 2016)

 
 

"Summer school: Various Aspects of Mathematical Physics, Euler Institute, St. Petersburg, Russia." (August 2016)

 
 

Workshop ”Mathflows 2015”, Porquerolles, France. Plenary speaker. 

 
 

Summer school ”Transport, fluids and mixing”, Levico Terme, Italy (one of the four plenary lecturers, four 90 minute talks)

 
 

Analysis Partial Differential Equations summer school at the University of Chicago, five 60 minute talks, one of principal lecturers.

 
 Lectures
 

University of Houston, Colloquium

 
 

Avron Douglis Memorial Lecture, Department of Mathematics, University of Maryland

 
 Seminar Speaker
 

"Caltech, Mathematical Physics seminar." (February 2016)

 
 

"Duke University, Analysis and Applied Mathematics Seminar." (October 2016)

 
 

"Texas A&M University, PDE seminar." (February 2016)

 
 

University of Texas-Austin, PDE seminar

 
 

Partial Differential Equations seminar, University of Houston

 
 

Partial Differential Equations seminar, University of Zurich

 
 Workshops
 

A 50 minute talk at workshop on Turbulent Mixing, Institute for Pure and Applied Mathematics, Los Angeles

 
 

A 60 minute talk at Birman's conference on spectral theory, Euler Institute, St Petersburg, Russia

 
 

60 minute talk at a Special Week on Reaction-Diffusion Equations, International Center of Mathematics and INformation Science, Toulouse, France

 
 

A 30 minute talk at American Institute of Mathematics conference, special session on Fluid Mechanics, Madrid, Spain

 
 

50 minute talk at a conference "Analysis and numerical approximation of PDEs", Zurich, Switzerland

 
Editorial Positions
 Editor, Memoirs of the American Mathematical Society. American Mathematical Society. (2013 - 2015)

 Editor, Transactions of American Mathematical Society. American Mathematical Society. (2013 - 2015)

 Associate Editor, Memoirs of the American Mathematical Society. (2016 - 2021)

 Member of the Editorial Board, Nonlinearity. (2016 - 2016)

 Associate Editor, Communications in Mathematical Sciences. International Press. (2012 - 2015)

 Editor for Special Issue, Journal of Nonlinear Science. Springer. (2015 - 2016)

 Associate Editor, Transactions of the American Mathematical Society. (2016 - 2021)