Professor Guo's research is concerned with the rigorous mathematical study of partial differential equations arising in various scientific applications, such as kinetic and fluid models for plasma physics, vortices in classical field theory (superconductivity and superfluidity), and stability problems in stellar dynamics, and other physical problems.

Professor Guo's research is concerned with the rigorous mathematical study of partial differential equations arising in various scientific applications. More specifically, he has been working on PDE arising in
the kinetic theory of statistical physics, especially in connection with the nonlinear stability of their steady states. Kinetic theory is concerned with the study of the dynamics of a large ensemble of 'particles'. Interestingly, such abstract 'particles' can be tiny
gas molecules, or enormous stars in a galaxy. The most fundamental equation in the kinetic theory for describing gas molecules is the celebrated Boltzmann equation. Many fundamental macroscopic fluid
equations, such as the Euler and Navier-Stokes equations, can be derived from the Boltzmann theory. He has been working on stability of Maxwellian states in the Boltzmann theory. In a kinetic theory of stars, collisions among stars are sufficiently rare to be ignored. Therefore, a galaxy or a globular cluster can be modeled as an
ensemble of particles, i.e., stars, which interact only by the gravitational field which they create collectively. The time evolution of a galaxy can then be described by the Vlasov theory. There are many well known steady state galaxy models. Professor Guo has been developing mathematical tools to analyze the dynamical stability of these steady galaxy models. Instabilities of equilibria in many physical and biological sciences has always attracted great attention. It is important, from a scientific point of view, to understand the rate, time scale, structure, pattern and dynamics of various instabilities in a fully nonlinear setting. Professor Guo has been working on developing general mathematical framework to prove and characterize such nonlinear instabilities.

CURRENT GRANTS

 `PDE methods in kinetic theory and their applications.'
 

Simon Research Fellow, 2015-2016

P. Sloan Research Fellow, 1998-2003

NSF Postdoctoral Fellowship, 1995-1998

Honorable Mention in SIAM Student Paper Competition for [1], 1992

A. P. Sloan Dissertation Fellowship, 1992

1. Managing Editor, Journal of Partial Differential Equations. 2009-

2. Associate Editor, SIAM Journal of Mathematical Analysis. 2009-

3. Associate Editor, Comm. Math. Sci. 2009-

4. Associate Editor, Acta Applicandae Mathematicae. 2009-

5. Associate Editor, Kinetic and Related Models. 2008-

6. Associate Editor, Discrete and Continuous Dynamics (B). 2008-

7. Editor of `Nonlinear Wave Equations,' special volume, Quart. Appl. Math. 2009.

8. Organizer of FRG meeting, May, 2010, Brown.

9. Co-organizer of `Kinetic and fluids,' Peking University, July, 2010.

10. Organizer of `Nonlinear Wave Equations' Brown, May 9-13, 2008.

11. Co-organizer of the international conference on mathematical fluid dynamics, Brown, April,
2006.

12. Co-organizer of the international PDE conference `Continuum Mechanics and Hyperbolic
Conservation Laws', May 2001, Brown University.

13. Associate Editor, Annale de la Faculte des Sciences de Toulouse. 2002-2006

14. Assistant Managing Editor, SIAM Jounral of Mathematical Analysis, 01/02-06/02.

15. Co-organizers for the international conference `Nonlinear analysis 2000', in May, 2000, Courant.

16. Editor of Contem. Math., volume 235. 1999.

17. Organizer of the conf. on `Nonlinear Wave Equations', May 2 - 3, Brown Univ., 1998.

18. Reviewer for Mathematical Reviews.

19. Referee for various professional journals including Acta. Math. (2), Invent. Math. (5), J. AMS (5), Comm. Pure Appl. Math. (7), Commun. Math. Phys. (14), and Arch. Rational Mech. Anal. (10), Comm. PDE (5), IAHP (3).

20. Reviewer for NSF grant proposals.