### Yan Guo

Office Hours: Tues Thurs 7 - 8 pm

Room 210, 170 Hope Street

## Biography

Ph.D., Brown University, 1993

Yan Guo’s research concerns with PDE study of basic physical models, which include kinetic theory modeling dilute gases (Boltzmann equation), plasma physics (Landau equation or Vlasov-Maxwell systems) and dynamics of galaxies and stars (Vlasov-Poissin system); classical fluid models governed by Euler and Navier-Stokes equations, such as the two-fluid theory (Euler-Maxwell system) for describing plasmas.

### Biography

Professor Guo received his B.S. from Peking University in 1987. He received his Ph.D in Mathematics from Brown University in 1993. He was a Courant Instructor at the Courant Institute of Mathematical Sciences for 1993-95. He joined the faculty of the Division of Applied Mathematics at Brown University as an Assistant Professor in September 1995. He was an Assistant Professor at Princeton University for 1996-97. His professional awards include an Honorable Mention in SIAM Student Paper Competition in 1992, an A. P. Sloan Dissertation Fellowship in 1993, an NSF Postdoctoral Fellowship for 1995-98. Professor Guo is an A. P. Sloan Research Fellow for 1998-2000. He was named a Manning Assistant Professor at Brown for 1998 to 1999, and was promoted to an Associate Professor in 1999 and then Professor in 2004.

### Research Interests

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.

### Affiliations

ICM PDE Panelist, 2016.

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

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

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

Associate Editor, *Acta Applicandae Mathematicae*. 2009-

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

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

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

Assistant Managing Editor, *SIAM Journal of Mathematical Analysis,* 01/02-06/02.

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

Reviewer for *Mathematical Reviews*.

Referee for various professional journals including *Annals of Math*. (1), *Acta. Math. *(2), *Invent. Math.* (12), *J. AMS* (7), *J. EMS* (1), *Comm. Pure Appl. Math*. (7), *Commun. **Math. Phys.* (20), and *Arch. Rational Mech. Anal*. (14), *Comm. PDE* (5).

Reviewer for NSF grant proposals. NSF panelist: Applied PDE (2) and Fluids (1), Applied Analysis (2).

### Links

## Awards

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

## Publications

Guo, Yan; Strauss, Walter A.: Instability of periodic BGK equilibria. Comm. Pure Appl. Math. 48 (1995), no. 8, 861–894.

Guo, Yan; Rein, Gerhard: Stable steady states in stellar dynamics. Arch. Ration. Mech. Anal. 147 (1999), no. 3, 225–243.

Guo, Yan: The Vlasov-Maxwell-Boltzmann system near Maxwellians. Invent. Math. 153 (2003), no. 3, 593–630.

Guo, Yan: The Vlasov-Poisson-Landau system in a periodic box. J. Amer. Math. Soc. 25 (2012), no. 3, 759–812.

Guo, Yan; Ionescu, Alexandru D.; Pausader, Benoit: Global solutions of the Euler-Maxwell two-fluid system in 3D. Ann. of Math. (2) 183 (2016), no. 2, 377–498.

Guo, Yan; Kim, Chanwoo; Tonon, Daniela; Trescases, Ariane: Regularity of the Boltzmann equation in convex domains. Invent. Math. 207 (2017), no. 1, 115–290.

Grenier, Emmanuel; Guo, Yan; Nguyen, Toan T.: Spectral instability of general symmetric shear flows in a two-dimensional channel. Adv. Math. 292 (2016), 52–110.