Yan Guo, Walter Strauss, and I organized two special issues on Nonlinear Waves and Kinetic Theory dedicated to the memory of Bob Glassey, who sadly passed away recently (I wrote an eulogy of his passing on this blog). The special issues are now published on the Kinetic and Related Models journal, Issue 1 and Issue 2.
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Bob Glassey
I am sadden to learn that Bob Glassey, a Professor Emeritus at Indiana, passed away this weekend after a long illness. Bob was a pioneer in the mathematical study of kinetic theory and nonlinear wave equations. He, together with Walter Strauss, was the first to initiate the mathematical study of Vlasov-Maxwell systems that describe the dynamics of a collisionless plasma. One of his fundamental theorems, known as Glassey-Strauss’ theorem (ARMA 1986), is to assert that solutions to the relativistic Vlasov-Maxwell system in the three dimensional space do not develop singularities as long as the velocity support remains bounded. The latter condition was subsequently verified by him and his former PhD student Jack Schaeffer for the case of low dimensions; namely, when particles are limited to one or two spatial domains. Their work has inspired several attempts from the mathematical community to tackle the full three dimensional case, which remains an outstanding open problem in the field.
Together with J. Schaeffer, Bob was also one of the first to initiate the mathematical study of Landau damping for Vlasov-Poisson systems in the presence of low frequency (or unconfined spatial domain). More precisely, for confined plasma (say, plasma on a torus), it was discovered and fully understood by Landau in the 40s that at the linearized level near a Gaussian, the electric field decays exponentially or polynomially depending on the regularity of initial data in the large time. The linear Landau damping remains to hold for more general spatially homogenous equilibria, known as Penrose stable equilibria. Later, Mouhot and Villani (Acta Math, 2011) verified this damping for data with analyticity for the nonlinear equations. In the unconfined case, Glassey and Schaeffer proved that the linear damping holds and is optimal at a much slower rate, which is surprisingly worse for Gaussians, due to the failure of the Penrose stability condition that holds in the confined case.
Bob also made fundamental and beautiful studies on the blowup issue for semilinear Heat, Wave, and Schr\”odinger (e.g., the Glassey’s trick), among other things. His book “The Cauchy problem in kinetic theory (SIAM 1996)” remains a fundamental textbook in the field.
Although Bob was already retired when I came to Indiana for my graduate study, he kindly participated and generously offered valuable guidances in a working seminar that I ran on the DiPerna-Lions theory for Boltzmann equations in the summer of 2008.
Dafermos and Rodnianski’s r^p-weighted approach to decay for wave equations
Dafermos and Rodnianski introduced an 
L-infinity instability of Prandtl layers
In 1904, Prandtl introduced his famous boundary layer theory in order to describe the behavior of solutions of incompressible Navier Stokes equations near a boundary as the viscosity goes to 
Graduate Student Seminar: Topics in Fluid Dynamics
Today, I give a Graduate Student Seminar lecture whose goal is to introduce to the first and second year graduate students at Penn State a few topics of research in Fluid Dynamics. There are many recent exciting developments in the field, which I only have time to present a few (many students haven’t taken any PDE course!). You may find the slides of my lecture here (up to many details delivered on the board!). You may also enjoy my similar lecture on Kinetic Theory of Gases, also aiming at first and second year students.
Sublayer of Prandtl boundary layers
The aim of this paper (arXiv:1705.04672), with E. Grenier, is to investigate the stability of Prandtl boundary layers in the vanishing viscosity limit: 
Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms
I’ve just uploaded this paper, with E. Grenier, on the arXiv (arXiv:1703.00881), entitled Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms, aiming a better understanding of the classical Prandtl’s boundary layers. Indeed, one of the key difficulties in dealing with boundary layers is the creation of (unbounded) vorticity in the inviscid limit.
Graduate student seminar: Kinetic theory of gases
Last week, I gave a graduate student seminar, whose purpose is to introduce to first and second year graduate students (at Penn State) an active and beautiful topics of research, and suggest a few possible ideas for students’ presentation later in the semester. Here are slides of my talk, which focuses on Kinetic Theory of Gases, a topics that I will teach as a graduate topics course, next fall (2017).
The Maxwell-Boltzmann approximation for ion kinetic modeling
In this short paper with C. Bardos, F. Golse, and R. Sentis, we are aimed to justify the Maxwell-Boltzmann approximation from kinetic models, widely used in the literature for electrons density distribution; namely, the relation
in which 
Math 505, Mathematical Fluid Mechanics: Notes 1
This Spring ’16 semester, I am teaching a graduate Math 505 course, whose goal is to introduce the basic concepts and the fundamental mathematical problems in Fluid Mechanics for students both in math and engineering. The difficulty is to assume no background in both fluids and analysis of PDEs from the students. That’s it!
On the spectral instability of parallel shear flows
This short note is to be published as the proceeding of a Laurent Schwartz PDE seminar talk that I gave last May at IHES, announcing our recent results (on channel flows and boundary layers), which provide a complete mathematical proof of the viscous destabilization phenomenon, pointed out by Heisenberg (1924), C.C. Lin and Tollmien (1940s), among other prominent physicists. Precisely, we construct growing modes of the linearized Navier-Stokes equations about general stationary shear flows in a bounded channel (channel flows) or on a half-space (boundary layers), for sufficiently large Reynolds number $R \to \infty$. Such an instability is linked to the emergence of Tollmien-Schlichting waves in describing the early stage of the transition from laminar to turbulent flows. In fact, the material in this note is only the first half of what I spoke on that day, skipping the steady case!
On wellposedness of Prandtl: a contradictory claim?
Yesterday, Nov 17, Xu and Zhang posted a preprint on the ArXiv, entitled “Well-posedness of the Prandtl equation in Sobolev space without monotonicity” (arXiv:1511.04850), claiming to prove what the title says. This immediately causes some concern or possible contrary to what has been known previously! Here, monotonicity is of the horizontal velocity component in the normal direction to the boundary. It’s well-known that monotonicity implies well-posedness of Prandtl (e.g., Oleinik in the 60s; see this previous post for Prandtl equations). It is then first proved by Gerard-Varet and Dormy that without monotonicity, the Prandtl equation is linearly illposed (and some followed-up works on the nonlinear case that I wrote with Gerard-Varet, and then with Guo). Is there a contradictory to what it’s known and this new preprint of Xu and Zhang? The purpose of this blog post is to clarify this.
Ill-posedness of the hydrostatic Euler and singular Vlasov equations
Daniel Han-Kwan and I have just submitted a paper (also, available on arxiv): “Ill-posedness of the hydrostatic Euler and singular Vlasov equations”, dedicated to Claude Bardos on the occasion of his 75th birthday, as a token of friendship and admiration.
Hello World !
This is a test to see if latex works here. Consider the Poisson equation: 
$latex
\begin{aligned}
u_t + u \cdot \nabla u + \nabla p &= \nu \Delta u
\\
\nabla \cdot u & =0
\end{aligned}$
wow !