I am sadden to learn that Bob Glassey 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.