Jian Wang （王健）

I am a Postdoctoral Research Associate at the Department of Mathematics, UNC-Chapel Hill. I obtained my Ph.D. at Berkeley under the supervision of Maciej Zworski.

I am interested in Microlocal Analysis, PDE, and Fluid Mechanics.

I am co-mentoring the Reseach Playground at UNC.

Contact: wangjian at email dot unc dot edu

Here is my CV.

Papers and Preprints

We provide a description in spaces of low regularity of the transport of  oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze the boundary of a convex obstacle to arbitrarily high finite or infinite order. The fundamental motivating example is the case where the spacetime manifold is the exterior of a spatial convex obstacle with smooth boundary, and the governing hyperbolic operator is the wave operator.
Motivated by numerically modeling surface waves for inviscid Euler equations, we analyze linear models for damped water waves and establish decay properties for the energy for sufficiently regular initial configurations.  Our findings give the explicit decay rates for the energy. For a subset of the models considered, this represents the first result proving the decay of the energy of the surface wave models.
We introduce the control conditions for 0th order pseudodifferential operators whose real parts satisfy the Morse—Smale dynamical condition. We obtain microlocal control estimates under the control conditions. As a result, we show that there are no singular profiles in the solution to the forced wave equation when the operator has a damping term that satisfies the control condition. This is motivated by the study of a microlocal model for the damped internal waves.
The scattering phase is the analogue of the counting function for eigenvalues when dealing with exterior domains and is closely related to Kreɪ̆n’s spectral shift function. We revisit classical results on asymptotics of the scattering phase and point out that it is never monotone in the case of strong trapping of waves. We provide the first numerical calculations of scattering phases for non-radial scatterers. They show that the asymptotic Weyl law is accurate even at low frequencies and reveal effects of trapping such as lack of monotonicity. This is achieved by using the recent high level multiphysics finite element software FreeFEM.
Following theoretical and experimental work of Maas et al we consider a linearized model for internal waves in effectively two dimensional aquaria. We provide a precise description of singular profiles appearing in long time wave evolution and associate them to classical attractors. That is done by microlocal analysis of the spectral Poincaré problem,  leading in particular to a limiting absorption principle. Some aspects of the paper can be considered as a natural microlocal continuation of the work of John on the Dirichlet problem for hyperbolic equations in two dimensions.
We study the dynamics of resonances of analytic perturbations of 0th order pseudodifferential operators. In particular, we prove a Fermi golden rule for embedded eigenvalues. We answer the question on the generic absence of eigenvalues asked by Colin de Verdière. We also study the convergence rate for eigenvalues in the viscosity limit.
Microlocal radial estimates were introduced by Melrose as a generalization of outgoing estimates of scattering theory. In this note we prove an end point version of these estimates in Besov spaces. This work has been cited by the works of Guedes Bonthonneau—Lefeuvre and Guillarmou—de Poyferré (with an appendix by Guedes Bonthonneau). 
We define the scattering matrix for 0th order pseudodifferential operators satisfying Morse—Smale dynamical conditions. After conjugation with natural reference operators, the scattering matrix becomes a 0th order Fourier integral operator with a canonical relation associated to the bicharacteristics of the operator. Such 0th order operators give a microlocal model of internal waves in stratified fluids as illustrated in the paper of Colin de Verdière—Saint-Raymond.
Using recent work of Bourgain—Dyatlov we show that for any convex co-compact hyperbolic surface Strichartz estimates for the Schrödinger equation hold with an arbitrarily small loss of regularity.

Teaching

In Spring 2024, I am teaching "Discrete Mathematics" and "First Course in Differential Equations"