We solve the Fu-Yau equation for arbitrary dimension and arbitrary slope $\alpha'$. Actually we obtain at the same time a solution of the open case $\alpha'>0$, an improved solution of the known case $\alpha'<0$, and solutions for a family of Hessian equations which includes the Fu-Yau equation as a special case. The method is based on the introduction of a more stringent ellipticity condition than the usual $\Gamma_k$ admissible cone condition, and which can be shown to be preserved by precise estimates with scale.

In the present paper, we prove an improved Combes–Thomas estimate, viz. the Combes–Thomas estimate in trace-class norms, for magnetic Schrödinger operators under general assumptions. In particular, we allow for unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.

The nonlinear Schrdinger equation (NLS) has been a fundamental model for understanding vortex motion in superfluids. The vortex motion law has been formally derived on various physical grounds and has been around for almost half a century. We study the nonlinear Schrdinger equation in the incompressible fluid limit on a bounded domain with Dirichlet or Neumann boundary condition. The initial condition contains any finite number of degree 1 vortices. We prove that the NLS linear momentum weakly converges to a solution of the incompressible Euler equation away from the vortices. If the initial NLS energy is almost minimizing, we show that the vortex motion obeys the classical Kirchhoff law for fluid point vortices. Similar results hold for the entire plane and periodic cases, and a related complex GinzburgLandau equation. We treat as well the semi-classical (WKB) limit of NLS in the presence of

Nonlinear hyperbolic systems with relaxations may encounter different scales of relaxation time, which is a prototype multiscale phenomenon that arises in many applications. In such a problem the relaxation time is of O(1) in part of the domain and very small in the remaining domain in which the solution can be approximated by the zero relaxation limit which can be solved numerically much more efficiently. For the Jin–Xin relaxation system in such a two-scale setting, we establish its wellposedness and singular limit as the (smaller) relaxation time goes to zero. The limit is a multiscale coupling problem which couples the original Jin–Xin system on the domain when the relaxation time is O(1) with its relaxation limit in the other domain through interface conditions which can be derived by matched interface layer analysis.As a result, we also establish the well-posedness and regularity (such as boundedness in sup norm with bounded total variation and L 1-contraction) of the coupling problem, thus providing a rigorous mathematical foundation, in the general nonlinear setting, to the multiscale domain decomposition method for this two-scale problem originally proposed in Jin et al. in Math. Comp. 82, 749–779, 2013.

During the past year a group of 12 graduate students, a visiting researcher and a postdoctoral fellow, supported in part by this contract, have studied various nonlinear partial differential equations. Seminars were held twice weekly on the topic of soliton theory. The graduate students involved have been studying the problem of singularity that arises in nonlinear equations.