We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated by Dyatlov-Zahl and proved for porous sets in Bourgain-Dyatlov.

A basic, simple energy method for the Boltzmann equation is presented here. It is based on a new macromicro decomposition of the Boltzmann equation as well as the H-theorem. This allows us to make use of the ideas from hyperbolic conservation laws and viscous conservation laws to yield the direct energy method. As an illustration, we apply the method for the study of the time-asymptotic, nonlinear stability of the global Maxwellian states. Previous energy method, starting with Grad and finishing with Ukai, involves the spectral analysis and regularity of collision operator through sophisticated weighted norms.

This work is intended as both an introductory text and a reference book for those interested in studying several complex variables in the context of partial differential equations. Significant progress has been made in the study of Cauchy-Riemann and tangential Cauchy-Riemann operators; this progress greatly influenced the development of PDEs and several complex variables.

In this paper we prove global regularity for the full water-wave system in three dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water-wave models. To construct global solutions, we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than |t|−5/6) and the presence of a large, codimension-1, set of quadratic time-resonances. To deal with such a situation, we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the non-linearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions, we use harmonic analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.

We consider the following mean field equation: $$\Delta_{g}v+\rho\left(\frac{h^*e^v}{\int_Mh^*e^v}-1\right)=4\pi\sum_{j=1}^N\alpha_j(\delta_{q_j}-1)$$ on M, where M is a compact Riemann surface with volume 1, $h^*$ is a positive $C^1$ function on M, and  $\alpha_j$ are constants satisfying $\alpha_j>-1$. In this paper, we derive the topological degree counting formula for noncritical values of $\rho$. We also give several applications of this formula, including existence of curvature +1 metric with conic singularities, doubly periodic solutions of electroweak theory, and a special case of self-gravitating strings.