In this paper, we consider the three-dimensional Schrdinger operator with a -interaction of strength > 0 supported on an unbounded surface parametrized by the mapping R2x(x,f(x)), where 0, and f:R2R, f 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrdinger operator coincides with 142,+. We prove that for all sufficiently small > 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit 0+. In particular, this eigenvalue tends to 142 exponentially fast as 0+.

We consider a strictly hyperbolic, genuinely nonlinear system of conservation laws in one space dimension. A sharp decay estimate is proved for the positive waves in an entropy weak solution. The result is stated in terms of a partial ordering among positive measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources.

This work is devoted to study the global behavior of viscous flows contained in a symmetric domain with complete slip boundary. In such scenario the boundary no longer provides friction and therefore the perturbation of angular velocity lacks decaying structure. In fact, we show the existence of uniformly rotating solutions as steady states for the compressible Navier-Stokes equations. By manipulating the conservation law of angular momentum, we establish a suitable Korn's type inequality to control the perturbation and show the asymptotic stability of the uniformly rotating solutions with small angular velocity. In particular, the initial perturbation which preserves the angular momentum would decay exponentially in time and the solution to the Navier-Stokes equations converges to the steady state as time grows up.

In this paper we study the distribution of scattering resonances for a multi-dimensional semi-classical Schrödinger operator, associated to a potential well in an island at energies close to the maximal one that limits the separation of the well and the surrounding sea.

Since the work by Guo (Invent Math 153(3):593630, 2003), it has remained an open problem to establish the global existence of perturbative classical solutions around a global Maxwellian to the VlasovMaxwellBoltzmann system with the whole range of soft potentials. This is mainly due to the complex structure of the system, in particular, the degenerate dissipation at large velocity, the velocity-growth of the nonlinear term induced by the Lorentz force, and the regularity-loss of the electromagnetic fields. This paper solves this problem in the whole space provided that initial perturbation has sufficient regularity and velocity-integrability.