Wave and Klein-Gordon equations on hyperbolic spaces

We consider the Klein--Gordon equation associated with the Laplace--Beltrami operator $\Delta$ on real hyperbolic spaces of dimension $n\!\ge\!2$; as $\Delta$ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well--posedness results for the corresponding semilinear equation with low regularity data.Comment: 50 pages, 30 figures. arXiv admin note: text overlap with arXiv:1010.237

SPHERICAL HARMONICS AND HARDY'S INEQUALITIES (Mathematical aspects of quantum fields and related topics)

We consider the derivative operators for radial direction and spherical direction. We also investigate the operator which takes the spherical average for functions. We reconfirm those properties with particular attention to orthogonality. As an application, the Hardy type inequality is presented with spherical derivatives in the framework of equalities. This clarifies the difference between contribution by radial and spherical derivatives in the improved Hardy inequality as well as nonexistence of nontrivial extremizers without compactness arguments

Scattering Theory for the Dirac Equation with a Nonlocal Term

Consider a scattering problem for the Dirac equation with a nonlocal term including he Hartree type. We show the existence of scattering operators for small initial data n the subcritical and critical Sobolev spaces

A supersolutions perspective on hypercontractivity

The purpose of this article is to expose an algebraic closure property of supersolutions to certain diffusion equations. This closure property quickly gives rise to a monotone quantity which generates a hypercontractivity inequality. Our abstract argument applies to a general Markov semigroup whose generator is a diffusion and satisfies a curvature condition.Comment: 7 page

On global solution to the Klein-Gordon-Hartree equation below energy space

In this paper, we consider the Cauchy problem for Klein-Gordon equation with a cubic convolution nonlinearity in $\R^3$. By making use of Bourgain's method in conjunction with a precise Strichartz estimate of S.Klainerman and D.Tataru, we establish the $H^s (s<1)$ global well-posedness of the Cauchy problem for the cubic convolution defocusing Klein-Gordon-Hartree equation. Before arriving at the previously discussed conclusion, we obtain global solution for this non-scaling equation with small initial data in $H^{s_0}\times H^{s_0-1}$ where $s_0=\frac\gamma 6$ but not $\frac\gamma2-1$, for this equation that we consider is a subconformal equation in some sense. In doing so a number of nonlinear prior estimates are already established by using Bony's decomposition, flexibility of Klein-Gordon admissible pairs which are slightly different from that of wave equation and a commutator estimate. We establish this commutator estimate by exploiting cancellation property and utilizing Coifman and Meyer multilinear multiplier theorem. As far as we know, it seems that this is the first result on low regularity for this Klein-Gordon-Hartree equation.Comment: 31 page

The well-posedness of the stochastic nonlinear Schr\"odinger equations in H2(Rd)H^2(\mathbb{R}^d)

The Cauchy problem for the stochastic nonlinear Schr\"odinger equation with a multiplicable noise is considered where the nonlinear term is of a power type and its coefficients are complex numbers. In particular, it is extremely important to consider the complex coefficients in the noise which cover non-conservative case, because they include measurement effects in quantum physics. The main purpose of this paper is to construct classical solutions in $H^2(\mathbb{R}^d)$ for the problem in question. The time local well-posedness in $L^2(\mathbb{R}^d)$ and $H^1(\mathbb{R}^d)$ was investigated in the papers [7,8]. In this paper we study the well-posedness in $H^2(\mathbb{R}^d)$ by making use of the rescaling approach as a main tool for dealing with the multiple noise, where we need to take advantage of a slight modification of the deterministic Strichartz estimate to fit into requirements under the setting of $H^2(\mathbb{R}^d)$. The other difficulty lies on the discussion on smoothness of functions in the nonlinear term, where the proof of time local well-posedness for the case of $H^2$-solutions does not go similarly as in the cases of $L^2$-solutions or $H^1$-solutions, because of the complexity in the computation of the nonlinear term with lower exponent $\alpha$. The techniques of Kato [18,19] work well on this difficulty even for the stochastic equations. We use the stochastic Strichartz estimate [4,16,17] as well to deal with white noise which did not appear in the proof for $L^2$-solutions or $H^1$-solutions. We also discuss time-global solutions in $H^2(\mathbb{R}^d)$.Comment: 31 pages, no figures. arXiv admin note: text overlap with arXiv:1404.5039 by other author

Global solutions of stochastic nonlinear Schr\"odinger system with quadratic interaction

The time-global existence of solutions to a system of stochastic Schr\"odinger equations with multiplicative noise and the quadratic nonlinear terms are discussed in this paper. The same system in the deterministic treatment was studied in [18] where the mass and energy are conserved. In our stochastic situation, those are not conserved and which causes several difficulties in the arguments of composing a-priori estimate.Comment: 28 pages, no figure