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In this article, we study long time dynamics for defocusing cubic nonlinear Schrödinger equations (NLS) on three dimensional product space. First, we apply the decoupling method in Bour-gain and Demeter [Ann. of Math. (2), 182 (2015), pp. 351-389] to establish a bilinear Strichartz estimate. Moreover, we prove global well-posedness for defocusing, cubic NLS on a three dimensional product space with rough initial data (Hs, s > 5/ 6 ) based on the I-method and the bilinear estimate. At last, we discuss the growth of the higher Sobolev norm problem which is tightly linked to the weak turbulence phenomenon. [ABSTRACT FROM AUTHOR]

In this paper, we study the global well-posedness and scattering theory for the defocusing fourth-order nonlinear Schrödinger equation (FNLS) in dimensions . We prove that if the solution u is apriorily bounded in the critical Sobolev space, that is, with all if p is an even integer or otherwise, then u is global and scatters. We will give a uniform way to treat the energy-subcritical, energy-critical and energy-supercritical FNLS by making use of the strategy derived from concentration compactness ideas, and we are able to overcome the logarithmic blowup in the double Duhamel trick in dimension eight by exploiting the refined dispersive estimate which is in sharp contrast to the Schrödinger equation. [ABSTRACT FROM AUTHOR]

It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

We reprove Wolff's L5/2-bound for the R³-Kakeya maximal function without appealing to the argument of induction on scales. The main ingredient in our proof is an adaptation of Sogge's strategy used in the work on Nikodym-type sets in curved spaces. Although the equivalence between these two type maximal functions is well known, our proof may shed light on some new geometric observations which is interesting in its own right. [ABSTRACT FROM AUTHOR]

We consider a class of defocusing energy-supercritical nonlinear Schrödinger equations in four space dimensions. Following a concentration-compactness approach, we show that for 1

In this paper, we study the Strichartz-type estimates of the solution for the linear wave equation with inverse square potential. Assuming the initial data possesses additional angular regularity, especially the radial initial data, the range of admissible pairs is improved. As an application, we show the global well-posedness of the semi-linear wave equation with inverse-square potential ... for power p being in some regime when the initial data are radial. This result extends the well-posedness result in Planchón, Stalker, and Tahvildar-Zadeh. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the restriction problem in harmonic analysis. Based on the spherical harmonics expansion and analyzing the asymptotic behavior of the Bessel function, we show that a modified linear adjoint restriction estimate holds for all Schwartz functions compactly supported on the cone, which generalizes Shao's result. [ABSTRACT FROM AUTHOR]