In this paper we study the small-data scattering of Hartree type fractional Schrödinger equations in space dimension 2, 3. It has Lévy index α between 1 and 2, and Hartree type nonlinearity F(u) = μ(|x|-γ * |u|2)u with 2d/(2d-1) < γ < 2, γ ≥ α > 1. This equation is scaling-critical in Ḣsc with sc = (γ-α)/2. We show that the solution scatters in Ḣsc,1, where Ḣsc,1 is also a scaling critical space of Sobolev type taking in angular regularity with norm defined by ||ϕ||Ḣsc,1 = ||ϕ||Ḣsc + ||∇Sϕ||Ḣsc. For this purpose we use the recently developed Strichartz estimate which is Lθ2-averaged on the unit sphere Sd-1.
Holonomic D-modules associated with certain kind of non-isolated hypersurface singularities, introduced by T. de Jong, are considered. Structures of these holonomic D-modules such as characteristic variety, monodromy are explicitly determined. The key of our approach is the use of the notion of algebraic local cohomology. An application to Betti numbers of local Milnor fibers is also discussed.
We investigate the asymptotic behavior in time of solutions to the Cauchy problem for the one-dimensional dissipative wave equation where the far field states are prescribed. In particular, we study the case where the flux function is convex but linearly degenerate on some interval. When the corresponding Riemann problem admits a Riemann solution which consists of rarefaction waves and contact discontinuity, it is proved that the solution of the Cauchy problem tends toward the linear combination of the rarefaction waves and viscous contact wave as time goes to infinity. The proof is given by a technical energy method under the sub-characteristic condition. We also show that the similar arguments are applicable to the initial-boundary value problem on the half space.
The multivariate Krawtchouk polynomials are orthogonal polynomials for the multinomial distribution, first defined by Griffiths in 1971. We construct infinite-variate extensions of them as complete orthogonal systems of specific weighted l2-spaces. We also give realizations of our infinite-variate extensions as zonal spherical functions on groups over a non-Archimedean local field. Some typical properties of Krawtchouk polynomials like duality, orthogonality and completeness are thus shed light from the point of view of zonal spherical functions.