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Register a new userStrichartz estimates for the Schrodinger equation on polygonal domains
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We prove Strichartz estimates with a loss of derivatives for the Schrodinger equation on polygonal domains with either Dirichlet or Neumann homogeneous boundary conditions. Using a standard doubling procedure, estimates the on polygon follow from those on Euclidean surfaces with conical singularities. We develop a Littlewood-Paley squarefunction estimate with respect to the spectrum of the Laplacian on these spaces. This allows us to reduce matters to proving estimates at each frequency scale. The problem can be localized in space provided the time intervals are sufficiently small. Strichartz estimates then follow from a result of the second author regarding the Schrodinger equation on the Euclidean cone.
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Using a new local smoothing estimate of the first and third authors, we prove local-in-time Strichartz and smoothing estimates without a loss exterior to a large class of polygonal obstacles with arbitrary boundary conditions and global-in-time Strichartz estimates without a loss exterior to a large class of polygonal obstacles with Dirichlet boundary conditions. In addition, we prove a global-in-time local smoothing estimate in exterior wedge domains with Dirichlet boundary conditions and discuss some nonlinear applications.
We prove global weighted Strichartz estimates for radial solutions of linear Schrodinger equation on a class of rotationally symmetric noncompact manifolds, generalizing the known results on hyperbolic and Damek-Ricci spaces. This yields classical Strichartz estimates with a larger class of exponents than in the Euclidian case and improvements for the scattering theory. The manifolds, whose volume element grows polynomially or exponentially at infinity, are characterized essentially by negativity conditions on the curvature, which shows in particular that the rich algebraic structure of the Hyperbolic and Damek-Ricci spaces is not the cause of the improved dispersive properties of the equation. The proofs are based on known dispersive results for the equation with potential on the Euclidean space, and on a new one, valid for C^1 potentials decaying like 1/r^2 at infinity.
We prove generalized Strichartz estimates with weaker angular integrability for the Schrodinger equation. Our estimates are sharp except some endpoints. Then we apply these new estimates to prove the scattering for the 3D Zakharov system with small data in the energy space with low angular regularity. Our results improve the results obtained recently in cite{GLNW}.
We study the Schrodinger equation on a flat euclidean cone $mathbb{R}_+ times mathbb{S}^1_rho$ of cross-sectional radius $rho > 0$, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the geometric front, the part of the solution coming from formal application of the method of images, and the diffractive front emerging from the cone tip. As an application, we prove Strichartz estimates for the Schrodinger propagator on this class of cones.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
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