No Arabic abstract
We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius $rho > 0$, the manifold $mathbb{R}_+ times mathbb{R} / 2 pi rho mathbb{Z}$ equipped with the metric $g(r,theta) = dr^2 + r^2 dtheta^2$. Using explicit representations of the solution operator in regions related to flat wave propagation and diffraction by the cone point, we prove dispersive estimates and hence scale invariant Strichartz estimates for the wave equation on flat cones. We then show that this yields corresponding inequalities on wedge domains, polygons, and Euclidean surfaces with conic singularities. This in turn yields well-posedness results for the nonlinear wave equation on such manifolds. Morawetz estimates on the cone are also treated.
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 study the hyperboloidal initial value problem for the one-dimensional wave equation perturbed by a smooth potential. We show that the evolution decomposes into a finite-dimensional spectral part and an infinite-dimensional radiation part. For the radiation part we prove a set of Strichartz estimates. As an application we study the long-time asymptotics of Yang-Mills fields on a wormhole spacetime.
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.
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.
We establish Strichartz estimates in similarity coordinates for the radial wave equation in three spatial dimensions with a (time-dependent) self-similar potential. As an application we consider the critical wave equation and prove the asymptotic stability of the ODE blowup profile in the energy space.