Strichartz estimates for the one-dimensional wave equation

Characteristic Strichartz Estimates for the Wave Equation

Strichartz estimates for the wave equation on Riemannian symmetric manifolds

We prove Strichartz type estimates for solutions of the homogeneous wave equation on Riemannian symmetric spaces. Our results generalize those of Ginibre and Velo in [7].

Counterexamples to Strichartz estimates for the wave equation in domains

Let Ω be the upper half plane {(x, y) ∈ R, x > 0, y ∈ R}. Define the Laplacian on Ω to be ∆D = ∂ 2 x + (1 + x)∂ 2 y , together with Dirichlet boundary conditions on ∂Ω: one may easily see that Ω, with the metric inherited from ∆D, is a strictly convex domain. We shall prove that, in such a domain Ω, Strichartz estimates for the wave equation suffer losses when compared to the usual case Ω = R, ...

Strichartz Estimates for the Wave Equation on Manifolds with Boundary

Strichartz estimates are well established on flat Euclidean space, where M = R and gij = δij . In that case, one can obtain a global estimate with T = ∞; see for example Strichartz [27], Ginibre and Velo [9], Lindblad and Sogge [16], Keel and Tao [14], and references therein. However, for general manifolds phenomena such as trapped geodesics and finiteness of volume can preclude the development...

Strichartz Estimates for the Wave Equation on Flat Cones

We consider the solution operator for the wave equation on the flat Euclidean cone over the circle of radius ρ > 0, the manifold R+ × ( R / 2πρZ ) equipped with the metric g(r, θ) = dr2 + r2 dθ2. 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...