An optimal decay estimate for the linearized water wave equation in 2D

Document Type

Article

Publication Date

1-1-2016

Abstract

We obtain a decay estimate for solutions to the linear dispersive equation iut − (−Δ) u = 0 for (t, x) ∈ ℝ × ℝ. This corresponds to a factorization of the linearized water wave equation utt + (−Δ) u = 0. In particular, by making use of the Littlewood-Paley decomposition and stationary phase estimates, we obtain decay of order |t|− for solutions corresponding to data u(0) = ϕ, assuming only bounds on (Formula presented). As another application of these ideas, we give an extension to equations of the form iut − (−Δ) u = 0 for a wider range of α. 1/4 1/2 1/2 α/2

Publication Source (Journal or Book title)

Proceedings of the American Mathematical Society

First Page

4733

Last Page

4742

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