Strichartz estimates in Wiener amalgam spaces and applications to nonlinear wave equations

TL;DR

This paper establishes new Strichartz estimates for the wave propagator in Wiener amalgam spaces, enabling refined analysis of nonlinear wave equations by leveraging oscillatory integral techniques and Bessel function asymptotics.

Contribution

It introduces the first Strichartz estimates for the wave propagator in Wiener amalgam spaces, expanding the analytical toolkit for nonlinear wave equations.

Findings

New Strichartz estimates for wave propagator in Wiener amalgam spaces

Application of estimates to analyze nonlinear wave equations

Extension of techniques to Schrödinger case

Abstract

In this paper we obtain some new Strichartz estimates for the wave propagator $e^{it\sqrt{-\Delta}}$ in the context of Wiener amalgam spaces. While it is well understood for the Schr\"odinger case, nothing is known about the wave propagator. This is because there is no such thing as an explicit formula for the integral kernel of the propagator unlike the Schr\"odinger case. To overcome this lack, we instead approach the kernel by rephrasing it as an oscillatory integral involving Bessel functions and then by carefully making use of cancellation in such integrals based on the asymptotic expansion of Bessel functions. Our approach can be applied to the Schr\"odinger case as well. We also obtain some corresponding retarded estimates to give applications to nonlinear wave equations where Wiener amalgam spaces as solution spaces can lead to a finer analysis of the local and global behavior of the solution.