Strichartz Estimates for the Vibrating Plate Equation

TL;DR

This paper investigates dispersive properties of the linear vibrating plate equation, establishing Strichartz estimates by relating it to Schrödinger equations, and proves well-posedness in homogeneous Sobolev spaces with sharpness results.

Contribution

It introduces Strichartz estimates for the vibrating plate equation and demonstrates well-posedness using homogeneous Sobolev spaces, extending dispersive PDE analysis.

Findings

Established Strichartz estimates for the LVP equation.

Proved well-posedness of the Cauchy problem with time-dependent potentials.

Demonstrated the sharpness of the estimates through stationary solutions.

Abstract

We study the dispersive properties of the linear vibrating plate (LVP) equation. Splitting it into two Schr\"odinger-type equations we show its close relation with the Schr\"odinger equation. Then, the homogeneous Sobolev spaces appear to be the natural setting to show Strichartz-type estimates for the LVP equation. By showing a Kato-Ponce inequality for homogeneous Sobolev spaces we prove the well-posedness of the Cauchy problem for the LVP equation with time-dependent potentials. Finally, we exhibit the sharpness of our results. This is achieved by finding a suitable solution for the stationary homogeneous vibrating plate equation.