A commuting-vector-field approach to some dispersive estimates

TL;DR

This paper introduces a novel vector field method combined with conservation laws to establish pointwise decay estimates for solutions of linear dispersive equations, including the Vlasov, Schrödinger, and Airy equations, improving upon traditional explicit formula approaches.

Contribution

It develops a unified approach using commuting vector fields and conservation laws to derive decay estimates for multiple linear dispersive equations, offering sharper results than classical methods.

Findings

Recovered sharp pointwise decay for Vlasov and Schrödinger equations.

Obtained local energy decay and Strichartz estimates for Schrödinger.

Achieved near-sharp local energy decay for the Airy equation.

Abstract

We prove the pointwise decay of solutions to three linear equations: (i) the transport equation in phase space generalizing the classical Vlasov equation, (ii) the linear Schrodinger equation, (iii) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain $L^1$---$L^\infty$ decay through directly estimating the fundamental solution in physical space, or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines "vector field" commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space-time weighted energy estimates. Combined with a simple version of Sobolev's inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrodinger equations we can recover sharp pointwise decay; in the Schrodinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on $L^2$---$L^\infty$ decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.