Heat-flow monotonicity of Strichartz norms

TL;DR

This paper proves that for dimensions 1 and 2, the classical Strichartz norm related to the free Schrödinger equation increases monotonically as the initial data undergoes a specific quadratic heat-flow, revealing new monotonicity properties.

Contribution

It establishes the monotonicity of the Strichartz norm under quadratic heat-flow for low dimensions, a novel insight into the behavior of solutions to the Schrödinger equation.

Findings

Strichartz norm is nondecreasing under heat-flow in 1D and 2D

Provides new monotonicity properties for Schrödinger solutions

Enhances understanding of dispersive PDE behavior

Abstract

Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i s\Delta}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the free Schr\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow.