Semi-classical propagation of singularity for Stokes system

TL;DR

This paper investigates the high-energy behavior of solutions to the Stokes system, showing that their concentration aligns with bi-characteristics and remains invariant under a specific flow, advancing understanding of singularity propagation.

Contribution

It introduces a semi-classical analysis of the Stokes system, demonstrating measure concentration on bi-characteristics and invariance under Melrose-Sj"ostrand flow, which is novel in this context.

Findings

High energy solutions concentrate on bi-characteristics.

The support of the measure is invariant under Melrose-Sj"ostrand flow.

Provides a semi-classical framework for analyzing the Stokes system.

Abstract

We study the quasi-mode of Stokes system posed on a smooth bounded domain with Dirichlet boundary condition. We prove that the high energy L2 norm of solutions concentrate on the bi-characteristic of Laplace operator as matrix-valued Radon measure. Moreover, we prove that the support of such measure is invariant under Melrose-Sj\"ostrand flow.