Uniqueness of Steady Navier-Stokes under Large Data by Continuous Data Assimilation

TL;DR

This paper introduces a continuous data assimilation method that ensures the uniqueness and well-posedness of steady Navier-Stokes equations even with large data, by incorporating spatial observations.

Contribution

The paper presents a novel CDA approach that guarantees solution uniqueness for steady NSE under large data conditions, addressing a key non-uniqueness challenge.

Findings

CDA ensures well-posedness of steady NSE with sufficient observations.

The method effectively addresses non-uniqueness in large data regimes.

The approach can be generalized to other non-uniqueness PDEs.

Abstract

We propose a continuous data assimilation (CDA) method to address the uniqueness problem for steady Navier-Stokes equations(NSE). The CDA method incorporates spatial observations into the NSE, and we prove that with sufficient observations, the CDA-NSE system is well-posed even for large data where multiple solutions may exist. This CDA idea is in general helpful to determine solution for non-uniqueness partial differential equations(PDEs).