A remark on ill-posedness

TL;DR

This paper explores the nuanced relationship between ill-posedness, norm inflation, and stability in the context of the Navier-Stokes equations, challenging traditional notions of well-posedness and stability in certain function spaces.

Contribution

It constructs a non-Gauss flow function to demonstrate that well-posedness, norm inflation, and stability can coexist in the Navier-Stokes equations, offering new insights into their interplay.

Findings

Norm inflation implies discontinuous dependence on initial data.

Stability in ${ m BMO}^{-1}$ differs from that in $L^{ abla}(({ m BMO}^{-1})^{n})$.

Well-posedness and norm inflation may not be mutually exclusive.

Abstract

Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${\rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tataru's solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{\infty}(({\rm BMO}^{-1})^{n})$.