On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces

Abstract

It is proved in \cite{IO21} that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in ${\dot{B}}_{p,q}^{2/p}({\mathbb{R}}^2)\times {\dot{B}}_{p,q}^{2/p-1}({\mathbb{R}}^2)\times {\dot{B}}_{p,q}^{2/p-2}({\mathbb{R}}^2)$ with $1\le p\le \infty$ and $1\le q<\infty$. In this paper, we aim to solve the end-point case left in \cite{IO21} and prove that the Cauchy problem is ill-posed in ${\dot{B}}_{p,\infty }^{d/p}({\mathbb{R}}^d)\times {\dot{B}}_{p,\infty }^{d/p-1}({\mathbb{R}}^d)\times {\dot{B}}_{p,\infty }^{d/p-2}({\mathbb{R}}^d)$ with $1\le p\le \infty$ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in ${\dot{B}}_{p,\infty }^{d/p-1}({\mathbb{R}}^d)$, which is an interesting open problem in itself.