Erratum to: Local ill-posedness of the incompressible Euler equations in $C^1$ and $B^1_{\infty ,1}$

1 Erratum to: Math. Ann. DOI 10.1007/s00208-015-1213-0

The proof of Theorem 3 in the original article relies on the following bound proved by Kato and Ponce [1], Theorem I, for solutions of the incompressible Euler equations

$$\begin{aligned} \sup _{[0,T]} \Vert u(t)\Vert _{s,p} \le K(T, \Vert u_0\Vert _{s,p}) \end{aligned}$$

for any $T>0$ and any divergence free vector field $u_0 \in W^{s,p}(\mathbb {R}^2)$ where $s>1+2/p$ and $1<p<\infty$. However, a careful examination indicates that the constant K may also depend directly on p itself and as a result our Lemma 4 is insufficient to guarantee that the above bound is uniform in p. Consequently, the statement of Theorem 3 should be amended as follows

Theorem 3

Let $2 < p < \infty$. Assume that the Euler equations (1.1)–(1.2) are well-posed in X. Let $\omega _0 \in C^\infty _c(\mathbb {R}^2)$ be the initial vorticity defined in (2.2) below. Then there exists $T>0$ and a sequence of initial vorticities $\omega _{0,n} \in C^\infty _c(\mathbb {R}^2)$ with the following properties, either

• 1. there exists a constant $C>0$ such that $\Vert \omega _{0,n} \Vert _{W^{1.p}} \le C$ for all $n \in \mathbb {Z}_+$ and

• 2. for any $M \gg 1$ there is $0 < t_0 \le T$ such that the solution $\omega _n(t)$ of the vorticity equations (1.4)–(1.5) with initial data $\omega _{0,n}$ satisfies $\Vert \omega _{n}(t_0)\Vert _{W^{1,p}} \ge M^{1/3}$ for all sufficiently large n and all p sufficiently close to 2 or we have $\sup _{0\le t \le T} \Vert \omega _n(t)\Vert _{W^{s-1,p}} \rightarrow \infty$ as $p\searrow 2$.

and Theorems 1 and 2 should be reformulated accordingly

Theorem 1

The 2D incompressible Euler equations (1.1) are locally ill-posed in the space $C^1$ provided that the bound $\sup _{0\le t \le T} \Vert \omega (t)\Vert _{W^{s-1,p}} \le K$ is uniform as $p \searrow 2$.

Theorem 2

The 2D incompressible Euler equations are locally ill-posed in the Besov space $B^1_{\infty , 1}$ provided that the bound $\sup _{0\le t \le T} \Vert \omega (t)\Vert _{W^{s-1,p}} \le K$ is uniform as $p \searrow 2$.

Lastly, we observe that if we could find a suitable function space $X \subset C^1$ in which the Euler equations are locally well-posed and the estimates of Lemmas 10, 11 and 12 hold (with appropriate modifications) then the last condition of Theorem 3 would follow. We do not yet know whether this is the case.