Correction to: Arch Rational Mech Analhttps://doi.org/10.1007/s00205-023-01947-9

The estimates stated in the original version of Proposition 5.2 are incorrect by a factor of \(\beta (\tau )\). This then propagates to Corollary 5.3 and Proposition 5.5 and therefore invalidates the proofs (not the statements) of Proposition 6.1 and Proposition 7.1, which are used to establish the main result Theorem 1.1.

We fix the problem by first proving a new variant of Proposition 5.2. More precisely, we prove an \(X^{k} \rightarrow X^{k+l}\) smoothing estimate for the free semigroup \(S_0(\tau )\). Based on this, we prove a corrected version of smoothing estimates for the semigroup \(S(\tau )\), which are now contained in Proposition 5.4. The proof is based on Lemma 5.3 and semigroup methods. In addition, we provide in Appendix D an alternative proof based on energy methods. These adjustments do not require Corollary 5.3, hence we removed it. The proof of Proposition 6.1 now remains essentially the same modulo a factor of \(\beta (\tau -s)\) in the estimate for \(K_u(\psi )\), which nevertheless yields the same bound.

Lastly, we fix the proof of Proposition 7.1. In particular, we exploit higher order smoothing for \(S_0(\tau )\) proved in Proposition 5.2 together with a generalization of estimate (5.1) to factional \(X^k\)-spaces.

We thank Zexing Li for bringing this mistake to our attention.

The original article has been corrected.