The Euler equations in a critical case of the generalized Campanato space

Abstract

In this paper we prove local in time well-posedness for the incompressible Euler equations in ${\mathbb{R}}^n$ for the initial data in ${\mathcal{L}}_{1(1)}^1({\mathbb{R}}^n)$, which corresponds to a critical case of the generalized Campanato spaces ${\mathcal{L}}_{q(N)}^s({\mathbb{R}}^n)$. The space is studied extensively in our companion paper [9], and in the critical case we have embeddings $B_{\infty ,1}^1({\mathbb{R}}^n)\hookrightarrow {\mathcal{L}}_{1(1)}^1({\mathbb{R}}^n)\hookrightarrow C^{0,1}({\mathbb{R}}^n)$, where $B_{\infty ,1}^1({\mathbb{R}}^n)$ and $C^{0,1}({\mathbb{R}}^n)$ are the Besov space and the Lipschitz space respectively. In particular ${\mathcal{L}}_{1(1)}^1({\mathbb{R}}^n)$ contains non-$C^1({\mathbb{R}}^n)$ functions as well as linearly growing functions at spatial infinity. We can also construct a class of simple initial velocity belonging to ${\mathcal{L}}_{1(1)}^1({\mathbb{R}}^n)$, for which the solution to the Euler equations blows up in finite time.