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

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