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