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

TL;DR

This paper establishes local well-posedness of the incompressible Euler equations for initial data in a critical generalized Campanato space, which includes non-smooth and linearly growing functions, and demonstrates finite-time blow-up for some initial velocities.

Contribution

It proves local existence and uniqueness of solutions in a critical generalized Campanato space and constructs initial data leading to finite-time blow-up, extending the understanding of Euler equations in critical function spaces.

Findings

Well-posedness in the critical space _{1(1)}(br^n)

Inclusion relations with Besov and Lipschitz spaces

Existence of initial data causing finite-time blow-up

Abstract

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.