Well-posedness of the Euler equation in Triebel–Lizorkin–Morrey spaces

ABSTRACT

This paper establishes the local existence, uniqueness and a blow-up criterion for the solutions to the inviscid incompressible Euler equation in Triebel–Lizorkin–Morrey space ${\mathcal{F}}_{p,q,r}^s\left({\mathbf{R}}^d\right),$ $d\ge 2$. As an application, we also derive the global persistence of the initial regularity in Triebel–Lizorkin–Morrey space for the solutions of 2-D Euler equation. These results are established using the logarithmic inequality of Beal–Kato–Majda type, the Moser type of inequality and commutator estimates in Triebel–Lizorkin–Morrey space.

COMMUNICATED BY:

• Ming Mei

KEYWORDS:

• Euler equation
• Triebel–Lizorkin–Morrey space
• local well-posedness
• blow-up criterion
• paraproduct

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary: 35Q35
• Secondary: 76B03
• 35E15

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors are partially supported by NSF of China [grant numbers 11261023 and 11461033, 11401269], the Natural Science Foundation of Jiangxi Province [grant number 20171BAB201003] and the foundation of CSC [grant number 201609470010] and CSC [grant number 201608360133].