Well-Posedness for the Euler Equations in Function Spaces of Generalized Smoothness

TL;DR

This work advances the well-posedness theory for the incompressible Euler equations by establishing local existence in generalized Besov spaces $B^{s,\psi}_{p,q}$ and generalized Triebel–Lizorkin spaces $F^{s,\psi}_{p,q}$ at the critical exponent $s=\frac{d}{p}+1$, provided the slowly varying function $\psi$ grows sufficiently fast. A Beale–Kato–Majda type continuation criterion is proved, showing that global existence in two dimensions follows from controlling $\int_0^T \|\omega(t)\|_{\dot{B}^0_{\infty,1}} \,dt$, and the vorticity transport structure in 2D is leveraged to obtain persistence of the norm. The analysis develops parallel results in the TL (Triebel–Lizorkin) scale, with careful handling of commutator estimates, pressure terms, and multiplier theorems for generalized smoothness spaces. Together, these results broaden the landscape of critical-function-space well-posedness for Euler and illuminate how generalized smoothness and slowly varying weights influence the dynamics.

Abstract

We consider the question of well-posedness for the incompressible Euler equations in generalized function spaces of the type $B^{s,ψ}_{p,q}(\mathbb{R}^d)$ and $F^{s,ψ}_{p,q}(\mathbb{R}^d)$ where $ψ$ is a slowly varying function in the Karamata sense and $s=d/p+1$. We prove that if $ψ$ grows fast enough, then there is a local in time solution to the Euler equations. We also establish a BKM-type criterion that allows us to obtain global existence in two dimensions.

Theorems & Definitions (50)

• Definition 1.1
• Theorem 1.2: Local existence in $\mathbb{R}^d$ and blow up criterion
• Corollary 1.3: Global existence in $\mathbb{R}^2$
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 2.5
• Lemma 2.6
• Definition 2.7