Transport equation in generalized Campanato spaces

Abstract

In this paper we study the transport equation in ${\mathbb{R}}^n\times (0,T)$, $T>0$, $n\ge 2$,

in generalized Campanato spaces ${\mathcal{L}}_{q(p,N)}^s({\mathbb{R}}^n)$. The critical case is particularly interesting, and is applied to the local well-posedness problem for the incompressible Euler equations in a space close to the Lipschitz space in our companion paper [Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201–241]. In the critical case $s=q=N=1$, we have the embeddings $B_{\infty ,1}^1({\mathbb{R}}^n)\hookrightarrow {\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n)\hookrightarrow C^{0,1}({\mathbb{R}}^n)$, where $B_{\infty ,1}^1({\mathbb{R}}^n)$ and $C^{0,1}({\mathbb{R}}^n)$ are the Besov and Lipschitz spaces, respectively. For $f_0\in {\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n)$, $v\in L^1(0,T;{\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n)))$ and $g\in L^1(0,T;{\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n)))$, we prove the existence and uniqueness of solutions to the transport equation in $L^{\infty }(0,T;{\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n))$ such that

Similar results for the other cases are also proved.