Transport equation in generalized Campanato spaces

Abstract

In this paper we study the transport equation in ${\mathbb{R}}^n\times (0,T)$, $T>0$, \[ \partial _t f + v\cdot \nabla f = g, \quad f(\cdot ,0)= f_0 \quad \text{in}\quad \mathbb{R}^n \] 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 in a space close to the Lipschitz space in our companion paper\cite{cw}. More specifically, in the critical case $s=q=N=1$ we have the embedding relations, $B_{\infty ,1}^1({\mathbb{R}}^n)\hookrightarrow {\mathcal{L}}_{1(p,1)}^1({\mathbb{R}}^n)\hookrightarrow C^{0,1}({\mathbb{R}}^n)$, where $B_{\infty ,1}^1({\mathbb{R}}^n)$ and $C^{0,1}({\mathbb{R}}^n)$ are the Besov space and the Lipschitz space 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 \[ \|f\|_{L^\infty(0,T; \mathscr{L}^1_{ 1(p, 1)} (\mathbb{R}^n)))} \le C \Big( \|v\|_{L^1(0,T; \mathscr{L}^1_{1(p, 1)} (\mathbb{R}^n)))}, \|g\|_{ L^1(0,T; \mathscr{L}^1_{ 1(p, 1)}(\mathbb{R}^n)))}\Big). \] Similar results in the other cases are also proved.

52 pages