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 $\mathscr{L}^s{ q(p, N)}(\mathbb{R}^n)$. 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, $B^1_{\infty, 1}(\Bbb R^n) \hookrightarrow \mathscr{L}^{ 1}{ 1(p, 1)}(\mathbb{R}^n) \hookrightarrow C^{0, 1} (\Bbb R^n)$, where $B^1{\infty, 1} (\Bbb R^n)$ and $C^{0, 1} (\Bbb R^n)$ are the Besov space and the Lipschitz space respectively. For $f_0\in \mathscr {L}^{ 1}{ 1(p, 1)}(\mathbb {R}^{n})$, $v\in L^1(0,T; \mathscr {L}^{ 1}{ 1(p, 1)}(\mathbb {R}^{n}))),$ and $ g\in L^1(0,T; \mathscr {L}^{ 1}{ 1(p, 1)}(\mathbb {R}^{n})))$, we prove the existence and uniqueness of solutions to the transport equation in $ L^\infty(0,T; \mathscr {L}^{ 1}{ 1(p, 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.