Transport equation in generalized Campanato spaces (1904.08215v2)
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, $B1_{\infty, 1}(\Bbb Rn) \hookrightarrow \mathscr{L}{ 1}{ 1(p, 1)}(\mathbb{R}n) \hookrightarrow C{0, 1} (\Bbb Rn)$, where $B1{\infty, 1} (\Bbb Rn)$ and $C{0, 1} (\Bbb Rn)$ are the Besov space and the Lipschitz space respectively. For $f_0\in \mathscr {L}{ 1}{ 1(p, 1)}(\mathbb {R}{n})$, $v\in L1(0,T; \mathscr {L}{ 1}{ 1(p, 1)}(\mathbb {R}{n}))),$ and $ g\in L1(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|{L1(0,T; \mathscr{L}1{1(p, 1)} (\mathbb{R}n)))}, |g|{ L1(0,T; \mathscr{L}1{ 1(p, 1)}(\mathbb{R}n)))}\Big). ] Similar results in the other cases are also proved.