Transport equation in generalized Campanato spaces

In this paper we study the transport equation in $\mathbb{R}^{n} \times (0,T)$, $T >0$, $n\ge 2$, $$ \partial \_t f + v\cdot \nabla = g, \quad f(\cdot,0)= f\_0 \text{in }\mathbb{R}^{n}, generalized Campanato spaces $\mathscr{L}^{s}{{q(p, N)}}(\mathbb{R}^{n})$. The critical case is particularly interesting, and applied to local well-posedness problem for incompressible Euler equations a space close Lipschitz our companion \[Ann. Inst. H. Poincaré Anal. Non Linéaire 38 (2021), no. 2, 201–241]. $s=q=N=1$, have embeddings $B^{1}{\infty, 1} (\mathbb R^n) \hookrightarrow \mathscr{L}^{1}{1(p, 1)}(\mathbb {R}^{n}) C^{0, R^n)$, where R^n)$ $C^{0, are Besov spaces, respectively. For $f\_0\in {R}^{n})$, $v\in L^1(0,T; \mathscr {L}^{1}{1(p, {R}^{n})))$ $g\in {R}^{n})))$, prove existence uniqueness of solutions $L^\infty(0,T; \smash{\mathscr{L}^{1}{1(p, {R}^{n})})$ such that |f|{L^\infty(0,T; {R}^{n})))} \le C \big( |v|{L^1(0,T; {R}^{n})))}, |g|{L^1(0,T; 1)} {R}^{n})))}\big). Similar results other cases also proved.