will heavily use also the following statement about composition of paravector-field and paraproduct operators (see the appendix in [@D1999] for its proof).

\[l:pvec-pprod\] Fix $s\in\,]0,1[$. There exist constants $C$, depending only on $s$, such that, for all $t_1<0$ and $t_2\in{\mathbb{R}}$, $$\begin{aligned}
 \left\|T_X\,T_u\,v\right\|_{{\mathcal}{C}^{s-1+t_1+t_2}} & \leq & C\bigl(\|X\|_{{\mathcal}{C}^s}\,\|u\|_{{\mathcal}{C}^{t_1}}\,
\|v\|_{{\mathcal}{C}^{t_2}}\,+ \\
& & \qquad+\,\|v\|_{{\mathcal}{C}^{t_2}}\,\|T_Xu\|_{{\mathcal}{C}^{s-1+t_1}}\,+\,\|u\|_{{\mathcal}{C}^{t_1}}\,\|T_Xv\|_{{\mathcal}{C}^{s-1+t_2}}\bigr)\,,\end{aligned}$$ and this is still true in the case $t_1=0$ with $\|u\|_{L^\infty}$ instead of $\|u\|_{{\mathcal}{C}^0_*}$.

Moreover, if $\,s-1+t_1+t_2\,>\,0$, then we have also $$\begin{aligned}
 \left\|T_X\,R(u,v)\right\|_{{\mathcal}{C}^{s-1+t_1+t_2}} & \leq & C\bigl(\|X\|_{{\mathcal}{C}^s}\,\|u\|_{{\mathcal}{C}^{t_1}}\,
\|v\|_{{\mathcal}{C}^{t_2}}\,+ \\
& & \qquad+\,\|v\|_{{\mathcal}{C}^{t_2}}\,\|T_Xu\|_{{\mathcal}{C}^{s-1+t_1}}\,+\,\|u\|_{{\mathcal}{C}^{t_1}}\,\|T_Xv\|_{{\mathcal}{C}^{s-1+t_2}}\bigr)\,.\end{aligned}$$

Transport and elliptic equations
--------------------------------

System is basically a coupling of transport equations of the type $$\left\{\begin{array}{l}
\d_tf\,+\,v\cdot\nabla f\,=\,g\,,\\[1ex]
f_{|t=0}\,=\,f_0\,.
\end{array}\right.
\leqno(T)$$ So, we often need to use the following result, which enables us to solve $(T)$ in the Hölder spaces framework.

\[p:transport\] Let $\sigma>0$ ($\sigma>-1$ if $\,{{\rm div}\,}v=0$).\
Let $f_0\in{\mathcal}{C}^\sigma$, $g\in L^1([0,T];{\mathcal}{C}^\sigma)$ and $v$ be a time dependent vector field in ${\mathcal}{C}_b([0,T]\times{\mathbb{R}}^N)$ such that $$\begin{array}{lllll}
\nabla v&\in&  L^1([0,T];L^\infty) &\hbox{\rm if} &\sigma<1\,,\\[2ex]
\nabla v&\in& L^1([0,T];{\mathcal}{C}^{\sigma-1}) &\hbox{\rm if} &\sigma>1\,.
\end{array}$$

Then equation $(T)$ has a unique solution $f$ in the space $\Bigl(\bigcap_{\sigma'<\sigma}{\mathcal}{C}([0,T];{\mathcal}{C}^{\sigma'})\Bigr)\bigcap{\mathcal}{C}_w([0,T];{\mathcal}{C}^{\sigma})$.

Moreover, for all $t\in[0,T]$ we have $$\label{est:no-loss-1}
e^{-CV(t)}\,\|f(t)\|_{{\mathcal}{C}^\sigma}\,\leq\,
\|f_0\|_{{\mathcal}{C}^\sigma}\,+\,\int_0^t e^{-CV(\tau)}\,\|g(\tau)\|_{{\mathcal}{C}^\sigma}\,d\tau$$ $$\displaylines{
\mbox{with}\quad\!\! V'(t):=\left\{
\begin{array}{ll}
\!\|\nabla v(t)\|_{L^\infty}\! & \mbox{if}\!\!\!\quad\sigma<1\,,\\[2ex]
\!\|\nabla v(t)\|_{{\mathcal}{C}^{\sigma-1}}\! & \mbox{if}\!\!\!\quad\sigma>1\,. 
\end{array}\right.\hfill}$$ If $f\equiv v$ 