with the estimates in \[H2\], this allows us to control the right-hand-side of equation in a weighted negative Sobolev space.

-   A prototypical example of a mildly degenerate flow is given by $U(y)=\e^{\alpha y}, B(y)=C \e^{\alpha y}$, where condition \[H1\] imposes a constraint on $\alpha$. Considering that $\sinh(ky) \sin(kx)$ is in the kernel of the Laplacian this condition seems necessary. Condition \[H3\] then corresponds to a partition of $I$ using (dyadic) level sets of $U'$. Using a newly introduced localization procedure, we establish our main results by constructing localized pseudodifferential weights adapted to this covering.

-   As we discuss in following, this prototypical setting also allows us to consider circular flows with power law singularities or zeros as $r\downarrow
      0$ or $r \uparrow \infty$ on $\R^{2}\setminus \{0\}$.

-   If for instance $U(y)=\cos(y)$ or $U(y)=y^{2}$, the condition can relaxed to allow shrinking dyadic intervals. However, in that case further cancellation or decay due to symmetry, Hardy’s inequality or higher decay of $B(y)$ has to be assumed. The remaining obstacle to treat such non-mildly degenerate flows is then in improving \[H2\] to $$\begin{aligned}
        |B'(y)| \leq \eps_{0} |U'(y)|,
      \end{aligned}$$ which would require improved commutator estimates.

-   The smallness condition \[P\] quantifies closeness to 