= Stokes resolvent kernel with advection
{c}
{title2=$R_p(s)$}
For $\operatorname{Re}p>0$, let $r_-$ and $r_1,r_2$ be the <characteristic roots> of $r^3+r+p=0$, split by the signs of their <real parts>. Put $d_j=3r_j^2+1$. The real-line <Green function> of $p+\partial_x+\partial_x^3$ is
$$
R_p(s)=\begin{cases}e^{r_-s}/d_-,&s\ge0,\\-\sum_{j=1}^2e^{r_js}/d_j,&s<0.\end{cases}
$$
The <reciprocal-polynomial root identities> imply continuity of $R_p,R_p'$ and jump $R_p''(0+)-R_p''(0-)=1$. The <distributional derivative> therefore gives $(p+\partial_s+\partial_s^3)R_p=\delta_0$. The transport term changes the roots: the pure <Airy resolvent kernel> with $r^3+p=0$ cannot be substituted unchanged.
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