# Solve $(A(x).\nabla)u+cu=0$

ِDoes the equation $$y\partial_x u(x,y)-x\partial_y u(x,y)+cu=0$$ have complex-valued compact-supported or vanishing-at-infinity $$C^1$$ solution defined on the whole plane without any singularity? Here $$c$$ is a complex number. I have used the method of characteristics but the solution is $$u(x,y)=g(\sqrt{x^2+y^2})exp(-c arctg(y/x))$$ where g is a given function. But this solution has singularity at x=0. More generally, a similar question for the equation $$(A(x).\nabla)u+cu=0$$ where $$A(x)$$ is a real vector field over $$R^n$$ and $$c$$ is a complex number. Is there a complex-valued compact-supported or vanishing-at-infinity $$C^1$$ solution on whole $$R^n$$?

• Did you also want non-trivial? Since $u\equiv 0$is a solution. Jul 14, 2021 at 15:41
• In any case, your equation can be written in polar coordinates $(r,\theta)$ as $$\partial_\theta u + c u = 0$$ and the existence of non-singular solution boils down to asking whether the equation $g' = -c g$ has a $2\pi$ periodic solution. Jul 14, 2021 at 15:49
• Since you mentioned the similar equation and method of characteristics: it depends on the geometry of the integral curves of $A$. If $A$ has closed curves then there will be at most a discrete, countable set of $c$ that will work. OTOH, if all integral curves of $A$ are unbounded, then the only compactly supported solution will be the 0 solution. Jul 14, 2021 at 15:53
• @ Willie Thanks for useful comments. Jul 18, 2021 at 12:40