# Liouville theorem for elliptic equation with advection term

How can one prove that any $$L^2$$ solution of $$- \Delta \phi(x) + a(x) \cdot \nabla \phi(x)=0 \qquad \mbox{in } \mathbb R^N$$ is zero if $$a(x)$$ is a divergence-free vector field such that $$\int |\nabla a|^2 dx < \infty$$? If the statement above is not true, under which additional assumption of $$a$$ and $$\phi$$ is it? For example, do we also need to assume additional decay for $$a$$ if $$N\ge 2$$?

A similar question has been asked in A Liouville theorem involving an advection term