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


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