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How can I prove the following Liouville theorem for an elliptic equation with gradient perturbation?

Let $u \in L^2(\mathbb R^n;\mathbb R)$ be a smooth solution of $$ -\Delta u + v \cdot \nabla u = 0 \qquad \text{in } \mathbb R^n$$ where we assume $\int_{\mathbb{R}^n}|\nabla v|^2 dx \leq C$ and $\mathrm{div}v=0$ for some $C > 0$, then $u$ is constant.

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    $\begingroup$ I guess that $u$ is a function rather than a vector and that the conclusion should be $u=0$. Do you assume integrability only for $\nabla v$? $\endgroup$
    – Giorgio Metafune
    Commented Jan 6, 2022 at 23:29
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    $\begingroup$ a related question mathoverflow.net/questions/186856/… $\endgroup$
    – Math604
    Commented Jan 8, 2022 at 8:04

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