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Investigation of Liouville-type theorems for MHD equations shows trivial solutions under controlled conditions, suggesting new paths for analysis.
In this paper, we investigate a Liouville‐type theorem for the MHD equations using Saint‐Venant type estimates. We show that is a trivial solution if the growth of the mean oscillation of the potential functions for both the velocity and magnetic fields are controlled. Our growth assumption is weaker than those previously known for similar results. The key idea is to refine Saint‐Venant type estimates by employing an approach analogous to the Frullani integral.
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Loong et al. (2026) studied this question.
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