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In this paper, we are concerned with an incompressible and inviscid conducting fluid in the presence of surface tension and horizontal electric field. It is obtained the local-in-time estimates on the interface inand the velocity fields inby using the method introduced by Shatah and Zeng [A priori estimates for fluid interface problems. Commun Pure Appl Math. 2008;61(6):848–876]. The estimates of a horizontal electric field can be obtained from the estimates of the interface and elliptic estimates.

A priori estimates of the electrohydrodynamic waves with vorticity: horizontal electric field