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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP

Well-posedness of electrohydrodynamic waves under vertical electric field Jiaqi Yang

Abstract. This paper is devoted to the problem of wave propagation on a perfect conducting fluid under a vertical electric field, a kind of electrohydrodynamic waves. Three competing restoring forces resulting from gravity, surface tension, and normal electric field are taken into account. There are many numerical and experimental results on electrohydrodynamic waves, but the system’s well-posedness, which is a fundamental problem, has not been investigated in the past. We shall show the system considered is locally well posed based on energy estimates in proper Sobolev spaces and careful examinations of the Dirichlet–Neumann operators. Mathematics Subject Classification. Primary 35Q35, Secondary 76W05. Keywords. Local well-posedness, Electrohydrodynamic waves, Vertical electric field.

1. Introduction 1.1. Formulation In this paper, the evolution of the free surface of a conducting fluid in the presence of gravity, surface tension, and a vertical electric field is considered (see [11,22]). To be exact, we consider a fluid bounded below by a wall electrode at z = −h and bounded above by a free surface at z = η(t, X) (X = (X1 , . . . , Xd ), d = 1, 2), where h is the mean depth of the fluid. The region {(X, z) ∈ Rd : z > η(t, X)} is occupied by a hydrodynamically passive dielectric with permittivity p . We assume that there are no free charges or currents in {(X, z) ∈ Rd : z > η(t, X)}, so the electric field can be represented as a gradient of a potential function, E = −∇X,z V with ∇X,z = (∂X1 , . . . ∂Xd , ∂z ) . A vertical electrical field is imposed by requiring that V → E0 z as z → +∞, where E0 is a constant. Therefore, the voltage potential satisfies the Laplace equation, that is, ΔX,z V = 0. In our case, due to the existence of the electrical field, the Bernoulli equation on the free surface becomes σ 1 1 φt + (|∇φ|2 + φ2z ) + gη − n · Σ · n = − κ , 2 ρ ρ see [11,22], where ∇ = (∂X1 , . . . , ∂Xd ) is the gradient operator in the horizontal coordinates, and   1 2 Σij = p Ei Ej − |E| δij 2 (−∇η,1) is the Maxwell stress tensor, n = √



1+|∇η|2

is the unit normal vector pointing upward, ρ is the density of

the fluid, g accountsfor the gravitational acceleration, σ represents the surface tension coefficient of the  ∇η liquid, and κ = ∇ · √ denotes the mean curvature of the free surface. 2 1+|∇η|

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J. Yang

ZAMP

Now, from the above analyses, the governing system can be written as follows: ⎧ for − h < z < η(t, X), ⎪ ⎪ΔX,z φ = 0, ⎪ ⎪ ⎪ V = 0, for z > η(t, X), Δ ⎪ X,z ⎪ ⎪ ⎪ ⎪ ηt = φz − ∇η · ∇φ, ⎪ ⎪   at z = η(t, X), ⎨ φt + 12 (|∇φ|2 + φ2z ) + gη − ρ1 n · Σ · n = σρ ∇ · √ ∇η 2 , at z = η(t, X), 1+|∇η| ⎪ ⎪ ⎪ ⎪ ⎪ V = 0, at z = η(t, X), ⎪ ⎪ ⎪ ⎪ ⎪ = 0, at z = −h, φ ⎪ ⎪ z ⎩ as z → ∞, V → E0 z,

(1.1)

2 2 where ΔX,z := ∂X + · · · + ∂X + ∂z2 . We remark that, at the free surface z = η(t, X), noting that 1 d

∂Xi V + ∂Xi η∂z V = 0,

i = 1, . . . , d ,

one can easi