Well-posedness of the Water-waves Equations
نویسنده
چکیده
1.1. Presentation of the problem. The water-waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect, incompressible, irrotational fluid under the influence of gravity. In this paper, we restrict our attention to the case when the surface is a graph parameterized by a function ζ(t,X), where t denotes the time variable and X = (X1, . . . , Xd) ∈ R the horizontal spatial variables. The method developed here works equally well for any integer d ≥ 1, but the only physically relevant cases are of course d = 1 and d = 2. The layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time-independent function b(X). We denote by Ωt the fluid domain at time t. The incompressibility of the fluids is expressed by
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