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The zero surface tension limit two-dimensional water waves. (English) Zbl 1086.76004

Summary: We consider two-dimensional water waves of infinite depth, periodic in the horizontal direction. It has been proven by S. Wu [Invent. Math. 130, No. 1, 39–72 (1997; Zbl 0892.76009)] (in the slightly different nonperiodic setting) that solutions to this initial value problem exist in the absence of surface tension. Recently D. M. Ambrose [SIAM J. Math. Anal. 35, No. 1, 211–244 (2003; Zbl 1107.76010)] has proven that solutions exist when surface tension is taken into account. In this paper, we provide a shorter, more elementary proof of existence of solutions to the water wave initial value problem both with and without surface tension. Our proof requires estimating the growth of geometric quantities using a renormalized arc length parametrization of the free surface and using physical quantities related to the tangential velocity of the free surface. Using this formulation, we find that as surface tension goes to 0, the water wave without surface tension is the limit of the water wave with surface tension. Far from being a simple adaptation of previous works, our method requires a very original choice of variables; these variables turn out to be physical and well adapted to both cases.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45 Capillarity (surface tension) for incompressible inviscid fluids
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q35 PDEs in connection with fluid mechanics
35B10 Periodic solutions to PDEs
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