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Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity. (English) Zbl 1428.35438

Summary: In this paper, we study solitary waves propagating along the surface of an infinitely deep body of water in two or three dimensions. The waves are acted upon by gravity and capillary effects are allowed – but not required – on the interface. We assume that the vorticity is localized in the sense that it satisfies certain moment conditions, and we permit there to be finitely many point vortices in the bulk of the fluid in two dimensions. We also consider a two-fluid model with a vortex sheet. Under mild decay assumptions, we obtain precise asymptotics for the velocity field and free surface, and relate this to global properties of the wave. For instance, we rule out the existence of waves whose free surface elevations have a single sign and of vortex sheets with finite angular momentum. Building on the work of J. Shatah et al. [Nonlinearity 26, No. 6, 1529–1564 (2013; Zbl 1277.35280)], we also prove the existence of families of two-dimensional capillary-gravity waves with compactly supported vorticity satisfying the above assumptions. For these waves, we further show that the free surface is positive in a neighborhood of infinity, and that the asymptotics at infinity are linked to the net vorticity.

MSC:

35Q51 Soliton equations
76B25 Solitary waves for incompressible inviscid fluids
35Q31 Euler equations
76V05 Reaction effects in flows
76B55 Internal waves for incompressible inviscid fluids
35J25 Boundary value problems for second-order elliptic equations
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B45 Capillarity (surface tension) for incompressible inviscid fluids
35B40 Asymptotic behavior of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35B65 Smoothness and regularity of solutions to PDEs
35R35 Free boundary problems for PDEs

Citations:

Zbl 1277.35280
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References:

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