Asymptotic properties of deep water solitary waves with compactly supported vorticity

Abstract

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] In this thesis, we consider two-and three-dimensional gravity and capillary-gravity solitary waves propagating along the surface of a body of water with infinite depth. The waves are acted upon by gravity and we allow the effect of surface tension but do not require it. In addition, we assume that the flow is only rotational on some compact set inscribed in the domain of the fluid. Under some decay assumptions, we show that the velocity potential behaves like a dipole at infinity and obtain asymptotics form of the velocity potential and the free surface. We manage to rule out waves of pure elevation and depression. We also exhibit an equation relating the total momentum in the Cauchy principal value-sense with the dipole moment. This equation combined with the theorem on angular-momentum tells us that finite angular momentum implies zero total momentum in the Cauchy principal value sense. Moreover, if the horizontal component of the velocity field of a wave is non-zero, then such wave has an in finite angular momentum.