Results 1 
7 of
7
Steady periodic water waves with constant vorticity: regularity and local
, 2011
"... Abstract. This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the freeboundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The ne ..."
Abstract

Cited by 23 (4 self)
 Add to MetaCart
(Show Context)
Abstract. This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the freeboundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.
On the existence of extreme waves and the Stokes conjecture with vorticity
 J. Differential Equations
"... This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
This is a study of singular solutions of the problem of traveling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a corner of 120 ◦ or a horizontal tangent at any stagnation point about which it is supposed symmetric. Moreover, the profile necessarily has a corner of 120 ◦ if the vorticity is nonnegative near the free surface. 1
On some properties of travelling water waves with vorticity
 SIAM J. Math. Anal
"... We prove that for a large class of vorticity functions the crest of a corresponding travelling water wave is necessarily a point of maximal horizontal velocity. We also show that for waves with nonpositive vorticity the pressure in the flow is everywhere larger than the atmospheric pressure. A relat ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
We prove that for a large class of vorticity functions the crest of a corresponding travelling water wave is necessarily a point of maximal horizontal velocity. We also show that for waves with nonpositive vorticity the pressure in the flow is everywhere larger than the atmospheric pressure. A related a priori estimate for waves with nonnegative vorticity is also given. 1
ABCEB/E7EE#"EE$D BEDEDE%CB E BEE89:EEDEA"E$ E"E: CBE!E D E" CE!E8" CEDE$:EBEDEA"ECEBEECE " CECDBE4BEDEDBEE BEECBEEEDDEE
"... On some properties of traveling water waves with vorticity ABCDEFB ..."
(Show Context)
A geometric approach to . . .
, 2009
"... We consider Stokes’ conjecture concerning the shape of the extremal twodimensional water wave. By new geometric methods including a nonlinear frequency formula, we prove Stokes’ conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such a ..."
Abstract
 Add to MetaCart
We consider Stokes’ conjecture concerning the shape of the extremal twodimensional water wave. By new geometric methods including a nonlinear frequency formula, we prove Stokes’ conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.