| Stokes Waves (2007) | |||||||||||||
Abstract | |||||||||||||
| this paper we give complete proofs of the main results about the Stokeswave problem starting with its basic formulation as a free boundary problem for a harmonic function in an unknown domain in the plane. At the outset only the natural smoothness required to state the problem is assumed. (Indeed the boundary is supposed only to be continuously differentiable, so that the usual form of the Hopf boundary-point lemma [22], [25], [49], is not available for the initial analysis; see however [25], page 46.) Lewy's theorem shows that the boundary must be a real-analytic curve and that the complex potential must have an analytic extension across the boundary. With such regularity up to the boundary the equations can be manipulated at will and various estimates on the wave slope and speed emerge from calculations involving the Maximum Principle for harmonic functions in the plane. These matters are dealt with in Sections 2, 3, 4 and 5, where the question of the existence of non-trivial Stokes waves is ignored. (Uniform horizontal flow with any speed c is a trivial solution of the free boundary-value problem.) To formulate the boundary-value problem in a way which is amenable to existence theory we follow Levi-Civita and Nekrasov who, in the 1920s, used a hodograph transformation to map the unknown domain occupied by the water into a fixed semi-infinite strip in a complex plane where the variable is the complex potential of the fluid flow. As a function of this new independent variable, the velocity field of the fluid is written in polar co-ordinates and the angular variable ` is regarded as an unknown harmonic function on the strip which satisfies nonlinear boundary conditions. In Levi-Civita's treatment the nonlinear boundary conditions involve both ` and its complex conjugate... | |||||||||||||
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