2015/02/18 by Sijue Wu, Wu, Sijue
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #math.AP #physics.flu-dyn
paper · pdf · doi:10.48550/arxiv.1502.05342
arxiv created 2015/02/18 · arxiv updated 2015/02/19
We consider the two dimensional gravity water wave equation in the regime that includes free surfaces with angled crests. We assume that the fluid is inviscid, incompressible and irrotational, the air density is zero, and we neglect the surface tension. In \citekw it was shown that in this regime, only a degenerate Taylor inequality -\frac∂ P∂\boldn≥ 0 holds, with degeneracy at the singularities; an energy functional \frak E was constructed and an aprori estimate was proved. In this paper we show that a (generalized) solution of the water wave equation with smooth data will remain smooth so long as \frak E(t) remains finite; and for any data satisfying \frak E(0)<∞, the equation is solvable locally in time, for a period depending only on \frak E(0).