Low regularity Cauchy theory for the water-waves problem: canals and swimming pools

A Lagrangian Interior Regularity Result for the Incompressible Free Boundary Euler Equation with Surface Tension

“…The problem of well-posedness under the Rayleigh-Taylor condition and in the case of zero surface tension was solved by Wu [74,75]. Regarding optimal regularity of the initial data, Wang et al obtained in [73] the local existence under the sharp Sobolev regularity H 2.5+δ for the zero surface tension case, extending the previous result of Alazard et al [6], who considered irrotational data. For the Euler equations in R 2 or R 3 , the sharpness of the exponent 2.5 + δ was shown in [15].…”

“…Free-boundary problems constitute a very active and fast-growing area of research, and a complete, or even thorough review of prior works is beyond the scope of this paper. A partial list of references relevant to the above discussion and the results of this paper is [1,7,8,9,10,6,14,18,19,20,21,23,28,26,29,30,40,43,44,46,49,50,56,57,58,60,61,62,63,64,67,76,77].…”