Local well-posedness of the capillary-gravity water waves with acute contact angles

We consider the two-dimensional capillary-gravity water waves problem where the free surface Γt intersects the bottom Γb at two contact points. In our previous works [52, 53], the local wellposedness for this problem has been proved with the contact angles less than π/16. In this paper, we study the case where the contact angles belong to (0, π/2). It involves much worse singularities generated from corresponding elliptic systems, which have a strong influence on the regularities for the free surface and the velocity field. Combining the theory of singularity decompositions for elliptic problems with the structure of the water waves system, we obtain a priori energy estimates. Based on these estimates, we also prove the local well-posedness of the solutions in a geometric formulation.

[1]  Thomas Alazard,et al.  On the water-wave equations with surface tension , 2009, 0906.4406.

[2]  Nader Masmoudi,et al.  The zero surface tension limit of three-dimensional water waves , 2009 .

[3]  Masao Ogawa,et al.  FREE BOUNDARY PROBLEM FOR AN INCOMPRESSIBLE IDEAL FLUID WITH SURFACE TENSION , 2002 .

[4]  Jalal Shatah,et al.  Geometry and a priori estimates for free boundary problems of the Euler's equation , 2006 .

[5]  Yan Guo,et al.  Stability of contact lines in fluids: 2D Navier–Stokes flow , 2020, Journal of the European Mathematical Society.

[6]  J. K. Hunter,et al.  Two Dimensional Water Waves in Holomorphic Coordinates , 2014, 1401.1252.

[7]  Serge Nicaise,et al.  Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III: finite element methods on polygonal domains , 1992 .

[8]  Jacek Banasiak,et al.  On Mixed Boundary Value Problems of Dirichlet Oblique-Derivative Type in Plane Domains with Piecewise Differentiable Boundary , 1989 .

[9]  T. Alazard,et al.  Strichartz Estimates for Water Waves , 2010, 1002.0323.

[10]  V. Kondratiev,et al.  Elliptic Boundary Value Problems of Second Order in Piecewise Smooth Domains , 2006 .

[11]  Hans Lindblad.  Well-posedness for the motion of an incompressible liquid with free surface boundary , 2005 .

[12]  J. Roßmann,et al.  Elliptic Boundary Value Problems in Domains with Point Singularities , 2002 .

[13]  George Gabriel Stokes,et al.  On the theory of oscillatory waves , 2009 .

[14]  Ben Schweizer,et al.  On the three-dimensional Euler equations with a free boundary subject to surface tension , 2005 .

[15]  Chao Wang,et al.  Water Waves Problem with Surface Tension in a Corner Domain I: A Priori Estimates with Constrained Contact Angle , 2017, SIAM J. Math. Anal..

[16]  Thomas Alazard,et al.  Cauchy theory for the gravity water waves system with non localized initial data , 2013, 1305.0457.

[17]  Yu Deng,et al.  Global solutions of the gravity-capillary water-wave system in three dimensions , 2016 .

[18]  R. H. Kinsey,et al.  A Priori Estimates for Two-Dimensional Water Waves with Angled Crests , 2014, 1406.7573.

[19]  N. Masmoudi,et al.  Darcy’s Flow with Prescribed Contact Angle: Well-Posedness and Lubrication Approximation , 2012, 1204.2278.

[20]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 2-D , 1997 .

[21]  D. Lannes,et al.  The shoreline problem for the one-dimensional shallow water and Green-Naghdi equations , 2017, 1710.03651.

[22]  S. Agrawal.  Rigidity of singularities of 2D gravity water waves , 2018, Journal of Differential Equations.

[23]  Nader Masmoudi,et al.  The zero surface tension limit two‐dimensional water waves , 2005 .

[24]  David Lannes,et al.  Well-posedness of the water-waves equations , 2005 .

[25]  Nastasia Grubic,et al.  Local wellposedness for the free boundary incompressible Euler equations with interfaces that exhibit cusps and corners of nonconstant angle , 2021, 2107.09751.

[26]  Walter Craig,et al.  An existence theory for water waves and the boussinesq and korteweg-devries scaling limits , 1985 .

[27]  M. Ifrim,et al.  Two dimensional water waves in holomorphic coordinates II: global solutions , 2014, 1404.7583.

[28]  Well-Posedness and Uniform Bounds for a Nonlocal Third Order Evolution Operator on an Infinite Wedge , 2013 .

[29]  Xuecheng Wang.  Global Infinite Energy Solutions for the 2D Gravity Water Waves System , 2015, 1502.00687.

[30]  Chao Wang,et al.  Water‐Waves Problem with Surface Tension in a Corner Domain II: The Local Well‐Posedness , 2018, Communications on Pure and Applied Mathematics.

[31]  T. Levi-Civita,et al.  Détermination rigoureuse des ondes permanentes d'ampleur finie , 1925 .

[32]  T. Poyferr'e.  A Priori Estimates for Water Waves with Emerging Bottom , 2016, 1612.04103.

[33]  Zhifei Zhang,et al.  Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary , 2015, Memoirs of the American Mathematical Society.

[34]  G. Amberg,et al.  Modeling of dynamic wetting far from equilibrium , 2009 .

[35]  Daniel Coutand,et al.  Well-posedness of the free-surface incompressible Euler equations with or without surface tension , 2005 .

[36]  Klaus Beyer,et al.  On the Cauchy problem for a capillary drop. Part I: irrotational motion , 1998 .

[37]  Sijue Wu,et al.  Almost global wellposedness of the 2-D full water wave problem , 2009, 0910.2473.

[38]  Fabio Pusateri,et al.  Global solutions for the gravity water waves system in 2d , 2013, Inventiones mathematicae.

[39]  Ping Zhang,et al.  On the free boundary problem of three‐dimensional incompressible Euler equations , 2008 .

[40]  Thomas Alazard,et al.  Global solutions and asymptotic behavior for two dimensional gravity water waves , 2013, 1305.4090.

[41]  Thomas Young,et al.  An Essay on the Cohesion of Fluids , 1800 .

[42]  M. Günther,et al.  The Jacobi equation for irrotational free boundary flows , 2000 .

[43]  J. Shatah,et al.  Local Well-Posedness for Fluid Interface Problems , 2011 .

[44]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[45]  Charles Fefferman,et al.  Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves , 2011, 1102.1902.

[46]  Bartłomiej Dyda,et al.  A fractional order Hardy inequality , 2004 .

[47]  B. Andreotti,et al.  Moving Contact Lines: Scales, Regimes, and Dynamical Transitions , 2013 .

[48]  C. Fefferman,et al.  Finite time singularities for the free boundary incompressible Euler equations , 2011, 1112.2170.

[49]  Sijue Wu,et al.  Global wellposedness of the 3-D full water wave problem , 2011 .

[50]  D. Christodoulou,et al.  S E M I N A I R E E quations aux , 2008 .

[51]  Sijue Wu,et al.  A blow-up criteria and the existence of 2d gravity water waves with angled crests , 2015, 1502.05342.

[52]  Hideaki Yosihara,et al.  Gravity Waves on the Free Surface of an Incompressible Perfect Fluid of Finite Depth , 1982 .

[53]  J.-F. Gerbeau,et al.  Generalized Navier boundary condition and geometric conservation law for surface tension , 2008, 0804.1563.

[54]  E. Weinan,et al.  Boundary Conditions for the Moving Contact Line Problem , 2007 .

[55]  Yan Guo,et al.  Stability of Contact Lines in Fluids: 2D Stokes Flow , 2016, 1603.03721.

[56]  S. Shkoller,et al.  On the Finite-Time Splash and Splat Singularities for the 3-D Free-Surface Euler Equations , 2012, 1201.4919.

[57]  P. Grisvard.  Espaces intermédiaires entre espaces de Sobolev avec poids , 1963 .

[58]  Mei Ming,et al.  Well-posedness of the water-wave problem with surface tension , 2008, 0806.2433.

[59]  D. Lannes,et al.  Hyperbolic free boundary problems and applications to wave-structure interactions , 2018, Indiana University Mathematics Journal.

[60]  D. Bonn,et al.  Wetting and Spreading , 2009 .

[61]  David Lannes,et al.  On the Dynamics of Floating Structures , 2016, 1609.06136.

[62]  Sijue Wu,et al.  Well-posedness in Sobolev spaces of the full water wave problem in 3-D , 1999 .

[63]  Hideaki Yosihara.  Capillary-gravity waves for an incompressible ideal fluid , 1983 .

[64]  M. Ifrim,et al.  The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves , 2014, 1406.5471.