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

We consider the three-dimensional incompressible free boundary Euler equations in a bounded domain and with surface tension. Using Lagrangian coordinates, we establish a priori estimates for soluti...

[1]  D. Coutand.  Finite-Time Singularity Formation for Incompressible Euler Moving Interfaces in the Plane , 2018, Archive for Rational Mechanics and Analysis.

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

[3]  I. Kukavica,et al.  A priori estimates for the free-boundary Euler equations with surface tension in three dimensions , 2017, Nonlinearity.

[4]  Gustavo Ponce,et al.  Well-Posedness of the Euler and Navier-Stokes Equations in the Lebesgue Spaces $L^p_s(\mathbb R^2)$ , 1986 .

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

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

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

[8]  T. Alazard,et al.  The Water-Wave Equations: From Zakharov to Euler , 2012, 1212.0632.

[9]  I. Kukavica,et al.  On the Local Existence and Uniqueness for the 3D Euler Equation with a Free Interface , 2017 .

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

[11]  I. Kukavica,et al.  A Regularity Result for the Incompressible Euler Equation with a Free Interface , 2014 .

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

[13]  I. Kukavica,et al.  ON THE 2D FREE BOUNDARY EULER EQUATION , 2012 .

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

[15]  Y. Deng,et al.  Global solutions of the gravity-capillary water wave system in 3 dimensions , 2016, 1601.05685.

[16]  M. Disconzi.  On a linear problem arising in dynamic boundaries , 2014, 1405.6954.

[17]  Thomas Alazard,et al.  On the Cauchy problem for gravity water waves , 2012, 1212.0626.

[18]  I. Kukavica,et al.  Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces , 2016, Advances in Differential Equations.

[19]  Igor Kukavica,et al.  On the local existence of the free-surface Euler equation with surface tension , 2016, Asymptot. Anal..

[20]  Sijue Wu,et al.  On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary , 2015, 1511.00759.

[21]  D. Ebin.  The equations of motion of a perfect fluid with free boundary are not well posed. , 1987 .

[22]  W. Craig.  On the Hamiltonian for water waves , 2016, 1612.08971.

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

[24]  D. Ebin,et al.  On the Limit of Large Surface Tension for a Fluid Motion with Free Boundary , 2013, 1301.7507.

[25]  F. Pusateri.  On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations , 2009, 0912.3296.

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

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

[28]  S. Shkoller,et al.  A simple proof of well-posedness for the free-surfaceincompressible Euler equations , 2010 .

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

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

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

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

[33]  Jean Bourgain,et al.  Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces , 2013, 1307.7090.

[34]  Seungly Oh,et al.  The Kato-Ponce Inequality , 2013, 1303.5144.

[35]  L. E. Fraenkel,et al.  NAVIER-STOKES EQUATIONS (Chicago Lectures in Mathematics) , 1990 .

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

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

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

[39]  M. Tsutsumi,et al.  On the Generalized Korteweg-de Vries Equation , 1970 .

[40]  Nader Masmoudi,et al.  Global Existence for Capillary Water Waves , 2012, 1210.1601.

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

[42]  M. Ifrim,et al.  Two-dimensional gravity water waves with constant vorticity, I: Cubic lifespan , 2015, Analysis & PDE.

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

[44]  Thomas Y. Hou,et al.  Growth rates for the linearized motion of fluid interfaces away from equilibrium , 1993 .

[45]  P. Germain,et al.  Global solutions for the gravity water waves equation in dimension 3 , 2009, 0906.5343.

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

[47]  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.

[48]  Pietro Baldi,et al.  Gravity Capillary Standing Water Waves , 2014, 1405.1934.

[49]  C. Fefferman,et al.  Finite time singularities for water waves with surface tension , 2012, 1204.6633.

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

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

[52]  Well-Posedness for the Linearized Motion of a Compressible Liquid with Free Surface Boundary , 2001, math/0112030.

[53]  J. Marsden,et al.  Groups of diffeomorphisms and the motion of an incompressible fluid , 1970 .

[54]  A. Ionescu,et al.  Global Analysis of a Model for Capillary Water Waves in Two Dimensions , 2016 .

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

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

[57]  Haim Brezis,et al.  Remarks on the Euler equation , 1974 .

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

[59]  S. Shkoller,et al.  On the motion of vortex sheets with surface tension in three‐dimensional Euler equations with vorticity , 2008 .

[60]  Solvability and Regularity for an Elliptic System Prescribing the Curl, Divergence, and Partial Trace of a Vector Field on Sobolev-Class Domains , 2014, 1408.2469.

[61]  A. Ionescu,et al.  Global Regularity for 2d Water Waves with Surface Tension , 2014, Memoirs of the American Mathematical Society.

[62]  D. Ebin,et al.  The free boundary Euler equations with large surface tension , 2015, 1506.02094.

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

[64]  Luis Vega,et al.  Well-posedness of the initial value problem for the Korteweg-de Vries equation , 1991 .

[65]  C. Fefferman,et al.  On the absence of splash singularities in the case of two-fluid interfaces , 2013, 1312.2917.

[66]  Karl Håkan Nordgren,et al.  A PRIORI ESTIMATES FOR THE MOTION OF A SELF-GRAVITATING INCOMPRESSIBLE LIQUID WITH FREE SURFACE BOUNDARY , 2008, 0810.4517.

[67]  Stabilization of the Water-Wave Equations with Surface Tension , 2016, 1610.07917.

[68]  David M. Ambrose,et al.  Well-Posedness of Vortex Sheets with Surface Tension , 2003, SIAM J. Math. Anal..