We establish a priori estimates for local-in-time existence of solutions to the water wave model consisting of the 3D incompressible Euler equations on a domain with a free surface, without surface tension, under minimal regularity assumptions on the initial data and the Rayleigh–Taylor sign condition. The initial data are allowed to be rotational and they are assumed to belong to , where is arbitrary.
T.Alazard, N.Burq and C.Zuily, On the water-wave equations with surface tension, Duke Math. J.158(3) (2011), 413–499. doi:10.1215/00127094-1345653.
2.
T.Alazard, N.Burq and C.Zuily, Low regularity Cauchy theory for the water-waves problem: Canals and wave pools, in: Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 3, Morningside Lect. Math., Vol. 3, Int. Press, Somerville, MA, 2013, pp. 1–42.
3.
T.Alazard and J.-M.Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér. (4)48(5) (2015), 1149–1238. doi:10.24033/asens.2268.
4.
D.M.Ambrose and N.Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math.58(10) (2005), 1287–1315. doi:10.1002/cpa.20085.
5.
D.M.Ambrose and N.Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J.58(2) (2009), 479–521. doi:10.1512/iumj.2009.58.3450.
6.
J.T.Beale, The initial value problem for the Navier–Stokes equations with a free surface, Comm. Pure Appl. Math.34(3) (1981), 359–392. doi:10.1002/cpa.3160340305.
7.
J.T.Beale, T.Y.Hou and J.S.Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math.46(9) (1993), 1269–1301. doi:10.1002/cpa.3160460903.
8.
M.Bukač, S.Čanić, R.Glowinski, B.Muha and A.Quaini, A modular, operator-splitting scheme for fluid-structure interaction problems with thick structures, Internat. J. Numer. Methods Fluids74(8) (2014), 577–604. doi:10.1002/fld.3863.
9.
A.Castro, D.Córdoba, C.L.Fefferman, F.Gancedo and J.Gómez-Serrano, Splash singularity for water waves, Proc. Natl. Acad. Sci. USA109(3) (2012), 733–738. doi:10.1073/pnas.1115948108.
10.
A.Castro and D.Lannes, Well-posedness and shallow-water stability for a new Hamiltonian formulation of the water waves equations with vorticity, Indiana Univ. Math. J.64(4) (2015), 1169–1270. doi:10.1512/iumj.2015.64.5606.
D.Coutand and S.Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc.20(3) (2007), 829–930. doi:10.1090/S0894-0347-07-00556-5.
13.
D.Coutand and S.Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser3 (2010), 429–449. doi:10.3934/dcdss.2010.3.429.
14.
M.M.Disconzi and D.G.Ebin, On the limit of large surface tension for a fluid motion with free boundary, Comm. Partial Differential Equations39(4) (2014), 740–779. doi:10.1080/03605302.2013.865058.
15.
M.M.Disconzi and D.G.Ebin, The free boundary Euler equations with large surface tension, J. Differential Equations261(2) (2016), 821–889. doi:10.1016/j.jde.2016.03.029.
16.
M.M.Disconzi and I.Kukavica, A priori estimates for the free-boundary Euler equations with surface tension in three dimensions, (submitted).
17.
D.G.Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations12(10) (1987), 1175–1201. doi:10.1080/03605308708820523.
18.
T.Elgindi and D.Lee, Uniform regularity for free-boundary Navier–Stokes equations with surface tension, 2014, arXiv:1403.0980.
19.
P.Germain, N.Masmoudi and J.Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2)175(2) (2012), 691–754. doi:10.4007/annals.2012.175.2.6.
20.
J.K.Hunter, M.Ifrim and D.Tataru, Two dimensional water waves in holomorphic coordinates, Comm. Math. Phys.346(2) (2016), 483–552. doi:10.1007/s00220-016-2708-6.
21.
M.Ifrim and D.Tataru, Two dimensional water waves in holomorphic coordinates II: Global solutions, Bull. Soc. Math. France144(2) (2016), 369–394. doi:10.24033/bsmf.2717.
22.
M.Ignatova and I.Kukavica, On the local existence of the free-surface Euler equation with surface tension, Asymptot. Anal.100(1–2) (2016), 63–86. doi:10.3233/ASY-161386.
23.
T.Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac.44(2) (2001), 219–241.
24.
A.D.Ionescu and F.Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math.199(3) (2015), 653–804. doi:10.1007/s00222-014-0521-4.
25.
M.Iowa and A.Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci.12(12) (2002), 1725–1740. doi:10.1142/S0218202502002306.
26.
T.Kato and G.Ponce, Commutator estimates and the Euler and Navier–Stokes equations, Comm. Pure Appl. Math.41(7) (1988), 891–907. doi:10.1002/cpa.3160410704.
27.
I.Kukavica and A.Tuffaha, On the 2D free boundary Euler equation, Evol. Equ. Control Theory1(2) (2012), 297–314. doi:10.3934/eect.2012.1.297.
28.
I.Kukavica and A.Tuffaha, Well-posedness for the compressible Navier–Stokes–Lamé system with a free interface, Nonlinearity25(11) (2012), 3111–3137. doi:10.1088/0951-7715/25/11/3111.
29.
I.Kukavica and A.Tuffaha, A regularity result for the incompressible Euler equation with a free interface, Appl. Math. Optim.69(3) (2014), 337–358. doi:10.1007/s00245-013-9221-5.
30.
I.Kukavica, A.Tuffaha and V.Vicol, On the local existence and uniqueness for the 3D Euler equation with a free interface, Appl. Math. Optim. (2016). doi:10.1007/s00245-016-9360-6.
31.
I.Kukavica, A.Tuffaha, V.Vicol and F.Wang, On the existence for the free interface 2D Euler equation with a localized vorticity condition, Appl. Math. Optim.73(3) (2016), 523–544. doi:10.1007/s00245-016-9346-4.
32.
D.Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc.18(3) (2005), 605–654, (electronic). doi:10.1090/S0894-0347-05-00484-4.
33.
H.Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math.56(2) (2003), 153–197. doi:10.1002/cpa.10055.
34.
H.Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2)162(1) (2005), 109–194. doi:10.4007/annals.2005.162.109.
35.
N.Masmoudi and F.Rousset, Uniform regularity and vanishing viscosity limit for the free surface Navier–Stokes equations, Arch. Ration. Mech. Anal.223(1) (2017), 301–417. doi:10.1007/s00205-016-1036-5.
36.
B.Muha and S.Čanić, Existence of a weak solution to a nonlinear fluid-structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch. Ration. Mech. Anal.207(3) (2013), 919–968. doi:10.1007/s00205-012-0585-5.
37.
B.Muha and S.Čanić, A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: A constructive existence proof, Commun. Inf. Syst.13(3) (2013), 357–397.
38.
B.Muha and S.Čanić, Fluid-structure interaction between an incompressible, viscous 3D fluid and an elastic shell with nonlinear Koiter membrane energy, Interfaces Free Bound.17(4) (2015), 465–495. doi:10.4171/IFB/350.
39.
B.Muha and S.Čanić, Existence of a weak solution to a fluid-elastic structure interaction problem with the Navier slip boundary condition, J. Differential Equations260(12) (2016), 8550–8589. doi:10.1016/j.jde.2016.02.029.
40.
V.I.Nalimov, The Cauchy–Poisson problem, Dinamika Splošn. Sredy18 (1974), Dinamika Zidkost. so Svobod. Granicami, 104–210, 254.
41.
F.Pusateri, On the limit as the surface tension and density ratio tend to zero for the two-phase Euler equations, J. Hyperbolic Differ. Equ.8(2) (2011), 347–373. doi:10.1142/S021989161100241X.
42.
B.Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire22(6) (2005), 753–781. doi:10.1016/j.anihpc.2004.11.001.
43.
J.Shatah and C.Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math.61(5) (2008), 698–744. doi:10.1002/cpa.20213.
44.
J.Shatah and C.Zeng, Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal.199(2) (2011), 653–705. doi:10.1007/s00205-010-0335-5.
45.
M.Shinbrot, The initial value problem for surface waves under gravity. I. The simplest case, Indiana Univ. Math. J.25(3) (1976), 281–300. doi:10.1512/iumj.1976.25.25023.
46.
A.I.Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N. S.)128(170) (1985), 82–109, 144.
47.
A.Tani, Small-time existence for the three-dimensional Navier–Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal.133(4) (1996), 299–331. doi:10.1007/BF00375146.
48.
C.Wang, Z.Zhang, W.Zhao and Y.Zheng, Local well-posedness and break-down criterion of the incompressible Euler equations with free boundary, 2015, arXiv:1507.02478.
49.
S.Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math.130(1) (1997), 39–72. doi:10.1007/s002220050177.
50.
S.Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc.12(2) (1999), 445–495. doi:10.1090/S0894-0347-99-00290-8.
51.
S.Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math.184(1) (2011), 125–220. doi:10.1007/s00222-010-0288-1.
52.
L.Xu and Z.Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations248(5) (2010), 1044–1111. doi:10.1016/j.jde.2009.11.001.
53.
H.Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci.18(1) (1982), 49–96. doi:10.2977/prims/1195184016.
54.
H.Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ.23(4) (1983), 649–694. doi:10.1215/kjm/1250521429.
55.
P.Zhang and Z.Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math.61(7) (2008), 877–940. doi:10.1002/cpa.20226.
