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September  2019, 8(3): 503-542. doi: 10.3934/eect.2019025

A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid

Marcelo M. Disconzi 1,, and  Igor Kukavica 2,

1. 

Department of Mathematics, Vanderbilt University, Nashville, TN 37240, USA
2. 

Department of Mathematics, University of Southern California, Los Angeles, CA 91107, USA

* Corresponding author: marcelo.disconzi@vanderbilt.edu

Received  April 2018 Revised  January 2019 Published  May 2019

Fund Project: The first author is partially supported by the NSF grant DMS-1812826, a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Discovery grant administered by Vanderbilt University. The second author is partially supported by the NSF grant DMS-1615239

We derive a priori estimates for the compressible free-boundary Euler equations with surface tension in three spatial dimensions in the case of a liquid. These are estimates for local existence in Lagrangian coordinates when the initial velocity and initial density belong to H3, with an extra regularity condition on the moving boundary, thus lowering the regularity of the initial data. Our methods are direct and involve two key elements: the boundary regularity provided by the mean curvature and a new compressible Cauchy invariance.

Keywords: Compressible Euler, free-boundary, surface tension, liquid, regularity.
Mathematics Subject Classification: Primary: 35L45; Secondary: 35L60.
Citation: Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025
References:
[1]

T. Alazard, About global existence and asymptotic behavior for two dimensional gravity water waves, in Séminaire Laurent Schwartz–-Équations aux dérivées partielles et applications. Année 2012–2013, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2014, Exp. No. ⅩⅧ, 16.  Google Scholar

[2]

T. Alazard, Stabilization of the water-wave equations with surface tension, Ann. PDE, 3 (2017), Art. 17, 41 pp. doi: 10.1007/s40818-017-0032-x.  Google Scholar

[3]

T. Alazard and P. Baldi, Gravity capillary standing water waves, Arch. Ration. Mech. Anal., 217 (2015), 741-830.  doi: 10.1007/s00205-015-0842-5.  Google Scholar

[4]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  doi: 10.1215/00127094-1345653.  Google Scholar

[5]

T. Alazard, N. Burq and C. Zuily, Strichartz estimates for water waves, Ann. Sci. Éc. Norm. Supér. (4), 44 (2011), 855–903. doi: 10.24033/asens.2156.  Google Scholar

[6]

T. Alazard, N. Burq and C. Zuily, The water-wave equations: From Zakharov to Euler, in Studies in Phase Space Analysis with Applications to PDEs, vol. 84 of Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, New York, 2013, 1–20. doi: 10.1007/978-1-4614-6348-1_1.  Google Scholar

[7]

T. Alazard, N. Burq and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math., 198 (2014), 71-163.  doi: 10.1007/s00222-014-0498-z.  Google Scholar

[8]

T. Alazard, N. Burq and C. Zuily, Cauchy theory for the gravity water waves system with non-localized initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 337-395.  doi: 10.1016/j.anihpc.2014.10.004.  Google Scholar

[9]

T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1149–1238. doi: 10.24033/asens.2268.  Google Scholar

[10]

T. Alazard and J.-M. Delort, Sobolev estimates for two dimensional gravity water waves, Astérisque, 374 (2015), viii+241.  Google Scholar

[11]

D. M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211–244 (electronic). doi: 10.1137/S0036141002403869.  Google Scholar

[12]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  doi: 10.1002/cpa.20085.  Google Scholar

[13]

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 (1993), 1269–1301, http://dx.doi.org/10.1002/cpa.3160460903. doi: 10.1002/cpa.3160460903.  Google Scholar

[14]

L. Bieri, S. Miao, S. Shahshahani and S. Wu, On the motion of a self-gravitating incompressible fluid with free boundary and constant vorticity: An appendix, arXiv: 1511.07483 [math.AP]. Google Scholar

[15]

L. Bieri, S. Miao, S. Shahshahani and S. Wu, On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary, Comm. Math. Phys., 355 (2017), 161–243, http://dx.doi.org/10.1007/s00220-017-2884-z. doi: 10.1007/s00220-017-2884-z.  Google Scholar

[16]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Finite time singularities for water waves with surface tension, J. Math. Phys., 53 (2012), 115622, 26pp. doi: 10.1063/1.4765339.  Google Scholar

[17]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. G´omez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2), 178 (2013), 1061–1134. doi: 10.4007/annals.2013.178.3.6.  Google Scholar

[18]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Structural stability for the splash singularities of the water waves problem, Discrete Contin. Dyn. Syst., 34 (2014), 4997–5043, http://dx.doi.org/10.3934/dcds.2014.34.4997. doi: 10.3934/dcds.2014.34.4997.  Google Scholar

[19]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), 909–948. doi: 10.4007/annals.2012.175.2.9.  Google Scholar

[20]

C.-H. A. Chen, D. Coutand and S. Shkoller, Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains, J. Math. Fluid Mech., 19 (2017), 375-422.  doi: 10.1007/s00021-016-0289-y.  Google Scholar

[21]

G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369–408, http://dx.doi.org/10.1007/s00205-007-0070-8. doi: 10.1007/s00205-007-0070-8.  Google Scholar

[22]

C.-H. Cheng and S. Shkoller, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity, Comm. Pure Appl. Math., 61 (2008), 1715-1752.  doi: 10.1002/cpa.20240.  Google Scholar

[23]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.  Google Scholar

[24]

J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 85–139. doi: 10.24033/asens.2064.  Google Scholar

[25]

J.-F. Coulombel and P. Secchi, Uniqueness of 2-D compressible vortex sheets, Commun. Pure Appl. Anal., 8 (2009), 1439–1450, http://dx.doi.org/10.3934/cpaa.2009.8.1439. doi: 10.3934/cpaa.2009.8.1439.  Google Scholar

[26]

D. Coutand, Finite time singularity formation for moving interface Euler equations, arXiv: 1701.01699 [math.AP], 43 pages. Google Scholar

[27]

D. Coutand, J. Hole and S. Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45 (2013), 3690-3767.  doi: 10.1137/120888697.  Google Scholar

[28]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

[29]

D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.  doi: 10.1090/S0894-0347-07-00556-5.  Google Scholar

[30]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429–449, http://dx.doi.org/10.3934/dcdss.2010.3.429. doi: 10.3934/dcdss.2010.3.429.  Google Scholar

[31]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344.  Google Scholar

[32]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[33]

D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325 (2014), 143-183.  doi: 10.1007/s00220-013-1855-2.  Google Scholar

[34]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003.  doi: 10.1080/03605308508820396.  Google Scholar

[35]

W. Craig, On the Hamiltonian for water waves, arXiv: 1612.08971 [math.AP], 10 pages. Google Scholar

[36]

T. de Poyferré, A priori estimates for water waves with emerging bottom, arXiv: 1612.04103 [math.AP], 45 pages. Google Scholar

[37]

Y. Deng, A. D. Ionescu, B. Pausader and F. Pusateri, Global solutions of the gravity-capillary water wave system in 3 dimensions, Acta Math., 219 (2017), 213–402, arXiv: 1601.05685 [math.AP]. doi: 10.4310/ACTA.2017.v219.n2.a1.  Google Scholar

[38]

M. M. Disconzi, On a linear problem arising in dynamic boundaries, Evol. Equ. Control Theory, 3 (2014), 627-644.  doi: 10.3934/eect.2014.3.627.  Google Scholar

[39]

M. M. Disconzi and D. G. Ebin, On the limit of large surface tension for a fluid motion with free boundary, Comm. Partial Differential Equations, 39 (2014), 740-779.  doi: 10.1080/03605302.2013.865058.  Google Scholar

[40]

M. M. Disconzi and D. G. Ebin, The free boundary Euler equations with large surface tension, Journal of Differential Equations, 261 (2016), 821-889.  doi: 10.1016/j.jde.2016.03.029.  Google Scholar

[41]

M. M. Disconzi and D. G. Ebin, Motion of slightly compressible fluids in a bounded domain, Ⅱ, Commun. Contemp. Math., 19 (2017), 1650054, 57pp. doi: 10.1142/S0219199716500541.  Google Scholar

[42]

M. M. Disconzi and I. Kukavica, A priori estimates for the free-boundary Euler equations with surface tension in three dimensions, arXiv: 1708.00086 [math.AP], 40 pages. Google Scholar

[43]

H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3.  Google Scholar

[44]

S. Ebenfeld, $L^2 $-regularity theory of linear strongly elliptic Dirichlet systems of order $ 2m$ with minimal regularity in the coefficients, Quart. Appl. Math., 60 (2002), 547-576.  doi: 10.1090/qam/1914441.  Google Scholar

[45]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175–1201, http://dx.doi.org/10.1080/03605308708820523. doi: 10.1080/03605308708820523.  Google Scholar

[46]

L. C. Evans, Partial Differential Equations, American Mathematical Society (2nd edition), 2010. doi: 10.1090/gsm/019.  Google Scholar

[47]

C. Fefferman, A. D. Ionescu and V. Lie, On the absence of splash singularities in the case of two-fluid interfaces, Duke Math. J., 165 (2016), 417-462.  doi: 10.1215/00127094-3166629.  Google Scholar

[48]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691–754, http://dx.doi.org/10.4007/annals.2012.175.2.6. doi: 10.4007/annals.2012.175.2.6.  Google Scholar

[49]

P. Germain, N. Masmoudi and J. Shatah, Global existence for capillary water waves, Comm. Pure Appl. Math., 68 (2015), 625-687.  doi: 10.1002/cpa.21535.  Google Scholar

[50]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, arXiv: 1511.07467 [math.AP]. Google Scholar

[51]

Q. Han and J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, vol. 130 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/130.  Google Scholar

[52]

J. K. Hunter, M. Ifrim and D. Tataru, Two dimensional water waves in holomorphic coordinates, Comm. Math. Phys., 346 (2016), 483–552, http://dx.doi.org/10.1007/s00220-016-2708-6. doi: 10.1007/s00220-016-2708-6.  Google Scholar

[53]

M. Ifrim and D. Tataru, Two dimensional gravity water waves with constant vorticity: Ⅰ. cubic lifespan, Analysis and PDE, 12 (2019), 903–967, arXiv: 1510.07732 [math.AP]. doi: 10.2140/apde.2019.12.903.  Google Scholar

[54]

M. Ifrim and D. Tataru, Two dimensional water waves in holomorphic coordinates Ⅱ: Global solutions, Bull. Soc. Math. France, 144 (2016), 369-394.  doi: 10.24033/bsmf.2717.  Google Scholar

[55]

M. Ifrim and D. Tataru, The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves, Arch. Ration. Mech. Anal., 225 (2017), 1279–1346, http://dx.doi.org/10.1007/s00205-017-1126-z. doi: 10.1007/s00205-017-1126-z.  Google Scholar

[56]

M. Ignatova and I. Kukavica, On the local existence of the free-surface Euler equation with surface tension, Asymptot. Anal., 100 (2016), 63–86, http://dx.doi.org/10.3233/ASY-161386. doi: 10.3233/ASY-161386.  Google Scholar

[57]

T. Iguchi, N. Tanaka and A. Tani, On a free boundary problem for an incompressible ideal fluid in two space dimensions, Adv. Math. Sci. Appl., 9 (1999), 415-472.   Google Scholar

[58]

A. D. Ionescu and F. Pusateri, Global regularity for 2d water waves with surface tension, Memoirs of the American Mathematical Society, 256 (2018), arXiv: 1408.4428 [math.AP]. doi: 10.1090/memo/1227.  Google Scholar

[59]

A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.  Google Scholar

[60]

A. D. Ionescu and F. Pusateri, Global analysis of a model for capillary water waves in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2015–2071, http://dx.doi.org/10.1002/cpa.21654. doi: 10.1002/cpa.21654.  Google Scholar

[61]

J. Jang, P. G. LeFloch and N. Masmoudi, Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum, Journal of Differential Equations, 260 (2016), 5481-5509.  doi: 10.1016/j.jde.2015.12.004.  Google Scholar

[62]

J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math., 62 (2009), 1327–1385, http://dx.doi.org/10.1002/cpa.20285. doi: 10.1002/cpa.20285.  Google Scholar

[63]

J. Jang and N. Masmoudi, Vacuum in gas and fluid dynamics, in Nonlinear Conservation Laws and Applications, vol. 153 of IMA Vol. Math. Appl., Springer, New York, 2011,315–329, http://dx.doi.org/10.1007/978-1-4419-9554-4_17. doi: 10.1007/978-1-4419-9554-4_17.  Google Scholar

[64]

T. Kano and T. Nishida, Sur les ondes de surface de l'eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ., 19 (1979), 335-370.  doi: 10.1215/kjm/1250522437.  Google Scholar

[65]

I. Kukavica and A. Tuffaha, On the 2D free boundary Euler equation, Evol. Equ. Control Theory, 1 (2012), 297-314.  doi: 10.3934/eect.2012.1.297.  Google Scholar

[66]

I. Kukavica and A. Tuffaha, Well-posedness for the compressible Navier-Stokes-Lamé system with a free interface, Nonlinearity, 25 (2012), 3111–3137, https://doi.org/10.1088/0951-7715/25/11/3111. doi: 10.1088/0951-7715/25/11/3111.  Google Scholar

[67]

I. Kukavica and A. Tuffaha, A regularity result for the incompressible Euler equation with a free interface, Appl. Math. Optim., 69 (2014), 337-358.  doi: 10.1007/s00245-013-9221-5.  Google Scholar

[68]

I. Kukavica, A. Tuffaha and V. Vicol, On the local existence for the 3d Euler equation with a free interface, Applied Mathematics and Optimization, 76 (2017), 535-563.  doi: 10.1007/s00245-016-9360-6.  Google Scholar

[69]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605–654 (electronic). doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[70]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.  Google Scholar

[71]

H. Lindblad, The motion of the free surface of a liquid, in Séminaire: Équations aux Dérivées Partielles, 2000–2001, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2001, Exp. No. VI, 10.  Google Scholar

[72]

H. Lindblad, Well-posedness for the linearized motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 236 (2003), 281-310.  doi: 10.1007/s00220-003-0812-x.  Google Scholar

[73]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197.  doi: 10.1002/cpa.10055.  Google Scholar

[74]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[75]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109–194. doi: 10.4007/annals.2005.162.109.  Google Scholar

[76]

H. Lindblad and C. Luo, A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit, Comm. Pure Appl. Math., 71 (2018), 1273–1333, https://doi.org/10.1002/cpa.21734. doi: 10.1002/cpa.21734.  Google Scholar

[77]

H. Lindblad and K. H. Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary, J. Hyperbolic Differ. Equ., 6 (2009), 407-432.  doi: 10.1142/S021989160900185X.  Google Scholar

[78]

C. Luo, On the motion of a compressible gravity water wave with vorticity, Annals of PDE, 4 (2018), 20, arXiv: 1701.03987 [math.AP]. doi: 10.1007/s40818-018-0057-9.  Google Scholar

[79]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and Waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, 1986,459–479, http://dx.doi.org/10.1016/S0168-2024(08)70142-5. doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[80]

V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy, 104–210,254.  Google Scholar

[81]

T. Nishida, Equations of fluid dynamics–-free surface problems, Comm. Pure Appl. Math., 39 (1986), S221–S238, http://dx.doi.org/10.1002/cpa.3160390712, Frontiers of the mathematical sciences: 1985 (New York, 1985). doi: 10.1002/cpa.3160390712.  Google Scholar

[82]

M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740.  doi: 10.1142/S0218202502002306.  Google Scholar

[83]

M. Ogawa and A. Tani, Incompressible perfect fluid motion with free boundary of finite depth, Adv. Math. Sci. Appl., 13 (2003), 201-223.   Google Scholar

[84]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016, http://dx.doi.org/10.1007/978-3-319-27698-4. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[85]

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 (2011), 347-373.  doi: 10.1142/S021989161100241X.  Google Scholar

[86]

J. Reeder and M. Shinbrot, The initial value problem for surface waves under gravity. Ⅱ. The simplest 3-dimensional case, Indiana Univ. Math. J., 25 (1976), 1049-1071.  doi: 10.1512/iumj.1976.25.25085.  Google Scholar

[87]

J. Reeder and M. Shinbrot, The initial value problem for surface waves under gravity. Ⅲ. Uniformly analytic initial domains, J. Math. Anal. Appl., 67 (1979), 340–391, http://dx.doi.org/10.1016/0022-247X(79)90028-3. doi: 10.1016/0022-247X(79)90028-3.  Google Scholar

[88]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781.  doi: 10.1016/j.anihpc.2004.11.001.  Google Scholar

[89]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  doi: 10.1002/cpa.20213.  Google Scholar

[90]

J. Shatah and C. Zeng, Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal., 199 (2011), 653-705.  doi: 10.1007/s00205-010-0335-5.  Google Scholar

[91]

M. Shinbrot, The initial value problem for surface waves under gravity. I. The simplest case, Indiana Univ. Math. J., 25 (1976), 281-300.  doi: 10.1512/iumj.1976.25.25023.  Google Scholar

[92]

Y. Trakhinin, Existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331–366, http://dx.doi.org/10.1007/s00205-005-0364-7. doi: 10.1007/s00205-005-0364-7.  Google Scholar

[93]

Y. Trakhinin, Existence and stability of compressible and incompressible current-vortex sheets, in Analysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2007,229–246, http://dx.doi.org/10.1007/978-3-7643-7742-7_13. doi: 10.1007/978-3-7643-7742-7_13.  Google Scholar

[94]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math., 62 (2009), 1551–1594, http://dx.doi.org/10.1002/cpa.20282. doi: 10.1002/cpa.20282.  Google Scholar

[95]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[96]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[97]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.  doi: 10.1007/s00222-009-0176-8.  Google Scholar

[98]

S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.  doi: 10.1007/s00222-010-0288-1.  Google Scholar

[99]

H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96.  doi: 10.2977/prims/1195184016.  Google Scholar

[100]

H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694.  doi: 10.1215/kjm/1250521429.  Google Scholar

show all references

References:
[1]

T. Alazard, About global existence and asymptotic behavior for two dimensional gravity water waves, in Séminaire Laurent Schwartz–-Équations aux dérivées partielles et applications. Année 2012–2013, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2014, Exp. No. ⅩⅧ, 16.  Google Scholar

[2]

T. Alazard, Stabilization of the water-wave equations with surface tension, Ann. PDE, 3 (2017), Art. 17, 41 pp. doi: 10.1007/s40818-017-0032-x.  Google Scholar

[3]

T. Alazard and P. Baldi, Gravity capillary standing water waves, Arch. Ration. Mech. Anal., 217 (2015), 741-830.  doi: 10.1007/s00205-015-0842-5.  Google Scholar

[4]

T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499.  doi: 10.1215/00127094-1345653.  Google Scholar

[5]

T. Alazard, N. Burq and C. Zuily, Strichartz estimates for water waves, Ann. Sci. Éc. Norm. Supér. (4), 44 (2011), 855–903. doi: 10.24033/asens.2156.  Google Scholar

[6]

T. Alazard, N. Burq and C. Zuily, The water-wave equations: From Zakharov to Euler, in Studies in Phase Space Analysis with Applications to PDEs, vol. 84 of Progr. Nonlinear Differential Equations Appl., Birkhäuser/Springer, New York, 2013, 1–20. doi: 10.1007/978-1-4614-6348-1_1.  Google Scholar

[7]

T. Alazard, N. Burq and C. Zuily, On the Cauchy problem for gravity water waves, Invent. Math., 198 (2014), 71-163.  doi: 10.1007/s00222-014-0498-z.  Google Scholar

[8]

T. Alazard, N. Burq and C. Zuily, Cauchy theory for the gravity water waves system with non-localized initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 337-395.  doi: 10.1016/j.anihpc.2014.10.004.  Google Scholar

[9]

T. Alazard and J.-M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves, Ann. Sci. Éc. Norm. Supér. (4), 48 (2015), 1149–1238. doi: 10.24033/asens.2268.  Google Scholar

[10]

T. Alazard and J.-M. Delort, Sobolev estimates for two dimensional gravity water waves, Astérisque, 374 (2015), viii+241.  Google Scholar

[11]

D. M. Ambrose, Well-posedness of vortex sheets with surface tension, SIAM J. Math. Anal., 35 (2003), 211–244 (electronic). doi: 10.1137/S0036141002403869.  Google Scholar

[12]

D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315.  doi: 10.1002/cpa.20085.  Google Scholar

[13]

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 (1993), 1269–1301, http://dx.doi.org/10.1002/cpa.3160460903. doi: 10.1002/cpa.3160460903.  Google Scholar

[14]

L. Bieri, S. Miao, S. Shahshahani and S. Wu, On the motion of a self-gravitating incompressible fluid with free boundary and constant vorticity: An appendix, arXiv: 1511.07483 [math.AP]. Google Scholar

[15]

L. Bieri, S. Miao, S. Shahshahani and S. Wu, On the Motion of a Self-Gravitating Incompressible Fluid with Free Boundary, Comm. Math. Phys., 355 (2017), 161–243, http://dx.doi.org/10.1007/s00220-017-2884-z. doi: 10.1007/s00220-017-2884-z.  Google Scholar

[16]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Finite time singularities for water waves with surface tension, J. Math. Phys., 53 (2012), 115622, 26pp. doi: 10.1063/1.4765339.  Google Scholar

[17]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. G´omez-Serrano, Finite time singularities for the free boundary incompressible Euler equations, Ann. of Math. (2), 178 (2013), 1061–1134. doi: 10.4007/annals.2013.178.3.6.  Google Scholar

[18]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and J. Gómez-Serrano, Structural stability for the splash singularities of the water waves problem, Discrete Contin. Dyn. Syst., 34 (2014), 4997–5043, http://dx.doi.org/10.3934/dcds.2014.34.4997. doi: 10.3934/dcds.2014.34.4997.  Google Scholar

[19]

A. Castro, D. Córdoba, C. Fefferman, F. Gancedo and M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves, Ann. of Math. (2), 175 (2012), 909–948. doi: 10.4007/annals.2012.175.2.9.  Google Scholar

[20]

C.-H. A. Chen, D. Coutand and S. Shkoller, Solvability and regularity for an elliptic system prescribing the curl, divergence, and partial trace of a vector field on Sobolev-class domains, J. Math. Fluid Mech., 19 (2017), 375-422.  doi: 10.1007/s00021-016-0289-y.  Google Scholar

[21]

G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal., 187 (2008), 369–408, http://dx.doi.org/10.1007/s00205-007-0070-8. doi: 10.1007/s00205-007-0070-8.  Google Scholar

[22]

C.-H. Cheng and S. Shkoller, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity, Comm. Pure Appl. Math., 61 (2008), 1715-1752.  doi: 10.1002/cpa.20240.  Google Scholar

[23]

D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602.  doi: 10.1002/1097-0312(200012)53:12<1536::AID-CPA2>3.0.CO;2-Q.  Google Scholar

[24]

J.-F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 41 (2008), 85–139. doi: 10.24033/asens.2064.  Google Scholar

[25]

J.-F. Coulombel and P. Secchi, Uniqueness of 2-D compressible vortex sheets, Commun. Pure Appl. Anal., 8 (2009), 1439–1450, http://dx.doi.org/10.3934/cpaa.2009.8.1439. doi: 10.3934/cpaa.2009.8.1439.  Google Scholar

[26]

D. Coutand, Finite time singularity formation for moving interface Euler equations, arXiv: 1701.01699 [math.AP], 43 pages. Google Scholar

[27]

D. Coutand, J. Hole and S. Shkoller, Well-posedness of the free-boundary compressible 3-D Euler equations with surface tension and the zero surface tension limit, SIAM J. Math. Anal., 45 (2013), 3690-3767.  doi: 10.1137/120888697.  Google Scholar

[28]

D. Coutand, H. Lindblad and S. Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys., 296 (2010), 559-587.  doi: 10.1007/s00220-010-1028-5.  Google Scholar

[29]

D. Coutand and S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829-930.  doi: 10.1090/S0894-0347-07-00556-5.  Google Scholar

[30]

D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429–449, http://dx.doi.org/10.3934/dcdss.2010.3.429. doi: 10.3934/dcdss.2010.3.429.  Google Scholar

[31]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344.  Google Scholar

[32]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[33]

D. Coutand and S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations, Commun. Math. Phys., 325 (2014), 143-183.  doi: 10.1007/s00220-013-1855-2.  Google Scholar

[34]

W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits, Comm. Partial Differential Equations, 10 (1985), 787-1003.  doi: 10.1080/03605308508820396.  Google Scholar

[35]

W. Craig, On the Hamiltonian for water waves, arXiv: 1612.08971 [math.AP], 10 pages. Google Scholar

[36]

T. de Poyferré, A priori estimates for water waves with emerging bottom, arXiv: 1612.04103 [math.AP], 45 pages. Google Scholar

[37]

Y. Deng, A. D. Ionescu, B. Pausader and F. Pusateri, Global solutions of the gravity-capillary water wave system in 3 dimensions, Acta Math., 219 (2017), 213–402, arXiv: 1601.05685 [math.AP]. doi: 10.4310/ACTA.2017.v219.n2.a1.  Google Scholar

[38]

M. M. Disconzi, On a linear problem arising in dynamic boundaries, Evol. Equ. Control Theory, 3 (2014), 627-644.  doi: 10.3934/eect.2014.3.627.  Google Scholar

[39]

M. M. Disconzi and D. G. Ebin, On the limit of large surface tension for a fluid motion with free boundary, Comm. Partial Differential Equations, 39 (2014), 740-779.  doi: 10.1080/03605302.2013.865058.  Google Scholar

[40]

M. M. Disconzi and D. G. Ebin, The free boundary Euler equations with large surface tension, Journal of Differential Equations, 261 (2016), 821-889.  doi: 10.1016/j.jde.2016.03.029.  Google Scholar

[41]

M. M. Disconzi and D. G. Ebin, Motion of slightly compressible fluids in a bounded domain, Ⅱ, Commun. Contemp. Math., 19 (2017), 1650054, 57pp. doi: 10.1142/S0219199716500541.  Google Scholar

[42]

M. M. Disconzi and I. Kukavica, A priori estimates for the free-boundary Euler equations with surface tension in three dimensions, arXiv: 1708.00086 [math.AP], 40 pages. Google Scholar

[43]

H. Dong and D. Kim, On the $L_p$-solvability of higher order parabolic and elliptic systems with BMO coefficients, Arch. Ration. Mech. Anal., 199 (2011), 889-941.  doi: 10.1007/s00205-010-0345-3.  Google Scholar

[44]

S. Ebenfeld, $L^2 $-regularity theory of linear strongly elliptic Dirichlet systems of order $ 2m$ with minimal regularity in the coefficients, Quart. Appl. Math., 60 (2002), 547-576.  doi: 10.1090/qam/1914441.  Google Scholar

[45]

D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175–1201, http://dx.doi.org/10.1080/03605308708820523. doi: 10.1080/03605308708820523.  Google Scholar

[46]

L. C. Evans, Partial Differential Equations, American Mathematical Society (2nd edition), 2010. doi: 10.1090/gsm/019.  Google Scholar

[47]

C. Fefferman, A. D. Ionescu and V. Lie, On the absence of splash singularities in the case of two-fluid interfaces, Duke Math. J., 165 (2016), 417-462.  doi: 10.1215/00127094-3166629.  Google Scholar

[48]

P. Germain, N. Masmoudi and J. Shatah, Global solutions for the gravity water waves equation in dimension 3, Ann. of Math. (2), 175 (2012), 691–754, http://dx.doi.org/10.4007/annals.2012.175.2.6. doi: 10.4007/annals.2012.175.2.6.  Google Scholar

[49]

P. Germain, N. Masmoudi and J. Shatah, Global existence for capillary water waves, Comm. Pure Appl. Math., 68 (2015), 625-687.  doi: 10.1002/cpa.21535.  Google Scholar

[50]

M. Hadžić, S. Shkoller and J. Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, arXiv: 1511.07467 [math.AP]. Google Scholar

[51]

Q. Han and J.-X. Hong, Isometric Embedding of Riemannian Manifolds in Euclidean Spaces, vol. 130 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2006. doi: 10.1090/surv/130.  Google Scholar

[52]

J. K. Hunter, M. Ifrim and D. Tataru, Two dimensional water waves in holomorphic coordinates, Comm. Math. Phys., 346 (2016), 483–552, http://dx.doi.org/10.1007/s00220-016-2708-6. doi: 10.1007/s00220-016-2708-6.  Google Scholar

[53]

M. Ifrim and D. Tataru, Two dimensional gravity water waves with constant vorticity: Ⅰ. cubic lifespan, Analysis and PDE, 12 (2019), 903–967, arXiv: 1510.07732 [math.AP]. doi: 10.2140/apde.2019.12.903.  Google Scholar

[54]

M. Ifrim and D. Tataru, Two dimensional water waves in holomorphic coordinates Ⅱ: Global solutions, Bull. Soc. Math. France, 144 (2016), 369-394.  doi: 10.24033/bsmf.2717.  Google Scholar

[55]

M. Ifrim and D. Tataru, The Lifespan of Small Data Solutions in Two Dimensional Capillary Water Waves, Arch. Ration. Mech. Anal., 225 (2017), 1279–1346, http://dx.doi.org/10.1007/s00205-017-1126-z. doi: 10.1007/s00205-017-1126-z.  Google Scholar

[56]

M. Ignatova and I. Kukavica, On the local existence of the free-surface Euler equation with surface tension, Asymptot. Anal., 100 (2016), 63–86, http://dx.doi.org/10.3233/ASY-161386. doi: 10.3233/ASY-161386.  Google Scholar

[57]

T. Iguchi, N. Tanaka and A. Tani, On a free boundary problem for an incompressible ideal fluid in two space dimensions, Adv. Math. Sci. Appl., 9 (1999), 415-472.   Google Scholar

[58]

A. D. Ionescu and F. Pusateri, Global regularity for 2d water waves with surface tension, Memoirs of the American Mathematical Society, 256 (2018), arXiv: 1408.4428 [math.AP]. doi: 10.1090/memo/1227.  Google Scholar

[59]

A. D. Ionescu and F. Pusateri, Global solutions for the gravity water waves system in 2d, Invent. Math., 199 (2015), 653-804.  doi: 10.1007/s00222-014-0521-4.  Google Scholar

[60]

A. D. Ionescu and F. Pusateri, Global analysis of a model for capillary water waves in two dimensions, Comm. Pure Appl. Math., 69 (2016), 2015–2071, http://dx.doi.org/10.1002/cpa.21654. doi: 10.1002/cpa.21654.  Google Scholar

[61]

J. Jang, P. G. LeFloch and N. Masmoudi, Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum, Journal of Differential Equations, 260 (2016), 5481-5509.  doi: 10.1016/j.jde.2015.12.004.  Google Scholar

[62]

J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math., 62 (2009), 1327–1385, http://dx.doi.org/10.1002/cpa.20285. doi: 10.1002/cpa.20285.  Google Scholar

[63]

J. Jang and N. Masmoudi, Vacuum in gas and fluid dynamics, in Nonlinear Conservation Laws and Applications, vol. 153 of IMA Vol. Math. Appl., Springer, New York, 2011,315–329, http://dx.doi.org/10.1007/978-1-4419-9554-4_17. doi: 10.1007/978-1-4419-9554-4_17.  Google Scholar

[64]

T. Kano and T. Nishida, Sur les ondes de surface de l'eau avec une justification mathématique des équations des ondes en eau peu profonde, J. Math. Kyoto Univ., 19 (1979), 335-370.  doi: 10.1215/kjm/1250522437.  Google Scholar

[65]

I. Kukavica and A. Tuffaha, On the 2D free boundary Euler equation, Evol. Equ. Control Theory, 1 (2012), 297-314.  doi: 10.3934/eect.2012.1.297.  Google Scholar

[66]

I. Kukavica and A. Tuffaha, Well-posedness for the compressible Navier-Stokes-Lamé system with a free interface, Nonlinearity, 25 (2012), 3111–3137, https://doi.org/10.1088/0951-7715/25/11/3111. doi: 10.1088/0951-7715/25/11/3111.  Google Scholar

[67]

I. Kukavica and A. Tuffaha, A regularity result for the incompressible Euler equation with a free interface, Appl. Math. Optim., 69 (2014), 337-358.  doi: 10.1007/s00245-013-9221-5.  Google Scholar

[68]

I. Kukavica, A. Tuffaha and V. Vicol, On the local existence for the 3d Euler equation with a free interface, Applied Mathematics and Optimization, 76 (2017), 535-563.  doi: 10.1007/s00245-016-9360-6.  Google Scholar

[69]

D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605–654 (electronic). doi: 10.1090/S0894-0347-05-00484-4.  Google Scholar

[70]

D. Lannes, The Water Waves Problem, vol. 188 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2013, Mathematical analysis and asymptotics. doi: 10.1090/surv/188.  Google Scholar

[71]

H. Lindblad, The motion of the free surface of a liquid, in Séminaire: Équations aux Dérivées Partielles, 2000–2001, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 2001, Exp. No. VI, 10.  Google Scholar

[72]

H. Lindblad, Well-posedness for the linearized motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 236 (2003), 281-310.  doi: 10.1007/s00220-003-0812-x.  Google Scholar

[73]

H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197.  doi: 10.1002/cpa.10055.  Google Scholar

[74]

H. Lindblad, Well posedness for the motion of a compressible liquid with free surface boundary, Comm. Math. Phys., 260 (2005), 319-392.  doi: 10.1007/s00220-005-1406-6.  Google Scholar

[75]

H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109–194. doi: 10.4007/annals.2005.162.109.  Google Scholar

[76]

H. Lindblad and C. Luo, A priori estimates for the compressible Euler equations for a liquid with free surface boundary and the incompressible limit, Comm. Pure Appl. Math., 71 (2018), 1273–1333, https://doi.org/10.1002/cpa.21734. doi: 10.1002/cpa.21734.  Google Scholar

[77]

H. Lindblad and K. H. Nordgren, A priori estimates for the motion of a self-gravitating incompressible liquid with free surface boundary, J. Hyperbolic Differ. Equ., 6 (2009), 407-432.  doi: 10.1142/S021989160900185X.  Google Scholar

[78]

C. Luo, On the motion of a compressible gravity water wave with vorticity, Annals of PDE, 4 (2018), 20, arXiv: 1701.03987 [math.AP]. doi: 10.1007/s40818-018-0057-9.  Google Scholar

[79]

T. Makino, On a local existence theorem for the evolution equation of gaseous stars, in Patterns and Waves, vol. 18 of Stud. Math. Appl., North-Holland, Amsterdam, 1986,459–479, http://dx.doi.org/10.1016/S0168-2024(08)70142-5. doi: 10.1016/S0168-2024(08)70142-5.  Google Scholar

[80]

V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy, 104–210,254.  Google Scholar

[81]

T. Nishida, Equations of fluid dynamics–-free surface problems, Comm. Pure Appl. Math., 39 (1986), S221–S238, http://dx.doi.org/10.1002/cpa.3160390712, Frontiers of the mathematical sciences: 1985 (New York, 1985). doi: 10.1002/cpa.3160390712.  Google Scholar

[82]

M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740.  doi: 10.1142/S0218202502002306.  Google Scholar

[83]

M. Ogawa and A. Tani, Incompressible perfect fluid motion with free boundary of finite depth, Adv. Math. Sci. Appl., 13 (2003), 201-223.   Google Scholar

[84]

J. Prüss and G. Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, vol. 105 of Monographs in Mathematics, Birkhäuser/Springer, [Cham], 2016, http://dx.doi.org/10.1007/978-3-319-27698-4. doi: 10.1007/978-3-319-27698-4.  Google Scholar

[85]

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 (2011), 347-373.  doi: 10.1142/S021989161100241X.  Google Scholar

[86]

J. Reeder and M. Shinbrot, The initial value problem for surface waves under gravity. Ⅱ. The simplest 3-dimensional case, Indiana Univ. Math. J., 25 (1976), 1049-1071.  doi: 10.1512/iumj.1976.25.25085.  Google Scholar

[87]

J. Reeder and M. Shinbrot, The initial value problem for surface waves under gravity. Ⅲ. Uniformly analytic initial domains, J. Math. Anal. Appl., 67 (1979), 340–391, http://dx.doi.org/10.1016/0022-247X(79)90028-3. doi: 10.1016/0022-247X(79)90028-3.  Google Scholar

[88]

B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781.  doi: 10.1016/j.anihpc.2004.11.001.  Google Scholar

[89]

J. Shatah and C. Zeng, Geometry and a priori estimates for free boundary problems of the Euler equation, Comm. Pure Appl. Math., 61 (2008), 698-744.  doi: 10.1002/cpa.20213.  Google Scholar

[90]

J. Shatah and C. Zeng, Local well-posedness for fluid interface problems, Arch. Ration. Mech. Anal., 199 (2011), 653-705.  doi: 10.1007/s00205-010-0335-5.  Google Scholar

[91]

M. Shinbrot, The initial value problem for surface waves under gravity. I. The simplest case, Indiana Univ. Math. J., 25 (1976), 281-300.  doi: 10.1512/iumj.1976.25.25023.  Google Scholar

[92]

Y. Trakhinin, Existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331–366, http://dx.doi.org/10.1007/s00205-005-0364-7. doi: 10.1007/s00205-005-0364-7.  Google Scholar

[93]

Y. Trakhinin, Existence and stability of compressible and incompressible current-vortex sheets, in Analysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2007,229–246, http://dx.doi.org/10.1007/978-3-7643-7742-7_13. doi: 10.1007/978-3-7643-7742-7_13.  Google Scholar

[94]

Y. Trakhinin, Local existence for the free boundary problem for nonrelativistic and relativistic compressible Euler equations with a vacuum boundary condition, Comm. Pure Appl. Math., 62 (2009), 1551–1594, http://dx.doi.org/10.1002/cpa.20282. doi: 10.1002/cpa.20282.  Google Scholar

[95]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 2-D, Invent. Math., 130 (1997), 39-72.  doi: 10.1007/s002220050177.  Google Scholar

[96]

S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495.  doi: 10.1090/S0894-0347-99-00290-8.  Google Scholar

[97]

S. Wu, Almost global wellposedness of the 2-D full water wave problem, Invent. Math., 177 (2009), 45-135.  doi: 10.1007/s00222-009-0176-8.  Google Scholar

[98]

S. Wu, Global wellposedness of the 3-D full water wave problem, Invent. Math., 184 (2011), 125-220.  doi: 10.1007/s00222-010-0288-1.  Google Scholar

[99]

H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96.  doi: 10.2977/prims/1195184016.  Google Scholar

[100]

H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694.  doi: 10.1215/kjm/1250521429.  Google Scholar

[1]

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