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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

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.

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

T. AlazardN. 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.

[8]

T. AlazardN. 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.

[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.

[10]

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

[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.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[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.

[20]

C.-H. A. ChenD. 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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

D. CoutandJ. 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.

[28]

D. CoutandH. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[46]

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

[47]

C. FeffermanA. 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.

[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.

[49]

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

[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].

[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.

[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.

[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.

[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.

[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.

[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.

[57]

T. IguchiN. 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.

[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.

[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.

[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.

[61]

J. JangP. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[68]

I. KukavicaA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[80]

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

[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.

[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.

[83]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[100]

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

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.

[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.

[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.

[4]

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

[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.

[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.

[7]

T. AlazardN. 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.

[8]

T. AlazardN. 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.

[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.

[10]

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

[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.

[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.

[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.

[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].

[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.

[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.

[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.

[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.

[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.

[20]

C.-H. A. ChenD. 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.

[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.

[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.

[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.

[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.

[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.

[26]

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

[27]

D. CoutandJ. 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.

[28]

D. CoutandH. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[35]

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

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[46]

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

[47]

C. FeffermanA. 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.

[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.

[49]

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

[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].

[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.

[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.

[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.

[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.

[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.

[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.

[57]

T. IguchiN. 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.

[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.

[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.

[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.

[61]

J. JangP. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[68]

I. KukavicaA. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[80]

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

[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.

[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.

[83]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[100]

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

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