Traveling wave solutions to the free boundary incompressible Navier-Stokes equations. (English) Zbl 1532.35349
From the authors’ abstract: “In this paper we study a finite-depth layer of viscous incompressible fluid in dimension \(n \geq 2\), modeled by the Navier-Stokes equations. The fluid is assumed to be bounded below by a flat rigid surface and above by a free, moving interface. A uniform gravitational field acts perpendicularly to the flat surface, and we consider the cases with and without surface tension acting on the free interface. In addition to these gravity-capillary effects, we allow for a second force field in the bulk and an external stress tensor on the free interface, both of which are posited to be in traveling wave form, i.e., time-independent when viewed in a coordinate system moving at a constant velocity parallel to the rigid lower boundary.”
The main results of the paper can be summarised as follows: In dimension \(n \geq 2\), for the case surface tension, and for every nontrivial traveling velocity, it is proven that there exists a non empty open set of force and stress data that give rise to travelling wave solutions. A similar result is proven, in absence of surface tension, for dimension \(n=2\). The key novelty of the results stems from the fact the viscous traveling wave solutions are constructed.
It is a interesting and well written paper. One of the main difficulties of the problem stems from the fact that the actual domain is also unknown. To overcome this difficulty, a suitably reformulated system is defined. The reformulated (quasilinear) system is now posed on a fixed domain, but on the other hand, its overall nonlinear structure is worsen compared to the original system. The key ingredient of the proofs of the main results relies on a very detailed analysis of the reformulated system. Several novel analytical tools, of independent interest, are involved in the proofs. In particular, a new class of some speciliazed Sobolev spaces and the detailed study of an overdetermined Stokes problem (with inhomogenenous divergence condition) and its underdetermined adjoint problem play an important role throughout the analysis.
The main results of the paper can be summarised as follows: In dimension \(n \geq 2\), for the case surface tension, and for every nontrivial traveling velocity, it is proven that there exists a non empty open set of force and stress data that give rise to travelling wave solutions. A similar result is proven, in absence of surface tension, for dimension \(n=2\). The key novelty of the results stems from the fact the viscous traveling wave solutions are constructed.
It is a interesting and well written paper. One of the main difficulties of the problem stems from the fact that the actual domain is also unknown. To overcome this difficulty, a suitably reformulated system is defined. The reformulated (quasilinear) system is now posed on a fixed domain, but on the other hand, its overall nonlinear structure is worsen compared to the original system. The key ingredient of the proofs of the main results relies on a very detailed analysis of the reformulated system. Several novel analytical tools, of independent interest, are involved in the proofs. In particular, a new class of some speciliazed Sobolev spaces and the detailed study of an overdetermined Stokes problem (with inhomogenenous divergence condition) and its underdetermined adjoint problem play an important role throughout the analysis.
Reviewer: Konstantinos Chrysafinos (Athína)
MSC:
| 35Q30 | Navier-Stokes equations |
| 76D05 | Navier-Stokes equations for incompressible viscous fluids |
| 76D07 | Stokes and related (Oseen, etc.) flows |
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
| 35C07 | Traveling wave solutions |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35N25 | Overdetermined boundary value problems for PDEs and systems of PDEs |
| 35R35 | Free boundary problems for PDEs |
| 35R37 | Moving boundary problems for PDEs |
| 35S15 | Boundary value problems for PDEs with pseudodifferential operators |
Keywords:
incompressible Navier-Stokes equations; surface tension; traveling velocity; overdetermined Stokes problem; free boundary; moving upper boundariesReferences:
| [1] | Abe, T.; Shibata, Y.On a resolvent estimate of the Stokes equation on an infinite layer. J. Math. Soc. Japan55 (2003), no. 2, 469-497. https://doi.org/10.2969/jmsj/1191419127 · Zbl 1048.35052 · doi:10.2969/jmsj/1191419127 |
| [2] | Abe, T.; Shibata, Y.On a resolvent estimate of the Stokes equation on an infinite layer. II. 𝜆 = 0 case. J. Math. Fluid Mech.5 (2003), no. 3, 245-274. https://doi.org/10.1007/s00021-003-0075-5 · Zbl 1162.76328 · doi:10.1007/s00021-003-0075-5 |
| [3] | Abe, T.; Yamazaki, M.On a stationary problem of the Stokes equation in an infinite layer in Sobolev and Besov spaces. J. Math. Fluid Mech.12 (2010), no. 1, 61-100. https://doi.org/10.1007/s00021-008-0276-z · Zbl 1261.35117 · doi:10.1007/s00021-008-0276-z |
| [4] | Abels, H.Reduced and generalized Stokes resolvent equations in asymptotically flat layers. I. Unique solvability. J. Math. Fluid Mech.7 (2005), no. 2, 201-222. https://doi.org/10.1007/s00021-004-0116-8 · Zbl 1070.35020 · doi:10.1007/s00021-004-0116-8 |
| [5] | Abels, H. Reduced and generalized Stokes resolvent equations in asymptotically flat layers. II. H_∞‐calculus. J. Math. Fluid Mech. 7 (2005), no. 2, 223-260. 10.1007/s00021‐004‐0117‐7 · Zbl 1083.35085 |
| [6] | Abels, H.The initial‐value problem for the Navier‐Stokes equations with a free surface in L^q‐Sobolev spaces. Adv. Differential Equations10 (2005), no. 1, 45-64. · Zbl 1105.35072 |
| [7] | Abels, H.Generalized Stokes resolvent equations in an infinite layer with mixed boundary conditions. Math. Nachr.279 (2006), no. 4, 351-367. https://doi.org/10.1002/mana.200310365 · Zbl 1117.35060 · doi:10.1002/mana.200310365 |
| [8] | Abels, H.; Wiegner, M.Resolvent estimates for the Stokes operator on an infinite layer. Differential Integral Equations18 (2005), no. 10, 1081-1110. · Zbl 1212.35343 |
| [9] | Abraham, R.; Marsden, J. E.; Ratiu, T.Manifolds, tensor analysis, and applications. Second edition. Applied Mathematical Sciences, 75. Springer, New York, 1988. 10.1007/978‐1‐4612‐1029‐0 · Zbl 0875.58002 |
| [10] | Agmon, S.; Douglis, A.; Nirenberg, L.Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II. Comm. Pure Appl. Math.17 (1964), 35-92. https://doi.org/10.1002/cpa.3160170104 · Zbl 0123.28706 · doi:10.1002/cpa.3160170104 |
| [11] | Alazard, T.; Burq, N.; Zuily, C.On the water‐wave equations with surface tension. Duke Math. J.158 (2011), no. 3, 413-499. https://doi.org/10.1215/00127094-1345653 · Zbl 1258.35043 · doi:10.1215/00127094-1345653 |
| [12] | Alazard, T.; Burq, N.; Zuily, C.On the Cauchy problem for gravity water waves. Invent. Math.198 (2014), no. 1, 71-163. https://doi.org/10.1007/s00222-014-0498-z · Zbl 1308.35195 · doi:10.1007/s00222-014-0498-z |
| [13] | Alazard, T.; Burq, N.; Zuily, C.Cauchy theory for the gravity water waves system with non‐localized initial data. Ann. Inst. H. Poincaré Anal. Non Linéaire33 (2016), no. 2, 337-395. https://doi.org/10.1016/j.anihpc.2014.10.004 · Zbl 1339.35227 · doi:10.1016/j.anihpc.2014.10.004 |
| [14] | Alazard, T.; Delort, J.‐M. Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. Éc. Norm. Supér. (4)48 (2015), no. 5, 1149-1238. 10.24033/asens.2268 · Zbl 1347.35198 |
| [15] | Alazard, T.; Delort, J.‐M. Sobolev estimates for two dimensional gravity water waves. Astérisque (2015), no. 374, viii+241. · Zbl 1360.35002 |
| [16] | Amick, C. J.; Fraenkel, L. E.; Toland, J. F.On the Stokes conjecture for the wave of extreme form. Acta Math.148 (1982), 193-214. https://doi.org/10.1007/BF02392728 · Zbl 0495.76021 · doi:10.1007/BF02392728 |
| [17] | Amick, C. J.; Toland, J. F.On solitary water‐waves of finite amplitude. Arch. Rational Mech. Anal.76 (1981), no. 1, 9-95. https://doi.org/10.1007/BF00250799 · Zbl 0468.76025 · doi:10.1007/BF00250799 |
| [18] | Bae, H.Solvability of the free boundary value problem of the Navier‐Stokes equations. Discrete Contin. Dyn. Syst.29 (2011), no. 3, 769-801. https://doi.org/10.3934/dcds.2011.29.769 · Zbl 1222.35215 · doi:10.3934/dcds.2011.29.769 |
| [19] | Bae, H.‐O.; Cho, K. Free surface problem of stationary non‐Newtonian fluids. Nonlinear Anal. 41 (2000), no. 1‐2, Ser. A: Theory Methods, 243-258. 10.1016/S0362‐546X(98)00276‐4 · Zbl 0967.76004 |
| [20] | Beale, J. T.The existence of solitary water waves. Comm. Pure Appl. Math.30 (1977), no. 4, 373-389. https://doi.org/10.1002/cpa.3160300402 · Zbl 0379.35055 · doi:10.1002/cpa.3160300402 |
| [21] | Beale, J. T.The initial value problem for the Navier‐Stokes equations with a free surface. Comm. Pure Appl. Math.34 (1981), no. 3, 359-392. https://doi.org/10.1002/cpa.3160340305 · Zbl 0464.76028 · doi:10.1002/cpa.3160340305 |
| [22] | Beale, J. T.Large‐time regularity of viscous surface waves. Arch. Rational Mech. Anal.84 (1983/84), no. 4, 307-352. https://doi.org/10.1007/BF00250586 · Zbl 0545.76029 · doi:10.1007/BF00250586 |
| [23] | Beale, J. T.; Nishida, T. Large‐time behavior of viscous surface waves. Recent topics in nonlinear PDE, II (Sendai, 1984), 1-14. North‐Holland Math. Stud., 128. North‐Holland, Amsterdam, 1985. 10.1016/S0304‐0208(08)72355‐7 · Zbl 0642.76048 |
| [24] | Bergh, J.; Löfström, J.Interpolation spaces. An introduction. Grundlehren der mathematischen Wissenschaften, 223. Springer, Berlin-New York, 1976. · Zbl 0344.46071 |
| [25] | Bourgain, J.; Brezis, H.New estimates for elliptic equations and Hodge type systems. J. Eur. Math. Soc. (JEMS)9 (2007), no. 2, 277-315. https://doi.org/10.4171/JEMS/80 · Zbl 1176.35061 · doi:10.4171/JEMS/80 |
| [26] | Buffoni, B.; Groves, M. D.; Sun, S. M.; Wahlén, E.Existence and conditional energetic stability of three‐dimensional fully localised solitary gravity‐capillary water waves. J. Differential Equations254 (2013), no. 3, 1006-1096. https://doi.org/10.1016/j.jde.2012.10.007 · Zbl 1336.76008 · doi:10.1016/j.jde.2012.10.007 |
| [27] | Buffoni, B.; Groves, M. D.; Wahlén, E.A variational reduction and the existence of a fully localised solitary wave for the three‐dimensional water‐wave problem with weak surface tension. Arch. Ration. Mech. Anal.228 (2018), no. 3, 773-820. https://doi.org/10.1007/s00205-017-1205-1 · Zbl 1397.35199 · doi:10.1007/s00205-017-1205-1 |
| [28] | Bühler, O.; Shatah, J.; Walsh, S.; Zeng, C.On the wind generation of water waves. Arch. Ration. Mech. Anal.222 (2016), no. 2, 827-878. https://doi.org/10.1007/s00205-016-1012-0 · Zbl 1356.35165 · doi:10.1007/s00205-016-1012-0 |
| [29] | Chae, D.; DubovskiI, P.Traveling wave‐like solutions of the Navier‐Stokes and the related equations. J. Math. Anal. Appl.204 (1996), no. 3, 930-939. https://doi.org/10.1006/jmaa.1996.0477 · Zbl 0874.35084 · doi:10.1006/jmaa.1996.0477 |
| [30] | Chen, R. M.; Walsh, S.; Wheeler, M. H.Existence and qualitative theory for stratified solitary water waves. Ann. Inst. H. Poincaré Anal. Non Linéaire35 (2018), no. 2, 517-576. https://doi.org/10.1016/j.anihpc.2017.06.003 · Zbl 1408.35135 · doi:10.1016/j.anihpc.2017.06.003 |
| [31] | Chen, R. M.; Walsh, S.; Wheeler, M. H.Existence, nonexistence, and asymptotics of deep water solitary waves with localized vorticity. Arch. Ration. Mech. Anal.234 (2019), no. 2, 595-633. https://doi.org/10.1007/s00205-019-01399-0 · Zbl 1428.35438 · doi:10.1007/s00205-019-01399-0 |
| [32] | Cho, Y.; Diorio, J. D.; Akylas, T. R.; Duncan, J. H. Resonantly forced gravity-capillary lumps on deep water. Part 2. Theoretical model. J. Math. Fluid Mech. 672 (2011), 288-306. 10.1017/s0022112010006002 · Zbl 1225.76059 |
| [33] | Constantin, A.; Strauss, W.Exact steady periodic water waves with vorticity. Comm. Pure Appl. Math.57 (2004), no. 4, 481-527. https://doi.org/10.1002/cpa.3046 · Zbl 1038.76011 · doi:10.1002/cpa.3046 |
| [34] | Deng, Y.; Ionescu, A. D.; Pausader, B.; Pusateri, F.Global solutions of the gravity‐capillary water‐wave system in three dimensions. Acta Math.219 (2017), no. 2, 213-402. https://doi.org/10.4310/ACTA.2017.v219.n2.a1 · Zbl 1397.35190 · doi:10.4310/ACTA.2017.v219.n2.a1 |
| [35] | Diorio, J. D.; Cho, Y.; Duncan, J. H.; Akylas, T. R. Resonantly forced gravity-capillary lumps on deep water. Part 1. Experiments. J. Math. Fluid Mech. 672 (2011), 268-287. 10.1017/s0022112010005999 · Zbl 1225.76060 |
| [36] | Escher, J.; Lienstromberg, C.Travelling waves in dilatant non‐Newtonian thin films. J. Differential Equations264 (2018), no. 3, 2113-2132. https://doi.org/10.1016/j.jde.2017.10.015 · Zbl 1378.35064 · doi:10.1016/j.jde.2017.10.015 |
| [37] | Freistühler, H.Phase transitions and traveling waves in compressible fluids. Arch. Ration. Mech. Anal.211 (2014), no. 1, 189-204. https://doi.org/10.1007/s00205-013-0682-0 · Zbl 1290.35200 · doi:10.1007/s00205-013-0682-0 |
| [38] | Gellrich, R. S.Free boundary value problems for the stationary Navier‐Stokes equations in domains with noncompact boundaries. Z. Anal. Anwendungen12 (1993), no. 3, 425-455. https://doi.org/10.4171/ZAA/554 · Zbl 0778.76019 · doi:10.4171/ZAA/554 |
| [39] | Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3. Ann. of Math. (2)175 (2012), no. 2, 691-754. 10.4007/annals.2012.175.2.6 · Zbl 1244.35134 |
| [40] | Germain, P.; Masmoudi, N.; Shatah, J.Global existence for capillary water waves. Comm. Pure Appl. Math.68 (2015), no. 4, 625-687. https://doi.org/10.1002/cpa.21535 · Zbl 1314.35100 · doi:10.1002/cpa.21535 |
| [41] | Gravina, G.; Leoni, G. On the existence of non‐flat profiles for a bernoulli free boundary problem. Adv. Calc. Var., to appear. · Zbl 1486.35480 |
| [42] | Gravina, G.; Leoni, G.On the behavior of the free boundary for a one‐phase Bernoulli problem with mixed boundary conditions. Comm. Pure Appl. Anal.19 (2020), no. 10, 4853-4878. https://doi.org/10.3934/cpaa.2020215 · Zbl 1460.35397 · doi:10.3934/cpaa.2020215 |
| [43] | Groves, M. D.Steady water waves. J. Nonlinear Math. Phys.11 (2004), no. 4, 435-460. https://doi.org/10.2991/jnmp.2004.11.4.2 · Zbl 1064.35144 · doi:10.2991/jnmp.2004.11.4.2 |
| [44] | Groves, M. D.; Sun, S.‐M.Fully localised solitary‐wave solutions of the three‐dimensional gravity‐capillary water‐wave problem. Arch. Ration. Mech. Anal.188 (2008), no. 1, 1-91. https://doi.org/10.1007/s00205-007-0085-1 · Zbl 1133.76010 · doi:10.1007/s00205-007-0085-1 |
| [45] | Groves, M. D.; Wahlén, E.Small‐amplitude Stokes and solitary gravity water waves with an arbitrary distribution of vorticity. Phys. D237 (2008), no. 10‐12, 1530-1538. https://doi.org/10.1016/j.physd.2008.03.015 · Zbl 1143.76398 · doi:10.1016/j.physd.2008.03.015 |
| [46] | Guo, Y.; Tice, I.Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Ration. Mech. Anal.207 (2013), no. 2, 459-531. https://doi.org/10.1007/s00205-012-0570-z · Zbl 1320.35259 · doi:10.1007/s00205-012-0570-z |
| [47] | Guo, Y.; Tice, I.Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE6 (2013), no. 6, 1429-1533. https://doi.org/10.2140/apde.2013.6.1429 · Zbl 1292.35206 · doi:10.2140/apde.2013.6.1429 |
| [48] | Guo, Y.; Tice, I.Local well‐posedness of the viscous surface wave problem without surface tension. Anal. PDE6 (2013), no. 2, 287-369. https://doi.org/10.2140/apde.2013.6.287 · Zbl 1273.35209 · doi:10.2140/apde.2013.6.287 |
| [49] | Hataya, Y.Decaying solution of a Navier‐Stokes flow without surface tension. J. Math. Kyoto Univ.49 (2009), no. 4, 691-717. https://doi.org/10.1215/kjm/1265899478 · Zbl 1382.35190 · doi:10.1215/kjm/1265899478 |
| [50] | Hunter, J. K.; Ifrim, M.; Tataru, D.Two dimensional water waves in holomorphic coordinates. Comm. Math. Phys.346 (2016), no. 2, 483-552. https://doi.org/10.1007/s00220-016-2708-6 · Zbl 1358.35121 · doi:10.1007/s00220-016-2708-6 |
| [51] | Hur, V. M.Exact solitary water waves with vorticity. Arch. Ration. Mech. Anal.188 (2008), no. 2, 213-244. https://doi.org/10.1007/s00205-007-0064-6 · Zbl 1138.76025 · doi:10.1007/s00205-007-0064-6 |
| [52] | Ifrim, M.; Tataru, D. Two dimensional water waves in holomorphic coordinates II: Global solutions. Bull. Soc. Math. France144 (2016), no. 2, 369-394. 10.24033/bsmf.2717 · Zbl 1360.35179 |
| [53] | Inci, H.; Kappeler, T.; Topalov, P. On the regularity of the composition of diffeomorphisms. Mem. Amer. Math. Soc. 226 (2013), no. 1062, vi+60. 10.1090/S0065‐9266‐2013‐00676‐4 · Zbl 1293.58004 |
| [54] | Ionescu, A. D.; Pusateri, F.Global solutions for the gravity water waves system in 2d. Invent. Math.199 (2015), no. 3, 653-804. https://doi.org/10.1007/s00222-014-0521-4 · Zbl 1325.35151 · doi:10.1007/s00222-014-0521-4 |
| [55] | Ionescu, A. D.; Pusateri, F. Global regularity for 2D water waves with surface tension. Mem. Amer. Math. Soc. 256 (2018), no. 1227, v+124. 10.1090/memo/1227 · Zbl 1435.76002 |
| [56] | Iooss, G.; Plotnikov, P. I. Small divisor problem in the theory of three‐dimensional water gravity waves. Mem. Amer. Math. Soc. 200 (2009), no. 940, viii+128. 10.1090/memo/0940 · Zbl 1172.76005 |
| [57] | Jang, J.; Tice, I.; Wang, Y.The compressible viscous surface‐internal wave problem: stability and vanishing surface tension limit. Comm. Math. Phys.343 (2016), no. 3, 1039-1113. https://doi.org/10.1007/s00220-016-2603-1 · Zbl 1383.35163 · doi:10.1007/s00220-016-2603-1 |
| [58] | Jean, M.Free surface of the steady flow of a Newtonian fluid in a finite channel. Arch. Rational Mech. Anal.74 (1980), no. 3, 197-217. https://doi.org/10.1007/BF00280538 · Zbl 0451.76038 · doi:10.1007/BF00280538 |
| [59] | Kagei, Y.; Nishida, T.Traveling waves bifurcating from plane Poiseuille flow of the compressible Navier‐Stokes equation. Arch. Ration. Mech. Anal.231 (2019), no. 1, 1-44. https://doi.org/10.1007/s00205-018-1269-6 · Zbl 1419.76539 · doi:10.1007/s00205-018-1269-6 |
| [60] | Keady, G.; Norbury, J.On the existence theory for irrotational water waves. Math. Proc. Cambridge Philos. Soc.83 (1978), no. 1, 137-157. https://doi.org/10.1017/S0305004100054372 · Zbl 0393.76015 · doi:10.1017/S0305004100054372 |
| [61] | Krasovskiĭ, J. P. On the theory of steady‐state waves of finite amplitude. Ž. Vyčisl. Mat i Mat. Fiz. 1 (1961), 836-855. · Zbl 0158.12003 |
| [62] | Lannes, D.Well‐posedness of the water‐waves equations. J. Amer. Math. Soc.18 (2005), no. 3, 605-654. https://doi.org/10.1090/S0894-0347-05-00484-4 · Zbl 1069.35056 · doi:10.1090/S0894-0347-05-00484-4 |
| [63] | Leoni, G.A first course in Sobolev spaces. Second edition. Graduate Studies in Mathematics, 181. American Mathematical Society, Providence, RI, 2017. 10.1090/gsm/181 · Zbl 1382.46001 |
| [64] | Leoni, G.; Tice, I.Traces for homogeneous Sobolev spaces in infinite strip‐like domains. J. Funct. Anal.277 (2019), no. 7, 2288-2380. https://doi.org/10.1016/j.jfa.2019.01.005 · Zbl 1436.46039 · doi:10.1016/j.jfa.2019.01.005 |
| [65] | Leoni, G.; Tice, I.Traveling wave solutions to the free boundary incompressible Navier‐Stokes equations. Preprint.1912.10091 |
| [66] | Levi‐Civita, T. Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda. Rend. Accad. Lincei33, 141-150. · JFM 50.0680.01 |
| [67] | Masmoudi, N.; Rousset, F.Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations. Arch. Ration. Mech. Anal.223 (2017), no. 1, 301-417. https://doi.org/10.1007/s00205-016-1036-5 · Zbl 1359.35133 · doi:10.1007/s00205-016-1036-5 |
| [68] | Masnadi, N.; Duncan, J. H. The generation of gravity‐capillary solitary waves by a pressure source moving at a trans‐critical speed. J. Fluid Mech. 810 (2017), 448-474. 10.1017/jfm.2016.658 |
| [69] | McLeod, J. B.The Stokes and Krasovskii conjectures for the wave of greatest height. Stud. Appl. Math.98 (1997), no. 4, 311-333. https://doi.org/10.1111/1467-9590.00051 · Zbl 0882.76014 · doi:10.1111/1467-9590.00051 |
| [70] | Nazarov, S. A.; Pileckas, K.On the solvability of the Stokes and Navier‐Stokes problems in the domains that are layer‐like at infinity. J. Math. Fluid Mech.1 (1999), no. 1, 78-116. https://doi.org/10.1007/s000210050005 · Zbl 0941.35062 · doi:10.1007/s000210050005 |
| [71] | Nazarov, S. A.; Pileckas, K.The asymptotic properties of the solution to the Stokes problem in domains that are layer‐like at infinity. J. Math. Fluid Mech.1 (1999), no. 2, 131-167. https://doi.org/10.1007/s000210050007 · Zbl 0940.35155 · doi:10.1007/s000210050007 |
| [72] | Nekrasov, A. On steady waves. Izv. Ivanovo‐Voznesenk. Politekhn. 33 (1921). |
| [73] | Nishida, T.; Teramoto, Y.; Yoshihara, H.Global in time behavior of viscous surface waves: horizontally periodic motion. J. Math. Kyoto Univ.44 (2004), no. 2, 271-323. https://doi.org/10.1215/kjm/1250283555 · Zbl 1095.35028 · doi:10.1215/kjm/1250283555 |
| [74] | Park, B.; Cho, Y.Experimental observation of gravity-capillary solitary waves generated by a moving air suction. J. Math. Fluid Mech.808 (2016), 168-188. https://doi.org/10.1017/jfm.2016.639 · Zbl 1383.76011 · doi:10.1017/jfm.2016.639 |
| [75] | Park, B.; Cho, Y.Two‐dimensional gravity-capillary solitary waves on deep water: generation and transverse instability. J. Math. Fluid Mech.834 (2018), 92-124. https://doi.org/10.1017/jfm.2017.740 · doi:10.1017/jfm.2017.740 |
| [76] | Pileckas, K. Gliding of a flat plate of infinite span over the surface of a heavy viscous incompressible fluid of finite depth. Differentsial’nye Uravneniya i Primenen. (1983), no. 34, 60-74. · Zbl 0564.76052 |
| [77] | Pileckas, K. A remark on the paper: “Gliding of a flat plate of infinite span over the surface of a heavy viscous incompressible fluid of finite depth”. Differentsial’nye Uravneniya i Primenen. (1984), no. 36, 55-60, 139. · Zbl 0588.76046 |
| [78] | Pileckas, K.On the asymptotic behavior of solutions of a stationary system of Navier‐Stokes equations in a domain of layer type. Mat. Sb.193 (2002), no. 12, 69-104. https://doi.org/10.1070/SM2002v193n12ABEH000700 · Zbl 1069.35052 · doi:10.1070/SM2002v193n12ABEH000700 |
| [79] | Pileckas, K.; Zaleskis, L. On a steady three‐dimensional noncompact free boundary value problem for the Navier‐Stokes equations. Zap. Nauchn. Sem. S.‐Peterburg. Otdel. Mat. Inst. Steklov. (POMI)306 (2003), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funktsii. 34, 134-164, 230-231. |
| [80] | Plotnikov, P. I.Proof of the Stokes conjecture in the theory of surface waves. Stud. Appl. Math.108 (2002), no. 2, 217-244. https://doi.org/10.1111/1467-9590.01408 · Zbl 1152.76339 · doi:10.1111/1467-9590.01408 |
| [81] | Plotnikov, P. I.; Toland, J. F.Convexity of Stokes waves of extreme form. Arch. Ration. Mech. Anal.171 (2004), no. 3, 349-416. https://doi.org/10.1007/s00205-003-0292-3 · Zbl 1064.76017 · doi:10.1007/s00205-003-0292-3 |
| [82] | Remond‐Tiedrez, A.; Tice, I.The viscous surface wave problem with generalized surface energies. SIAM J. Math. Anal.51 (2019), no. 6, 4894-4952. https://doi.org/10.1137/18M1195851 · Zbl 1432.35167 · doi:10.1137/18M1195851 |
| [83] | Shibata, Y.; Shimizu, S. Free boundary problems for a viscous incompressible fluid. in Kyoto Conference on the Navier‐Stokes Equations and their Applications, pp. 356-358, RIMS Kôkyûroku Bessatsu, B1, Res. Inst. Math. Sci. (RIMS), Kyoto, 2007. · Zbl 1122.35100 |
| [84] | Shibata, Y.; Shimizu, S.Report on a local in time solvability of free surface problems for the Navier‐Stokes equations with surface tension. Appl. Anal.90 (2011), no. 1, 201-214. https://doi.org/10.1080/00036811003735899 · Zbl 1209.35158 · doi:10.1080/00036811003735899 |
| [85] | Stein, E. M.Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501 |
| [86] | Strauss, W. A. Steady water waves. Bull. Amer. Math. Soc. (N.S.)47 (2010), no. 4, 671-694. 10.1090/S0273‐0979‐2010‐01302‐1 · Zbl 1426.76078 |
| [87] | Strichartz, R. S. “Graph paper” trace characterizations of functions of finite energy. J. Anal. Math.128 (2016), 239-260. https://doi.org/10.1007/s11854-016-0008-x · Zbl 1342.28005 · doi:10.1007/s11854-016-0008-x |
| [88] | Tan, Z.; Wang, Y.Zero surface tension limit of viscous surface waves. Comm. Math. Phys.328 (2014), no. 2, 733-807. https://doi.org/10.1007/s00220-014-1986-0 · Zbl 1296.35125 · doi:10.1007/s00220-014-1986-0 |
| [89] | Tani, A.; Tanaka, N.Large‐time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Rational Mech. Anal.130 (1995), no. 4, 303-314. https://doi.org/10.1007/BF00375142 · Zbl 0844.76025 · doi:10.1007/BF00375142 |
| [90] | Tice, I. Asymptotic stability of shear‐flow solutions to incompressible viscous free boundary problems with and without surface tension. Z. Angew. Math. Phys. 69 (2018), no. 2, Art. 28, 39. 10.1007/s00033‐018‐0926‐9 · Zbl 1392.35212 |
| [91] | Tice, I.; Zbarsky, S. Decay of solutions to the linearized free surface Navier‐Stokes equations with fractional boundary operators. J. Math. Fluid Mech. 22 (2020), no. 48. 10.1007/s00021‐020‐00512‐8 · Zbl 1448.35371 |
| [92] | Toland, J. F.On the existence of a wave of greatest height and Stokes’s conjecture. Proc. Roy. Soc. London Ser. A363 (1978), no. 1715, 469-485. https://doi.org/10.1098/rspa.1978.0178 · Zbl 0388.76016 · doi:10.1098/rspa.1978.0178 |
| [93] | Toland, J. F. Stokes waves. Topol. Methods Nonlinear Anal. 7 (1996), no. 1, 1-48. 10.12775/TMNA.1996.001 · Zbl 0897.35067 |
| [94] | Triebel, H.Interpolation Theory,Function Spaces,Differential Operators, Huthig Pub Limited, 1995. Available at: https://books.google.com/books?id=ZzvvAAAAMAAJ · Zbl 0830.46028 |
| [95] | Wahlén, E.Steady periodic capillary‐gravity waves with vorticity. SIAM J. Math. Anal.38 (2006), no. 3, 921-943. https://doi.org/10.1137/050630465 · Zbl 1111.76007 · doi:10.1137/050630465 |
| [96] | Wahlén, E.Steady periodic capillary waves with vorticity. Ark. Mat.44 (2006), no. 2, 367-387. https://doi.org/10.1007/s11512-006-0024-7 · Zbl 1171.35312 · doi:10.1007/s11512-006-0024-7 |
| [97] | Walsh, S.Stratified steady periodic water waves. SIAM J. Math. Anal.41 (2009), no. 3, 1054-1105. https://doi.org/10.1137/080721583 · Zbl 1196.35173 · doi:10.1137/080721583 |
| [98] | Walsh, S.Steady stratified periodic gravity waves with surface tension I: Local bifurcation. Discrete Contin. Dyn. Syst.34 (2014), no. 8, 3241-3285. https://doi.org/10.3934/dcds.2014.34.3241 · Zbl 1284.35056 · doi:10.3934/dcds.2014.34.3241 |
| [99] | Walsh, S.Steady stratified periodic gravity waves with surface tension II: global bifurcation. Discrete Contin. Dyn. Syst.34 (2014), no. 8, 3287-3315. https://doi.org/10.3934/dcds.2014.34.3287 · Zbl 1511.35024 · doi:10.3934/dcds.2014.34.3287 |
| [100] | Walsh, S.; Bühler, O.; Shatah, J.Steady water waves in the presence of wind. SIAM J. Math. Anal.45 (2013), no. 4, 2182-2227. https://doi.org/10.1137/120880124 · Zbl 1277.37083 · doi:10.1137/120880124 |
| [101] | Wang, X.Global solution for the 3D gravity water waves system above a flat bottom. Adv. Math.346 (2019), 805-886. https://doi.org/10.1016/j.aim.2019.02.020 · Zbl 1412.35245 · doi:10.1016/j.aim.2019.02.020 |
| [102] | Wheeler, M. H.Large‐amplitude solitary water waves with vorticity. SIAM J. Math. Anal.45 (2013), no. 5, 2937-2994. https://doi.org/10.1137/120891460 · Zbl 1282.76078 · doi:10.1137/120891460 |
| [103] | Wheeler, M. H.Solitary water waves of large amplitude generated by surface pressure. Arch. Ration. Mech. Anal.218 (2015), no. 2, 1131-1187. https://doi.org/10.1007/s00205-015-0877-7 · Zbl 1457.35043 · doi:10.1007/s00205-015-0877-7 |
| [104] | Wu, L.Well‐posedness and decay of the viscous surface wave. SIAM J. Math. Anal.46 (2014), no. 3, 2084-2135. https://doi.org/10.1137/120897018 · Zbl 1304.35512 · doi:10.1137/120897018 |
| [105] | Wu, S.Well‐posedness in Sobolev spaces of the full water wave problem in 2‐D. Invent. Math.130 (1997), no. 1, 39-72. https://doi.org/10.1007/s002220050177 · Zbl 0892.76009 · doi:10.1007/s002220050177 |
| [106] | Wu, S.Well‐posedness in Sobolev spaces of the full water wave problem in 3‐D. J. Amer. Math. Soc.12 (1999), no. 2, 445-495. https://doi.org/10.1090/S0894-0347-99-00290-8 · Zbl 0921.76017 · doi:10.1090/S0894-0347-99-00290-8 |
| [107] | Wu, S.Almost global wellposedness of the 2‐D full water wave problem. Invent. Math.177 (2009), no. 1, 45-135. https://doi.org/10.1007/s00222-009-0176-8 · Zbl 1181.35205 · doi:10.1007/s00222-009-0176-8 |
| [108] | Wu, S.Global wellposedness of the 3‐D full water wave problem. Invent. Math.184 (2011), no. 1, 125-220. https://doi.org/10.1007/s00222-010-0288-1 · Zbl 1221.35304 · doi:10.1007/s00222-010-0288-1 |
| [109] | Zadrzyńska, E.Free boundary problems for nonstationary Navier‐Stokes equations. Dissertationes Math. (Rozprawy Mat.)424 (2004), 135. · Zbl 1062.35077 |
| [110] | Zhang, P.; Zhang, Z.On the free boundary problem of three‐dimensional incompressible Euler equations. Comm. Pure Appl. Math.61 (2008), no. 7, 877-940. https://doi.org/10.1002/cpa.20226 · Zbl 1158.35107 · doi:10.1002/cpa.20226 |
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