Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Derivation of Equations for Continuum Mechanics and Thermodynamics of Fluids Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

27. Water Waves with or Without Surface Tension

verfasst von : Diego Córdoba, Charles Fefferman

Erschienen in: Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this survey article, we present results from the last several decades on several systems of PDE that model water waves. Some of those results provide short-time existence of smooth solutions; other results establish existence of smooth solutions for all time; still other results assert that singularities form in finite time.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Vorheriges Kapitel Weak Solutions and Diffuse Interface Models for IncompressibleTwo-Phase Flows
Nächstes Kapitel Concepts of Solutions in the Thermodynamics of Compressible Fluids
Literatur
1.
H. Abels, The initial-value problem for the Navier-Stokes equations with a free surface in L q-Sobolev spaces. Adv. Differ. Equ. 10(1), 4564 (2005)
2.
3.
4.
T. Alazard, J.M. Delort, Global solutions and asymptotic behavior for two dimensional gravity water waves. Ann. Sci. Ec. Norm. Super. (4) 48(5), 1149–1238 (2015)
5.
T. Alazard, J.M. Delort, Sobolev estimates for two dimensional gravity water waves. Asterisque N. 374(viii), p. 241 (2015)
6.
D.M. Ambrose, N. Masmoudi, The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 12871315 (2005)
7.
C.J. Amick, L.E. Fraenkel, J.F. Toland, On the Stokes conjecture for the wave of extreme form. Acta Math. 148, 193–214 (1982)MathSciNetCrossRefMATH
8.
J. Beale, The initial value problem for the Navier-Stokes equations with a free surface. Commun. Pure Appl. Math. 34(3), 359–392 (1981)MathSciNetCrossRefMATH
9.
10.
J. Beale, T. Nishida, Large-time behavior of viscous surface waves, in Recent Topics in Nonlinear PDE, II, Sendai, 1984. North-Holland Mathematics Studies, vol. 128 (North-Holland, Amsterdam, 1985), pp. 1–14
11.
J.T. Beale, T.Y. Hou, J.S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium. Commun. Pure Appl. Math. 46(9), 1269–1301 (1993)MathSciNetCrossRefMATH
12.
J.T. Beale, T.Y. Hou, J. Lowengrub, Convergence of a boundary integral method for water waves. SIAM J. Numer. Anal. 33(5), 1797–1843 (1996)MathSciNetCrossRefMATH
13.
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano, Splash singularity for water waves. Proc. Natl. Acad. Sci. 109(3), 733–738 (2012)MathSciNetCrossRefMATH
14.
A. Castro, D. Córdoba, C.L. Fefferman, F. Gancedo, J. Gómez-Serrano, Finite time singularities for water waves with surface tension. J. Math. Phys. 53(11), 115622,26 (2012)
15.
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, M. López-Fernández, Rayleigh-Taylor breakdown for the Muskat problem with applications to water waves. Ann. Math. 175(2), 909–948 (2012)MathSciNetCrossRefMATH
16.
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano, Finite time singularities for the free boundary incompressible Euler equations. Ann. Math. 178(3), 1061–1134 (2013)MathSciNetCrossRefMATH
17.
A. Castro, D. Córdoba, C. Fefferman, F. Gancedo, J. Gómez-Serrano. Splash singularities for the free boundary Navier-Stokes equations. Preprint arXiv:1504.02775
18.
A. Castro, D. Córdoba, C.L. Fefferman, F. Gancedo, J. Gómez-Serrano, Structural stability for the splash singularities of the water wave problem. Discrete Contin. Dyn. Syst. 34(12), 4997–5043 (2014)MathSciNetCrossRefMATH
19.
D. Christodoulou, H. Lindblad, On the motion of the free surface of a liquid. Commun. Pure Appl. Math. 53(12), 1536–1602 (2000)MathSciNetCrossRefMATH
20.
21.
D. Córdoba, T. Pernas-Casta\( \tilde{n} \) o, Non-splat singularity for the one-phase Muskat problem. Trans. Am. Math. Soc. 369(1), 711–754 (2017, to appear)
22.
23.
D. Córdoba, A. Enciso, N. Grubic, On the existence of stationary splash singularities for the Euler equations. Adv. Math. 288, 922–941 (2016)MathSciNetCrossRefMATH
24.
D. Coutand, S. Shkoller, Well-posedness of the free-surface incompressible Euler equations with or without surface tension. J. Am. Math. Soc. 20(3), 829–930 (2007)MathSciNetCrossRefMATH
25.
D. Coutand, S. Shkoller, On the finite-time splash and splat singularities for the 3-D free-surface Euler equations. Commun. Math. Phys. 325(1), 143–183 (2014)MathSciNetCrossRefMATH
26.
D. Coutand, S. Shkoller, On the splash singularity for the free-surface of a Navier-Stokes fluid. Arxiv preprint arXiv:1505.01929
27.
D. Coutand, S. Shkoller, On the impossibility of finite-time splash singularities for vortex sheets. Arxiv preprint arXiv:1407.1479 (2014)
28.
W. Craig, An existence theory for water waves and the Boussinesq and Korteweg-de Vries scaling limits. Commun. Partial Differ. Equ. 10(8), 787–1003 (1985)MathSciNetCrossRefMATH
29.
30.
Y. Deng, A.D. Ionescu, B. Pausader, F. Pusateri, Global solutions of the gravity-capillary water waves system in 3D. Preprint arXiv:1601.05685
31.
C. Fefferman, A. Ionescu, V. Lie, On the absence of splash singularities in the case of two-fluid interfaces. Duke Math. 163(3), 417–462 (2016)MathSciNetCrossRefMATH
32.
P. Germain, N. Masmoudi, J. Shatah, Global solutions for the gravity water waves equation in dimension 3. C. R. Math. Acad. Sci. Paris 347(15–16), 897–902 (2009)MathSciNetCrossRefMATH
33.
P. Germain, N. Masmoudi, J. Shatah, Global solutions for capillary waves equation in dimension 3. Commun. Pure Appl. Math. 68(4), 625–687 (2015)CrossRefMATH
34.
Y. Guo, I. Tice, Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)MathSciNetCrossRefMATH
35.
Y. Guo, I. Tice, Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429–1533 (2013)MathSciNetCrossRefMATH
36.
Y. Guo, I. Tice, Almost exponential decay of periodic viscous surface waves without surface tension. ARch. Ration. Mech. Anal. 207(2), 459–531 (2013)MathSciNetCrossRefMATH
37.
Y. Hataya, Globlal solution of two-layer Navier-Stokes flow. Nonlinear Anal. 63, 1409–1420 (2005)CrossRef
38.
39.
M. Ifrim, D. Tataru, Two dimensional water waves in holomorphic coordinates II: global solutions. Bull. Soc. Math. Fr. 144(2), 369–394 (2016)MathSciNetCrossRefMATH
40.
M. Ifrim, D. Tataru, The lifespan of small data solutions in two dimensional capillary water waves. Preprint arXiv:1406.5471
41.
42.
43.
D. Lannes, The Water Waves Problem: Mathematical Analysis and Asymptotics. Mathematical Surveys and Monographs (American Mathematical Society, Providence, 2013)CrossRefMATH
44.
45.
46.
H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary. Ann. Math. (2) 162(1), 109–194 (2005)
47.
R. Moore, F. Bierbaum, Methods and Applications of Interval Analysis, vol. 2 (Society for Industrial & Applied Mathematics, Philadelphia, 1979)CrossRef
48.
M. Muskat, Two fluid systems in porous media. The encroachment of water into an oil sand. Physics 5, 250–264 (1934)MATH
49.
V.I. Nalimov, The Cauchy-Poisson problem. Dinamika Splo\( \breve{s} \)n. Sredy, (Vyp. 18 Dinamika Zidkost. so Svobod. Granicami), 104–210 (1974)
50.
A.I. Nekrasov, Tocnaya teoriya voln ustanovivsegocya vida na poverhnosti tyazeloy zidkosti. Izdat. Akad. Nauk SSSR, Moscow (1951)
51.
P.I. Plotnikov, Justification of the Stokes conjecture in the theory of surface waves. Dinamika Sploshn. Sredy 57, 41–76 (1982)MathSciNetMATH
52.
P.G. Saffman, G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. Ser. A 245, 312–329 (1958)MathSciNetCrossRefMATH
53.
V.A. Solonnikov, Solvability of a problem on the motion of a viscous incompressible fluid that is bounded by a free surface. Maht. USSR-Izu 6, 1323–1358 (1978)MATH
54.
V.A. Solonnikov, On an initial boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Proc. Steklov Inst. Maht. 3, 191–239 (1991)MATH
55.
V.A. Solonnikov, Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersburg Math. J. 3(1), 189–220 (1992)MathSciNetMATH
56.
G.G. Stokes, Considerations relative to the greatest height of oscillatory irrota- tional waves which can be propagated without change of form, in Mathematical and Physical Papers, ed. by G.G. Stokes, J. Larmor, J.W. Strutt Rayleigh (Cambridge University Press, Cambridge, 1880)MATH
57.
D.L.G. Sylvester, Large time existence of small viscous surface waves without surface tension. Commun. Partial Differ. Equ. 15(6), 823–903 (1990)MathSciNetCrossRefMATH
58.
59.
60.
61.
62.
63.
S. Wu, On a class of self-similar 2d surface water waves (2012). Arxiv preprint arXiv:1206.2208
64.
H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth. Publ. Res. Inst. Math. Sci. 18(1), 49–96 (1982)MathSciNetCrossRefMATH
65.
P. Zhang, Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations. Commun. Pure Appl. Math. 61(7), 877–940 (2008)MathSciNetCrossRefMATH
Metadaten
Titel
Water Waves with or Without Surface Tension
verfasst von
Diego Córdoba
Charles Fefferman
Copyright-Jahr
2018
Buch
Handbook of Mathematical Analysis in Mechanics of Viscous Fluids

Print ISBN: 978-3-319-13343-0

Electronic ISBN: 978-3-319-13344-7

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-13344-7_30