Abstract
We consider the free boundary problem for a layer of viscous, incompressible fluid in a uniform gravitational field, lying above a rigid bottom and below the atmosphere. For the “semi-small” initial data, we prove the zero surface tension limit of the problem within a local time interval. The unique local strong solution with surface tension is constructed as the limit of a sequence of approximate solutions to a special parabolic regularization. For the small initial data, we prove the global-in-time zero surface tension limit of the problem.
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Allain G.: Small-time existence for the Navier–Stokes equations with a free surface. Appl. Math. Optim. 16(1), 37–50 (1987)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of two-dimensional water waves. Commun. Pure Appl. Math. 58(10), 1287–1315 (2005)
Ambrose D.M., Masmoudi N.: The zero surface tension limit of three-dimensional water waves. Indiana Univ. Math. J. 58(2), 479–522 (2009)
Bae H.: Solvability of the free boundary value problem of the Navier–Stokes equations. Discret. Contin. Dyn. Syst. 29(3), 769–801 (2011)
Beale J.: The initial value problem for the Navier–Stokes equations with a free surface. Commun. Pure Appl. Math. 34(3), 359–392 (1981)
Beale J.: Large-time regularity of viscous surface waves. Arch. Ration. Mech. Anal. 84(4), 307–352 (1983)
Beale, J., Nishida, T.: Large-time behavior of viscous surface waves. In: Recent Topics in Nonlinear PDE, II (Sendai, 1984), North-Holland Mathematics Studies, Vol. 128, pp. 1–14. North-Holland, Amsterdam (1985)
Cheng C.H.A., Shkoller S.: The interaction of the 3D Navier–Stokes equations with a moving nonlinear Koiter elastic shell. SIAM J. Math. Anal. 42(3), 1094–1155 (2010)
Coutand, D., Shkoller, S.: Unique solvability of the free-boundary Navier–Stokes equations with surface tension (2003, preprint) (arXiv:math/0212116)
Coutand D., Shkoller S.: On the interaction between quasilinear elastodynamics and the Navier–Stokes equations. Arch. Ration. Mech. Anal. 179(3), 303–352 (2006)
Danchin R.: Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients. Rev. Mat. Iberoam. 21(3), 863–888 (2005)
Guo Y., Tice I.: Local well-posedness of the viscous surface wave problem without surface tension. Anal. PDE 6(2), 287–369 (2013)
Guo Y., Tice I.: Almost exponential decay of periodic viscous surface waves without surface tension. Arch. Ration. Mech. Anal. 207(2), 459–531 (2013)
Guo Y., Tice I.: Decay of viscous surface waves without surface tension in horizontally infinite domains. Anal. PDE 6(6), 1429–1533 (2013)
Hadzic, M., Shkoller, S.: Well-posedness for the classical Stefan problem and the zero surface tension limit (2011, preprint) (arXiv:math.AP/1112.5817)
Hataya Y.: Decaying solution of a Navier–Stokes flow without surface tension. J. Math. Kyoto Univ. 49(4), 691–717 (2009)
Ladyzhenskaya O.A.: The Mathematical Theory of Viscous Incompressible Flows. Gordon and Breach, New York (1969)
Nishida T., Teramoto Y., Yoshihara H.: Global in time behavior of viscous surface waves: horizontally periodic motion. J. Math. Kyoto Univ. 44(2), 271–323 (2004)
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, Research Institute for Mathematical Sciences (RIMS), Kyoto (2007)
Solonnikov V.A.: Solvability of a problem on the motion of a viscous incompressible fluid that is bounded by a free surface. Math. USSR Izv. 11(6), 1323–1358 (1978)
Solonnikov V.A.: On an initial boundary value problem for the Stokes systems arising in the study of a problem with a free boundary. Proc. Steklov Inst. Math. 3, 191–239 (1991)
Solonnikov V.A.: Solvability of the problem of evolution of a viscous incompressible fluid bounded by a free surface on a finite time interval. St. Petersb. Math. J. 3(1), 189–220 (1992)
Sylvester D.L.G.: Large time existence of small viscous surface waves without surface tension. Commun. Part. Differ. Equ. 15(6), 823–903 (1990)
Tani A.: Small-time existence for the three-dimensional Navier–Stokes equations for an incompressible fluid with a free surface. Arch. Ration. Mech. Anal. 133(4), 299–331 (1996)
Tani A., Tanaka N.: Large-time existence of surface waves in incompressible viscous fluids with or without surface tension. Arch. Ration. Mech. Anal. 130(4), 303–314 (1995)
Wehausen J., Laitone E.: Surface waves. Handbuch der Physik, Vol. 9, Part 3, pp. 446–778. Springer-Verlag, Berlin (1960)
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Communicated by L. Caffarelli
Y. J. Wang was supported by the National Natural Science Foundation of China (No. 11201389), the Natural Science Foundation of Fujian Province of China (No. 2012J05011), the Specialized Research Fund for the Doctoral program of Higher Education (No. 20120121120023), and the Fundamental Research Funds for the Central Universities (No. 2013121002). Z. Tan was supported by the National Natural Science Foundation of China (No. 11271305).
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Tan, Z., Wang, Y. Zero Surface Tension Limit of Viscous Surface Waves. Commun. Math. Phys. 328, 733–807 (2014). https://doi.org/10.1007/s00220-014-1986-0
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DOI: https://doi.org/10.1007/s00220-014-1986-0