# American Institute of Mathematical Sciences

July  2011, 29(3): 769-801. doi: 10.3934/dcds.2011.29.769

## Solvability of the free boundary value problem of the Navier-Stokes equations

 1 Center For Scientific Computation And Mathematical Modeling, University of Maryland, 4125 CSIC Building, Paint Branch Drive, College Park, MD 20742, United States

Received  January 2010 Revised  June 2010 Published  November 2010

In this paper, we study the incompressible Navier-Stokes equations on a moving domain in $\mathbb{R}^{3}$ of finite depth, bounded above by the free surface and bounded below by a solid flat bottom. We prove that there exists a unique, global-in-time solution to the problem provided that the initial velocity field and the initial profile of the boundary are sufficiently small in Sobolev spaces.
Citation: Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769
##### References:
 [1] R. Adams, "Sobolev Spaces," Academic Press, INC, 1975. [2] D. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci, 5 (2007), 391-430. [3] T. Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure. Appl. Math, 34 (1981), 359-392. doi: 10.1002/cpa.3160340305. [4] T. Beale, Large time regularity of viscous surface waves, Arch. Rational Mech. Anal, 84 (1983/84), 307-342. [5] A. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge University Press, 2002. [6] D. Coutand and S. Shkoller, Unique solvability of the free boundary Navier-Stokes equations with surface tension, preprint, arXiv:math/0212116. [7] 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. [8] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X. [9] R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoamericana, 21 (2005), 863-888. [10] D. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Part. Diff. Eq, 10 (1987), 1175-1201. [11] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc, 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4. [12] T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech, 382 (1999), 351-366. doi: 10.1017/S0022112098003978. [13] D. Sylvester, Large time existence of small viscous surface waves without surface tension, Comm. Partial Differential Equations, 15 (1990), 823-903. doi: 10.1080/03605309908820709. [14] 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. [15] J. Shatah and C. Zeng, Local wellposedness of interface problem, preprint. [16] V. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math, 40 (1988), 672-686. doi: 10.1007/BF01094193. [17] V. Solonnikov, Unsteady motion of an isolated volume of a viscous incompressible fluid, Math. USSR-Izv, 31 (1988), 381-405. doi: 10.1070/IM1988v031n02ABEH001081. [18] A. Tani and N. Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal, 30 (1995), 303-314. doi: 10.1007/BF00375142. [19] R. Temam, "Navier-Stokes Equations - Theory and Numerical Analysis," AMS, 2001. [20] 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. [21] 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. [22] P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math, 61 (2008), 877-940. doi: 10.1002/cpa.20226.

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##### References:
 [1] R. Adams, "Sobolev Spaces," Academic Press, INC, 1975. [2] D. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci, 5 (2007), 391-430. [3] T. Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure. Appl. Math, 34 (1981), 359-392. doi: 10.1002/cpa.3160340305. [4] T. Beale, Large time regularity of viscous surface waves, Arch. Rational Mech. Anal, 84 (1983/84), 307-342. [5] A. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge University Press, 2002. [6] D. Coutand and S. Shkoller, Unique solvability of the free boundary Navier-Stokes equations with surface tension, preprint, arXiv:math/0212116. [7] 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. [8] W. Craig, U. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X. [9] R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoamericana, 21 (2005), 863-888. [10] D. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Part. Diff. Eq, 10 (1987), 1175-1201. [11] D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc, 18 (2005), 605-654. doi: 10.1090/S0894-0347-05-00484-4. [12] T. Lundgren and P. Koumoutsakos, On the generation of vorticity at a free surface, J. Fluid Mech, 382 (1999), 351-366. doi: 10.1017/S0022112098003978. [13] D. Sylvester, Large time existence of small viscous surface waves without surface tension, Comm. Partial Differential Equations, 15 (1990), 823-903. doi: 10.1080/03605309908820709. [14] 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. [15] J. Shatah and C. Zeng, Local wellposedness of interface problem, preprint. [16] V. Solonnikov, Unsteady motion of a finite mass of fluid, bounded by a free surface, J. Soviet Math, 40 (1988), 672-686. doi: 10.1007/BF01094193. [17] V. Solonnikov, Unsteady motion of an isolated volume of a viscous incompressible fluid, Math. USSR-Izv, 31 (1988), 381-405. doi: 10.1070/IM1988v031n02ABEH001081. [18] A. Tani and N. Tanaka, Large-time existence of surface waves in incompressible viscous fluids with or without surface tension, Arch. Rational Mech. Anal, 30 (1995), 303-314. doi: 10.1007/BF00375142. [19] R. Temam, "Navier-Stokes Equations - Theory and Numerical Analysis," AMS, 2001. [20] 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. [21] 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. [22] P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math, 61 (2008), 877-940. doi: 10.1002/cpa.20226.
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