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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

Hantaek Bae 1,

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.
Keywords: Navier-Stokes equations, Free Boundary, Surface Tension..
Mathematics Subject Classification: 35K51, 76D0.
Citation: Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & 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.  Google Scholar

[2]

D. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci, 5 (2007), 391-430.  Google Scholar

[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.  Google Scholar

[4]

T. Beale, Large time regularity of viscous surface waves,, Arch. Rational Mech. Anal, 84 (): 307.   Google Scholar

[5]

A. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge University Press, 2002.  Google Scholar

[6]

D. Coutand and S. Shkoller, Unique solvability of the free boundary Navier-Stokes equations with surface tension,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoamericana, 21 (2005), 863-888.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Shatah and C. Zeng, Local wellposedness of interface problem,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Temam, "Navier-Stokes Equations - Theory and Numerical Analysis," AMS, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Adams, "Sobolev Spaces," Academic Press, INC, 1975.  Google Scholar

[2]

D. Ambrose and N. Masmoudi, Well-posedness of 3D vortex sheets with surface tension, Commun. Math. Sci, 5 (2007), 391-430.  Google Scholar

[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.  Google Scholar

[4]

T. Beale, Large time regularity of viscous surface waves,, Arch. Rational Mech. Anal, 84 (): 307.   Google Scholar

[5]

A. Bertozzi and A. Majda, "Vorticity and Incompressible Flow," Cambridge University Press, 2002.  Google Scholar

[6]

D. Coutand and S. Shkoller, Unique solvability of the free boundary Navier-Stokes equations with surface tension,, preprint, ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

R. Danchin, Estimates in Besov spaces for transport and transport-diffusion equations with almost Lipschitz coefficients, Rev. Mat. Iberoamericana, 21 (2005), 863-888.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. Shatah and C. Zeng, Local wellposedness of interface problem,, preprint., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

R. Temam, "Navier-Stokes Equations - Theory and Numerical Analysis," AMS, 2001.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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