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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 |
References:
[1] | |
[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. |
show all references
References:
[1] | |
[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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