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Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems
On the 2D free boundary Euler equation
1. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089 |
2. | Department of Mathematics, The Petroleum Institute, Abu Dhabi |
References:
[1] |
T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499. |
[2] |
T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the water-waves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin-10 Juin (2011), Exposé no. III, p. 20. |
[3] |
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. |
[4] |
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521. |
[5] |
J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359-392. |
[6] |
J. T. Beale, T. Y. Hou, and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301. |
[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. |
[8] |
D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449. |
[9] |
D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602. |
[10] |
D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201. |
[11] |
T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., 44 (2001), 219-241. |
[12] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. |
[13] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. |
[14] |
I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 1355-1389. |
[15] |
D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic). |
[16] |
H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197. |
[17] |
H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194. |
[18] |
V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104-210, 254 . |
[19] |
M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740. |
[20] |
B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781. |
[21] |
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. |
[22] |
A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170) (1985), 82-109, 144. |
[23] |
A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299-331. |
[24] |
T. Tao, Harmonic analysis,, , ().
|
[25] |
S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. |
[26] |
S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. |
[27] |
L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 1044-1111. |
[28] |
H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96. |
[29] |
H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694. |
[30] |
P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940. |
show all references
References:
[1] |
T. Alazard, N. Burq and C. Zuily, On the water-wave equations with surface tension, Duke Math. J., 158 (2011), 413-499. |
[2] |
T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the water-waves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin-10 Juin (2011), Exposé no. III, p. 20. |
[3] |
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of two-dimensional water waves, Comm. Pure Appl. Math., 58 (2005), 1287-1315. |
[4] |
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of three-dimensional water waves, Indiana Univ. Math. J., 58 (2009), 479-521. |
[5] |
J. Thomas Beale, The initial value problem for the Navier-Stokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359-392. |
[6] |
J. T. Beale, T. Y. Hou, and J. S. Lowengrub, Growth rates for the linearized motion of fluid interfaces away from equilibrium, Comm. Pure Appl. Math., 46 (1993), 1269-1301. |
[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. |
[8] |
D. Coutand and S. Shkoller, A simple proof of well-posedness for the free-surface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429-449. |
[9] |
D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 1536-1602. |
[10] |
D. G. Ebin, The equations of motion of a perfect fluid with free boundary are not well posed, Comm. Partial Differential Equations, 12 (1987), 1175-1201. |
[11] |
T. Iguchi, Well-posedness of the initial value problem for capillary-gravity waves, Funkcial. Ekvac., 44 (2001), 219-241. |
[12] |
T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. |
[13] |
C. E. Kenig, G. Ponce and L. Vega, Well-posedness of the initial value problem for the Korteweg-de Vries equation, J. Amer. Math. Soc., 4 (1991), 323-347. |
[14] |
I. Kukavica and A. Tuffaha, Solutions to a fluid-structure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 1355-1389. |
[15] |
D. Lannes, Well-posedness of the water-waves equations, J. Amer. Math. Soc., 18 (2005), 605-654 (electronic). |
[16] |
H. Lindblad, Well-posedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153-197. |
[17] |
H. Lindblad, Well-posedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109-194. |
[18] |
V. I. Nalimov, The Cauchy-Poisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104-210, 254 . |
[19] |
M. Ogawa and A. Tani, Free boundary problem for an incompressible ideal fluid with surface tension, Math. Models Methods Appl. Sci., 12 (2002), 1725-1740. |
[20] |
B. Schweizer, On the three-dimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753-781. |
[21] |
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. |
[22] |
A. I. Shnirelman, The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid, Mat. Sb. (N.S.), 128(170) (1985), 82-109, 144. |
[23] |
A. Tani, Small-time existence for the three-dimensional Navier-Stokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299-331. |
[24] |
T. Tao, Harmonic analysis,, , ().
|
[25] |
S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in $2$-D, Invent. Math., 130 (1997), 39-72. |
[26] |
S. Wu, Well-posedness in Sobolev spaces of the full water wave problem in 3-D, J. Amer. Math. Soc., 12 (1999), 445-495. |
[27] |
L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 1044-1111. |
[28] |
H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 49-96. |
[29] |
H. Yosihara, Capillary-gravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649-694. |
[30] |
P. Zhang and Z. Zhang, On the free boundary problem of three-dimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877-940. |
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