
Previous Article
On the exponential stabilization of a thermo piezoelectric/piezomagnetic system
 EECT Home
 This Issue

Next Article
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 waterwave equations with surface tension, Duke Math. J., 158 (2011), 413499. 
[2] 
T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the waterwaves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin10 Juin (2011), Exposé no. III, p. 20. 
[3] 
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of twodimensional water waves, Comm. Pure Appl. Math., 58 (2005), 12871315. 
[4] 
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of threedimensional water waves, Indiana Univ. Math. J., 58 (2009), 479521. 
[5] 
J. Thomas Beale, The initial value problem for the NavierStokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359392. 
[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), 12691301. 
[7] 
D. Coutand and S. Shkoller, Wellposedness of the freesurface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829930. 
[8] 
D. Coutand and S. Shkoller, A simple proof of wellposedness for the freesurface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429449. 
[9] 
D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 15361602. 
[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), 11751201. 
[11] 
T. Iguchi, Wellposedness of the initial value problem for capillarygravity waves, Funkcial. Ekvac., 44 (2001), 219241. 
[12] 
T. Kato and G. Ponce, Commutator estimates and the Euler and NavierStokes equations, Comm. Pure Appl. Math., 41 (1988), 891907. 
[13] 
C. E. Kenig, G. Ponce and L. Vega, Wellposedness of the initial value problem for the Kortewegde Vries equation, J. Amer. Math. Soc., 4 (1991), 323347. 
[14] 
I. Kukavica and A. Tuffaha, Solutions to a fluidstructure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 13551389. 
[15] 
D. Lannes, Wellposedness of the waterwaves equations, J. Amer. Math. Soc., 18 (2005), 605654 (electronic). 
[16] 
H. Lindblad, Wellposedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153197. 
[17] 
H. Lindblad, Wellposedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109194. 
[18] 
V. I. Nalimov, The CauchyPoisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104210, 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), 17251740. 
[20] 
B. Schweizer, On the threedimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753781. 
[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), 698744. 
[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), 82109, 144. 
[23] 
A. Tani, Smalltime existence for the threedimensional NavierStokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299331. 
[24] 
T. Tao, Harmonic analysis,, , (). 
[25] 
S. Wu, Wellposedness in Sobolev spaces of the full water wave problem in $2$D, Invent. Math., 130 (1997), 3972. 
[26] 
S. Wu, Wellposedness in Sobolev spaces of the full water wave problem in 3D, J. Amer. Math. Soc., 12 (1999), 445495. 
[27] 
L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 10441111. 
[28] 
H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 4996. 
[29] 
H. Yosihara, Capillarygravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649694. 
[30] 
P. Zhang and Z. Zhang, On the free boundary problem of threedimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877940. 
show all references
References:
[1] 
T. Alazard, N. Burq and C. Zuily, On the waterwave equations with surface tension, Duke Math. J., 158 (2011), 413499. 
[2] 
T. Alazard, N. Burq and C. Zuily, Low regularity Cauchy theory for the waterwaves problem: canals and swimming pools, Journeés Équations aux Dérivées Partielles, Biarritz 6 Juin10 Juin (2011), Exposé no. III, p. 20. 
[3] 
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of twodimensional water waves, Comm. Pure Appl. Math., 58 (2005), 12871315. 
[4] 
D. M. Ambrose and N. Masmoudi, The zero surface tension limit of threedimensional water waves, Indiana Univ. Math. J., 58 (2009), 479521. 
[5] 
J. Thomas Beale, The initial value problem for the NavierStokes equations with a free surface, Comm. Pure Appl. Math., 34 (1981), 359392. 
[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), 12691301. 
[7] 
D. Coutand and S. Shkoller, Wellposedness of the freesurface incompressible Euler equations with or without surface tension, J. Amer. Math. Soc., 20 (2007), 829930. 
[8] 
D. Coutand and S. Shkoller, A simple proof of wellposedness for the freesurface incompressible Euler equations, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 429449. 
[9] 
D. Christodoulou and H. Lindblad, On the motion of the free surface of a liquid, Comm. Pure Appl. Math., 53 (2000), 15361602. 
[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), 11751201. 
[11] 
T. Iguchi, Wellposedness of the initial value problem for capillarygravity waves, Funkcial. Ekvac., 44 (2001), 219241. 
[12] 
T. Kato and G. Ponce, Commutator estimates and the Euler and NavierStokes equations, Comm. Pure Appl. Math., 41 (1988), 891907. 
[13] 
C. E. Kenig, G. Ponce and L. Vega, Wellposedness of the initial value problem for the Kortewegde Vries equation, J. Amer. Math. Soc., 4 (1991), 323347. 
[14] 
I. Kukavica and A. Tuffaha, Solutions to a fluidstructure interaction free boundary problem, Discrete Contin. Dyn. Syst., 32 (2012), 13551389. 
[15] 
D. Lannes, Wellposedness of the waterwaves equations, J. Amer. Math. Soc., 18 (2005), 605654 (electronic). 
[16] 
H. Lindblad, Wellposedness for the linearized motion of an incompressible liquid with free surface boundary, Comm. Pure Appl. Math., 56 (2003), 153197. 
[17] 
H. Lindblad, Wellposedness for the motion of an incompressible liquid with free surface boundary, Ann. of Math. (2), 162 (2005), 109194. 
[18] 
V. I. Nalimov, The CauchyPoisson problem, Dinamika Splošn. Sredy no. Vyp. 18 Dinamika Zidkost. so Svobod. Granicami,(1974), 104210, 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), 17251740. 
[20] 
B. Schweizer, On the threedimensional Euler equations with a free boundary subject to surface tension, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 753781. 
[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), 698744. 
[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), 82109, 144. 
[23] 
A. Tani, Smalltime existence for the threedimensional NavierStokes equations for an incompressible fluid with a free surface, Arch. Rational Mech. Anal., 133 (1996), 299331. 
[24] 
T. Tao, Harmonic analysis,, , (). 
[25] 
S. Wu, Wellposedness in Sobolev spaces of the full water wave problem in $2$D, Invent. Math., 130 (1997), 3972. 
[26] 
S. Wu, Wellposedness in Sobolev spaces of the full water wave problem in 3D, J. Amer. Math. Soc., 12 (1999), 445495. 
[27] 
L. Xu and Z. Zhang, On the free boundary problem to the two viscous immiscible fluids, J. Differential Equations, 248 (2010), 10441111. 
[28] 
H. Yosihara, Gravity waves on the free surface of an incompressible perfect fluid of finite depth, Publ. Res. Inst. Math. Sci., 18 (1982), 4996. 
[29] 
H. Yosihara, Capillarygravity waves for an incompressible ideal fluid, J. Math. Kyoto Univ., 23 (1983), 649694. 
[30] 
P. Zhang and Z. Zhang, On the free boundary problem of threedimensional incompressible Euler equations, Comm. Pure Appl. Math., 61 (2008), 877940. 
[1] 
Chengchun Hao. Cauchy problem for viscous shallow water equations with surface tension. Discrete and Continuous Dynamical Systems  B, 2010, 13 (3) : 593608. doi: 10.3934/dcdsb.2010.13.593 
[2] 
Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible freeboundary Euler equations with surface tension in the case of a liquid. Evolution Equations and Control Theory, 2019, 8 (3) : 503542. doi: 10.3934/eect.2019025 
[3] 
Calin Iulian Martin. Dispersion relations for periodic water waves with surface tension and discontinuous vorticity. Discrete and Continuous Dynamical Systems, 2014, 34 (8) : 31093123. doi: 10.3934/dcds.2014.34.3109 
[4] 
Jie Wang, Xiaoqiang Wang. New asymptotic analysis method for phase field models in moving boundary problem with surface tension. Discrete and Continuous Dynamical Systems  B, 2015, 20 (9) : 31853213. doi: 10.3934/dcdsb.2015.20.3185 
[5] 
Calin I. Martin. On threedimensional free surface water flows with constant vorticity. Communications on Pure and Applied Analysis, , () : . doi: 10.3934/cpaa.2022053 
[6] 
Vera Mikyoung Hur. On the formation of singularities for surface water waves. Communications on Pure and Applied Analysis, 2012, 11 (4) : 14651474. doi: 10.3934/cpaa.2012.11.1465 
[7] 
Daniel Coutand, Steve Shkoller. A simple proof of wellposedness for the freesurface incompressible Euler equations. Discrete and Continuous Dynamical Systems  S, 2010, 3 (3) : 429449. doi: 10.3934/dcdss.2010.3.429 
[8] 
Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete and Continuous Dynamical Systems  S, 2011, 4 (1) : 209222. doi: 10.3934/dcdss.2011.4.209 
[9] 
Robert McOwen, Peter Topalov. Asymptotics in shallow water waves. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 31033131. doi: 10.3934/dcds.2015.35.3103 
[10] 
Hiroshi Matsuzawa. A free boundary problem for the FisherKPP equation with a given moving boundary. Communications on Pure and Applied Analysis, 2018, 17 (5) : 18211852. doi: 10.3934/cpaa.2018087 
[11] 
Jing Cui, Guangyue Gao, ShuMing Sun. Controllability and stabilization of gravitycapillary surface water waves in a basin. Communications on Pure and Applied Analysis, 2022, 21 (6) : 20352063. doi: 10.3934/cpaa.2021158 
[12] 
Chengchun Hao. Remarks on the free boundary problem of compressible Euler equations in physical vacuum with general initial densities. Discrete and Continuous Dynamical Systems  B, 2015, 20 (9) : 28852931. doi: 10.3934/dcdsb.2015.20.2885 
[13] 
Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete and Continuous Dynamical Systems  S, 2021, 14 (3) : 10631078. doi: 10.3934/dcdss.2020230 
[14] 
Harunori Monobe, Hirokazu Ninomiya. Multiple existence of traveling waves of a free boundary problem describing cell motility. Discrete and Continuous Dynamical Systems  B, 2014, 19 (3) : 789799. doi: 10.3934/dcdsb.2014.19.789 
[15] 
Aimin Huang, Roger Temam. The nonlinear 2D subcritical inviscid shallow water equations with periodicity in one direction. Communications on Pure and Applied Analysis, 2014, 13 (5) : 20052038. doi: 10.3934/cpaa.2014.13.2005 
[16] 
Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure and Applied Analysis, 2002, 1 (3) : 379415. doi: 10.3934/cpaa.2002.1.379 
[17] 
Denys Dutykh, Dimitrios Mitsotakis. On the relevance of the dam break problem in the context of nonlinear shallow water equations. Discrete and Continuous Dynamical Systems  B, 2010, 13 (4) : 799818. doi: 10.3934/dcdsb.2010.13.799 
[18] 
Madalina Petcu, Roger Temam. An interface problem: The twolayer shallow water equations. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 53275345. doi: 10.3934/dcds.2013.33.5327 
[19] 
Octavian G. Mustafa. On isolated vorticity regions beneath the water surface. Communications on Pure and Applied Analysis, 2012, 11 (4) : 15231535. doi: 10.3934/cpaa.2012.11.1523 
[20] 
Vincent Duchêne, Samer Israwi, Raafat Talhouk. Shallow water asymptotic models for the propagation of internal waves. Discrete and Continuous Dynamical Systems  S, 2014, 7 (2) : 239269. doi: 10.3934/dcdss.2014.7.239 
2020 Impact Factor: 1.081
Tools
Metrics
Other articles
by authors
[Back to Top]