January  2011, 29(1): 285-303. doi: 10.3934/dcds.2011.29.285

The domain of analyticity of solutions to the three-dimensional Euler equations in a half space

1. 

Department of Mathematics, University of Southern California, 3620 South Vermont Ave., Los Angeles, CA 90089-2532, United States, United States

Received  October 2009 Revised  May 2010 Published  September 2010

We address the problem of analyticity up to the boundary of solutions to the Euler equations in the half space. We characterize the rate of decay of the real-analyticity radius of the solution $u(t)$ in terms of exp$\int_{0}^{t} $||$ \nabla u(s) $|| L ds , improving the previously known results. We also prove the persistence of the sub-analytic Gevrey-class regularity for the Euler equations in a half space, and obtain an explicit rate of decay of the radius of Gevrey-class regularity.
Citation: Igor Kukavica, Vlad C. Vicol. The domain of analyticity of solutions to the three-dimensional Euler equations in a half space. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 285-303. doi: 10.3934/dcds.2011.29.285
References:
[1]

S. Alinhac and G. Métivier, Propagation de l'analyticité locale pour les solutions de l'équation d'Euler, Arch. Rational Mech. Anal., 92 (1986), 287-296. doi: doi:10.1007/BF00280434.

[2]

C. Bardos, Analyticité de la solution de l'équation d'Euler dans un ouvert de $R^n$, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A255-A258.

[3]

C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 647-687.

[4]

C. Bardos, S. Benachour and M. Zerner, Analycité des solutions périodiques de l'équation d'Euler en deux dimensions, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A995-A998.

[5]

C. Bardos and E. S. Titi, Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations, Discrete Contin. Dyn. Syst, to appear, Preprint available at arXiv:0906.2029v2.

[6]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: doi:10.1007/BF01212349.

[7]

S. Benachour, Analyticité des solutions périodiques de l'équation d'Euler en trois dimensions, C. R. Acad. Sci. Paris Sér. A-B,283 (1976), A107-A110.

[8]

J. L. Bona, Z. Grujić, Spatial analyticity for nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 1-15. doi: doi:10.1142/S0218202503002532.

[9]

J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.

[10]

J. L. Bona, Z. Grujić and H. Kalisch, Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip, J. Differential Equations, 229 (2006), 186-203. doi: doi:10.1016/j.jde.2006.04.013.

[11]

J. L. Bona, Z. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139. doi: doi:10.3934/dcds.2010.26.1121.

[12]

J. L. Bona and Yi A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl. (9), 76 (1997), 377-430. doi: doi:10.1016/S0021-7824(97)89957-6.

[13]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363. doi: doi:10.1016/0022-1236(74)90027-5.

[14]

P. Constantin, E. S. Titi and J. Vukadinović, Dissipativity and Gevrey regularity of a Smoluchowski equation, Indiana Univ. Math. J., 54 (2005), 949-969. doi: doi:10.1512/iumj.2005.54.2653.

[15]

S. A. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation, Discrete Contin. Dyn. Syst., 23 (2009), 755-764. doi: doi:10.3934/dcds.2009.23.755.

[16]

R. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: doi:10.1007/BF01214424.

[17]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid, Bull. Amer. Math. Soc., 75 (1969), 962-967. doi: doi:10.1090/S0002-9904-1969-12315-3.

[18]

A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.

[19]

C. Foias, U. Frisch and R. Temam, Existence de solutions $C^{\infty}$ des équations d'Euler, C. R. Acad. Sci. Paris Sér. A-B, 280 (1975), A505-A508.

[20]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: doi:10.1016/0022-1236(89)90015-3.

[21]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[22]

Z. Grujić and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466. doi: doi:10.1006/jfan.1997.3167.

[23]

Z. Grujić and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differential Equations, 154 (1999), 42-54. doi: doi:10.1006/jdeq.1998.3562.

[24]

W. D. Henshaw, H.-O. Kreiss and L. G. Reyna, Smallest scale estimates for the Navier-Stokes equations for incompressible fluids, Arch. Rational Mech. Anal., 112 (1990), 21-44. doi: doi:10.1007/BF00431721.

[25]

T. Kato, Nonstationary flows of viscous and ideal fluids in $mathbb{R}^{3}$, J. Functional Analysis, 9 (1972), 296-305. doi: doi:10.1016/0022-1236(72)90003-1.

[26]

I. Kukavica, Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity, 8 (1995), 113-129. doi: doi:10.1088/0951-7715/8/2/001.

[27]

I. Kukavica, On the dissipative scale for the Navier-Stokes equation, Indiana Univ. Math. J., 48 (1999), 1057-1081. doi: doi:10.1512/iumj.1999.48.1748.

[28]

I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677. doi: doi:10.1090/S0002-9939-08-09693-7.

[29]

D. Le Bail, Analyticité locale pour les solutions de l'équation d'Euler, Arch. Rational Mech. Anal., 95 (1986), 117-136. doi: doi:10.1007/BF00281084.

[30]

P. G. Lemarié-Rieusset, Une remarque sur l'analyticité des solutions milds des équations de Navier-Stokes dans $mathbb{R}^{3}$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 183-186.

[31]

P. G. Lemarié-Rieusset, Nouvelles remarques sur l'analyticité des solutions milds des équations de Navier-Stokes dans $mathbb{R}^{3}$, C. R. Math. Acad. Sci. Paris, 338 (2004), 443-446.

[32]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339. doi: doi:10.1006/jdeq.1996.3200.

[33]

J.-L. Lions and E. Magenes, "Problemès aux Limites non Homogènes et Applications," Vol. 3, Dunod, Paris, 1970.

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27 Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.

[35]

M. Oliver and E. S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74. doi: doi:10.1006/jdeq.2000.3927.

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition, McGraw-Hill Book Co., New York 1976.

[37]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Commun. Math. Phys., 192 (1998), 433-461. doi: doi:10.1007/s002200050304.

[38]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: doi:10.1016/0022-1236(75)90052-X.

[39]

V. I. Yudovich, Non stationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066.

[40]

V. I. Yudovich, On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid, Chaos, 10 (2000), 705-719. doi: doi:10.1063/1.1287066.

show all references

References:
[1]

S. Alinhac and G. Métivier, Propagation de l'analyticité locale pour les solutions de l'équation d'Euler, Arch. Rational Mech. Anal., 92 (1986), 287-296. doi: doi:10.1007/BF00280434.

[2]

C. Bardos, Analyticité de la solution de l'équation d'Euler dans un ouvert de $R^n$, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), A255-A258.

[3]

C. Bardos and S. Benachour, Domaine d'analycité des solutions de l'équation d'Euler dans un ouvert de $R^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 4 (1977), 647-687.

[4]

C. Bardos, S. Benachour and M. Zerner, Analycité des solutions périodiques de l'équation d'Euler en deux dimensions, C. R. Acad. Sci. Paris Sér. A-B, 282 (1976), A995-A998.

[5]

C. Bardos and E. S. Titi, Loss of smoothness and energy conserving rough weak solutions for the 3d Euler equations, Discrete Contin. Dyn. Syst, to appear, Preprint available at arXiv:0906.2029v2.

[6]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the $3$-D Euler equations, Comm. Math. Phys., 94 (1984), 61-66. doi: doi:10.1007/BF01212349.

[7]

S. Benachour, Analyticité des solutions périodiques de l'équation d'Euler en trois dimensions, C. R. Acad. Sci. Paris Sér. A-B,283 (1976), A107-A110.

[8]

J. L. Bona, Z. Grujić, Spatial analyticity for nonlinear waves, Math. Models Methods Appl. Sci., 13 (2003), 1-15. doi: doi:10.1142/S0218202503002532.

[9]

J. L. Bona, Z. Grujić and H. Kalisch, Algebraic lower bounds for the uniform radius of spatial analyticity for the generalized KdV equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 783-797.

[10]

J. L. Bona, Z. Grujić and H. Kalisch, Global solutions of the derivative Schrödinger equation in a class of functions analytic in a strip, J. Differential Equations, 229 (2006), 186-203. doi: doi:10.1016/j.jde.2006.04.013.

[11]

J. L. Bona, Z. Grujić and H. Kalisch, A KdV-type Boussinesq system: From the energy level to analytic spaces, Discrete Contin. Dyn. Syst., 26 (2010), 1121-1139. doi: doi:10.3934/dcds.2010.26.1121.

[12]

J. L. Bona and Yi A. Li, Decay and analyticity of solitary waves, J. Math. Pures Appl. (9), 76 (1997), 377-430. doi: doi:10.1016/S0021-7824(97)89957-6.

[13]

J. P. Bourguignon and H. Brezis, Remarks on the Euler equation, J. Functional Analysis, 15 (1974), 341-363. doi: doi:10.1016/0022-1236(74)90027-5.

[14]

P. Constantin, E. S. Titi and J. Vukadinović, Dissipativity and Gevrey regularity of a Smoluchowski equation, Indiana Univ. Math. J., 54 (2005), 949-969. doi: doi:10.1512/iumj.2005.54.2653.

[15]

S. A. Denisov, Infinite superlinear growth of the gradient for the two-dimensional Euler equation, Discrete Contin. Dyn. Syst., 23 (2009), 755-764. doi: doi:10.3934/dcds.2009.23.755.

[16]

R. DiPerna and A. Majda, Oscillations and concentrations in weak solutions of the incompressible fluid equations, Comm. Math. Phys., 108 (1987), 667-689. doi: doi:10.1007/BF01214424.

[17]

D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid, Bull. Amer. Math. Soc., 75 (1969), 962-967. doi: doi:10.1090/S0002-9904-1969-12315-3.

[18]

A. B. Ferrari and E. S. Titi, Gevrey regularity for nonlinear analytic parabolic equations, Comm. Partial Differential Equations, 23 (1998), 1-16.

[19]

C. Foias, U. Frisch and R. Temam, Existence de solutions $C^{\infty}$ des équations d'Euler, C. R. Acad. Sci. Paris Sér. A-B, 280 (1975), A505-A508.

[20]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations, J. Funct. Anal., 87 (1989), 359-369. doi: doi:10.1016/0022-1236(89)90015-3.

[21]

D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Reprint of the 1998 edition, Springer-Verlag, Berlin, 2001.

[22]

Z. Grujić and I. Kukavica, Space analyticity for the Navier-Stokes and related equations with initial data in $L^p$, J. Funct. Anal., 152 (1998), 447-466. doi: doi:10.1006/jfan.1997.3167.

[23]

Z. Grujić and I. Kukavica, Space analyticity for the nonlinear heat equation in a bounded domain, J. Differential Equations, 154 (1999), 42-54. doi: doi:10.1006/jdeq.1998.3562.

[24]

W. D. Henshaw, H.-O. Kreiss and L. G. Reyna, Smallest scale estimates for the Navier-Stokes equations for incompressible fluids, Arch. Rational Mech. Anal., 112 (1990), 21-44. doi: doi:10.1007/BF00431721.

[25]

T. Kato, Nonstationary flows of viscous and ideal fluids in $mathbb{R}^{3}$, J. Functional Analysis, 9 (1972), 296-305. doi: doi:10.1016/0022-1236(72)90003-1.

[26]

I. Kukavica, Hausdorff length of level sets for solutions of the Ginzburg-Landau equation, Nonlinearity, 8 (1995), 113-129. doi: doi:10.1088/0951-7715/8/2/001.

[27]

I. Kukavica, On the dissipative scale for the Navier-Stokes equation, Indiana Univ. Math. J., 48 (1999), 1057-1081. doi: doi:10.1512/iumj.1999.48.1748.

[28]

I. Kukavica and V. Vicol, On the radius of analyticity of solutions to the three-dimensional Euler equations, Proc. Amer. Math. Soc., 137 (2009), 669-677. doi: doi:10.1090/S0002-9939-08-09693-7.

[29]

D. Le Bail, Analyticité locale pour les solutions de l'équation d'Euler, Arch. Rational Mech. Anal., 95 (1986), 117-136. doi: doi:10.1007/BF00281084.

[30]

P. G. Lemarié-Rieusset, Une remarque sur l'analyticité des solutions milds des équations de Navier-Stokes dans $mathbb{R}^{3}$, C. R. Acad. Sci. Paris Sér. I Math., 330 (2000), 183-186.

[31]

P. G. Lemarié-Rieusset, Nouvelles remarques sur l'analyticité des solutions milds des équations de Navier-Stokes dans $mathbb{R}^{3}$, C. R. Math. Acad. Sci. Paris, 338 (2004), 443-446.

[32]

C. D. Levermore and M. Oliver, Analyticity of solutions for a generalized Euler equation, J. Differential Equations, 133 (1997), 321-339. doi: doi:10.1006/jdeq.1996.3200.

[33]

J.-L. Lions and E. Magenes, "Problemès aux Limites non Homogènes et Applications," Vol. 3, Dunod, Paris, 1970.

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27 Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2002.

[35]

M. Oliver and E. S. Titi, On the domain of analyticity of solutions of second order analytic nonlinear differential equations, J. Differential Equations, 174 (2001), 55-74. doi: doi:10.1006/jdeq.2000.3927.

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition, McGraw-Hill Book Co., New York 1976.

[37]

M. Sammartino and R. E. Caflisch, Zero viscosity limit for analytic solutions of the Navier-Stokes equation on a half-space. I. Existence for Euler and Prandtl equations, Commun. Math. Phys., 192 (1998), 433-461. doi: doi:10.1007/s002200050304.

[38]

R. Temam, On the Euler equations of incompressible perfect fluids, J. Functional Analysis, 20 (1975), 32-43. doi: doi:10.1016/0022-1236(75)90052-X.

[39]

V. I. Yudovich, Non stationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066.

[40]

V. I. Yudovich, On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid, Chaos, 10 (2000), 705-719. doi: doi:10.1063/1.1287066.

[1]

Jaeseop Ahn, Jimyeong Kim, Ihyeok Seo. On the radius of spatial analyticity for defocusing nonlinear Schrödinger equations. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 423-439. doi: 10.3934/dcds.2020016

[2]

Alain Haraux, Mitsuharu Ôtani. Analyticity and regularity for a class of second order evolution equations. Evolution Equations and Control Theory, 2013, 2 (1) : 101-117. doi: 10.3934/eect.2013.2.101

[3]

Valentin Keyantuo, Louis Tebou, Mahamadi Warma. A Gevrey class semigroup for a thermoelastic plate model with a fractional Laplacian: Between the Euler-Bernoulli and Kirchhoff models. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2875-2889. doi: 10.3934/dcds.2020152

[4]

Mei-Qin Zhan. Gevrey class regularity for the solutions of the Phase-Lock equations of Superconductivity. Conference Publications, 2001, 2001 (Special) : 406-415. doi: 10.3934/proc.2001.2001.406

[5]

Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 507-522. doi: 10.3934/dcds.1998.4.507

[6]

Fucai Li, Zhipeng Zhang. Zero viscosity-resistivity limit for the 3D incompressible magnetohydrodynamic equations in Gevrey class. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4279-4304. doi: 10.3934/dcds.2018187

[7]

Christophe Cheverry, Mekki Houbad. A class of large amplitude oscillating solutions for three dimensional Euler equations. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1661-1697. doi: 10.3934/cpaa.2012.11.1661

[8]

Qiaoyi Hu, Zhijun Qiao. Analyticity, Gevrey regularity and unique continuation for an integrable multi-component peakon system with an arbitrary polynomial function. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6975-7000. doi: 10.3934/dcds.2016103

[9]

A. Alexandrou Himonas, Gerson Petronilho. A $ G^{\delta, 1} $ almost conservation law for mCH and the evolution of its radius of spatial analyticity. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 2031-2050. doi: 10.3934/dcds.2020351

[10]

Hua Chen, Wei-Xi Li, Chao-Jiang Xu. Propagation of Gevrey regularity for solutions of Landau equations. Kinetic and Related Models, 2008, 1 (3) : 355-368. doi: 10.3934/krm.2008.1.355

[11]

Nan Chen, Cheng Wang, Steven Wise. Global-in-time Gevrey regularity solution for a class of bistable gradient flows. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1689-1711. doi: 10.3934/dcdsb.2016018

[12]

Marcel Oliver. The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$. Communications on Pure and Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221

[13]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[14]

Hideshi Yamane. Local and global analyticity for $\mu$-Camassa-Holm equations. Discrete and Continuous Dynamical Systems, 2020, 40 (7) : 4307-4340. doi: 10.3934/dcds.2020182

[15]

Josep M. Burgués, Joan Mateu. On the analyticity of the trajectories of the particles in the planar patch problem for some active scalar equations. Discrete and Continuous Dynamical Systems, 2022, 42 (6) : 2945-3003. doi: 10.3934/dcds.2022005

[16]

Feng Cheng, Chao-Jiang Xu. On the Gevrey regularity of solutions to the 3D ideal MHD equations. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6485-6506. doi: 10.3934/dcds.2019281

[17]

Wenmeng Geng, Kai Tao. Lyapunov exponents of discrete quasi-periodic gevrey Schrödinger equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2977-2996. doi: 10.3934/dcdsb.2020216

[18]

Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1409-1415. doi: 10.3934/dcdss.2013.6.1409

[19]

Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 1-10. doi: 10.3934/dcdsb.2009.11.1

[20]

Franco Flandoli, Dejun Luo. Euler-Lagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153-165. doi: 10.3934/jgm.2019008

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (138)
  • HTML views (0)
  • Cited by (17)

Other articles
by authors

[Back to Top]