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 & Continuous Dynamical Systems - A, 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. 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).

[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.

[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).

[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, ().

[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. 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).

[8]

J. L. Bona, Z. Grujić, Spatial analyticity for nonlinear waves,, Math. Models Methods Appl. Sci., 13 (2003), 1. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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.

[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).

[20]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. 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, (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. 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. 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. doi: doi:10.1007/BF00431721.

[25]

T. Kato, Nonstationary flows of viscous and ideal fluids in $\R^3$,, J. Functional Analysis, 9 (1972), 296. 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. 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. 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. 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. doi: doi:10.1007/BF00281084.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27, Cambridge Texts in Applied Mathematics, (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. doi: doi:10.1006/jdeq.2000.3927.

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition,, McGraw-Hill Book Co., (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. doi: doi:10.1007/s002200050304.

[38]

R. Temam, On the Euler equations of incompressible perfect fluids,, J. Functional Analysis, 20 (1975), 32. 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.

[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. 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. 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).

[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.

[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).

[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, ().

[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. 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).

[8]

J. L. Bona, Z. Grujić, Spatial analyticity for nonlinear waves,, Math. Models Methods Appl. Sci., 13 (2003), 1. 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.

[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. 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. 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. 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. 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. 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. 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. 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. 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.

[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).

[20]

C. Foias and R. Temam, Gevrey class regularity for the solutions of the Navier-Stokes equations,, J. Funct. Anal., 87 (1989), 359. 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, (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. 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. 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. doi: doi:10.1007/BF00431721.

[25]

T. Kato, Nonstationary flows of viscous and ideal fluids in $\R^3$,, J. Functional Analysis, 9 (1972), 296. 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. 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. 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. 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. doi: doi:10.1007/BF00281084.

[30]

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

[31]

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

[32]

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

[33]

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

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27, Cambridge Texts in Applied Mathematics, (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. doi: doi:10.1006/jdeq.2000.3927.

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition,, McGraw-Hill Book Co., (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. doi: doi:10.1007/s002200050304.

[38]

R. Temam, On the Euler equations of incompressible perfect fluids,, J. Functional Analysis, 20 (1975), 32. 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.

[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. doi: doi:10.1063/1.1287066.

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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 & Applied Analysis, 2002, 1 (2) : 221-235. doi: 10.3934/cpaa.2002.1.221

[10]

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

[11]

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

[12]

Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible Euler-Maxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503-518. doi: 10.3934/cpaa.2013.12.503

[13]

Yongcai Geng. Singularity formation for relativistic Euler and Euler-Poisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549-564. doi: 10.3934/cpaa.2015.14.549

[14]

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

[15]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[16]

Daria Bugajewska, Mirosława Zima. On the spectral radius of linearly bounded operators and existence results for functional-differential equations. Conference Publications, 2003, 2003 (Special) : 147-155. doi: 10.3934/proc.2003.2003.147

[17]

Moulay-Tahar Benameur, Alan L. Carey. On the analyticity of the bivariant JLO cocycle. Electronic Research Announcements, 2009, 16: 37-43. doi: 10.3934/era.2009.16.37

[18]

Benoît Pausader, Walter A. Strauss. Analyticity of the nonlinear scattering operator. Discrete & Continuous Dynamical Systems - A, 2009, 25 (2) : 617-626. doi: 10.3934/dcds.2009.25.617

[19]

Min He. A class of integrodifferential equations and applications. Conference Publications, 2005, 2005 (Special) : 386-396. doi: 10.3934/proc.2005.2005.386

[20]

Young-Pil Choi. Compressible Euler equations interacting with incompressible flow. Kinetic & Related Models, 2015, 8 (2) : 335-358. doi: 10.3934/krm.2015.8.335

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]