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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[8]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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).   Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27, Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[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.  Google Scholar

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition,, McGraw-Hill Book Co., (1976).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

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.  Google Scholar

[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).   Google Scholar

[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.   Google Scholar

[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).   Google Scholar

[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, ().   Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[8]

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

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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).   Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[33]

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

[34]

A. J. Majda and A. L. Bertozzi, "Vorticity and Incompressible Flow," Vol. 27, Cambridge Texts in Applied Mathematics, (2002).   Google Scholar

[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.  Google Scholar

[36]

W. Rudin, "Principles of Mathematical Analysis," 3rd edition,, McGraw-Hill Book Co., (1976).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[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.  Google Scholar

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