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October  2013, 6(5): 1409-1415. doi: 10.3934/dcdss.2013.6.1409

Analytic rates of solutions to the Euler equations

Okihiro Sawada 1,

1. 

Department of Mathematical and Design Engineering, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan

Received  November 2011 Revised  April 2012 Published  March 2013

The Cauchy problem of the Euler equations is considered with initial data with possibly less regularity. The time-local existence and the uniqueness of strong solutions were established by Pak-Park, when the initial velocity is in the Besov space $B^1_{\infty, 1}$. By treating non-decaying initial data, we are able to discuss the propagation of almost periodicity. It is also proved that if the initial data are real analytic, then the solutions become necessarily real analytic in space variables with an explicit convergence rate of the radius in Taylor's expansion. This result comes from the calculation of higher order derivatives, inductively.
Keywords: analyticity, Euler equations, almost periodicity, Besov space..
Mathematics Subject Classification: Primary: 35Q31; Secondary: 76B0.
Citation: 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
References:
[1]

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

[2]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  Google Scholar

[3]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles, Ann. École Norm. Sup. (4), 26 (1993), 517-542.  Google Scholar

[4]

Y. Giga, A. Mahalov and B. Nicolaenko, The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data, in "Mathematical Aspects of Nonlinear Dispersive Equations," Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007), 213-222.  Google Scholar

[5]

Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, in "Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday," 1, 2, Kluwer Acad. Publ., Dordrecht, (2003), 549-562.  Google Scholar

[6]

N. M. Günther, Über ein Hauptproblem der Hydrodynamik, Math. Z., 24 (1926), 448-499. doi: 10.1007/BF01216794.  Google Scholar

[7]

C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 33 (1969), 386-405.  Google Scholar

[8]

T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.  Google Scholar

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[10]

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: 10.1090/S0002-9939-08-09693-7.  Google Scholar

[11]

H. Miura and O. Sawada, On the regularizing rate estimates of Koch-Tataru's solution to the Navier-Stokes equations, Asymptot. Anal., 49 (2006), 1-15.  Google Scholar

[12]

H. C. Pak and Y. J. Park, Existence of solution for the Euler equations in a critical Besov space $B^1_{\infty,1} (\mathbbR^n)$, Comm. Partial Differential Equations, 29 (2004), 1149-1166. doi: 10.1081/PDE-200033764.  Google Scholar

[13]

O. Sawada and R. Takada, On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity, J. Funct. Anal., 260 (2011), 2148-2162. doi: 10.1016/j.jfa.2010.12.011.  Google Scholar

[14]

H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $\mathbbR^3$, Trans. Amer. Math. Soc., 157 (1971), 373-397.  Google Scholar

[15]

R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725. doi: 10.1007/s00028-008-0403-6.  Google Scholar

[16]

Y. Taniuchi, T. Tashiro and T. Yoneda, On the two-dimensional Euler equations with spatially almost periodic initial data, J. Math. Fluid Mech., 12 (2010), 594-612. doi: 10.1007/s00021-009-0304-7.  Google Scholar

[17]

H. Triebel, "Theory of Function Spaces," Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[18]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.  Google Scholar

[19]

V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066. 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. Ration. Mech. Anal., 92 (1986), 287-296. doi: 10.1007/BF00280434.  Google Scholar

[2]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94 (1984), 61-66.  Google Scholar

[3]

J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles, Ann. École Norm. Sup. (4), 26 (1993), 517-542.  Google Scholar

[4]

Y. Giga, A. Mahalov and B. Nicolaenko, The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data, in "Mathematical Aspects of Nonlinear Dispersive Equations," Ann. of Math. Stud., 163, Princeton Univ. Press, Princeton, NJ, (2007), 213-222.  Google Scholar

[5]

Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem, in "Nonlinear Analysis and Applications: To V. Lakshmikantham on his 80th Birthday," 1, 2, Kluwer Acad. Publ., Dordrecht, (2003), 549-562.  Google Scholar

[6]

N. M. Günther, Über ein Hauptproblem der Hydrodynamik, Math. Z., 24 (1926), 448-499. doi: 10.1007/BF01216794.  Google Scholar

[7]

C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 33 (1969), 386-405.  Google Scholar

[8]

T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.  Google Scholar

[9]

T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907. doi: 10.1002/cpa.3160410704.  Google Scholar

[10]

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: 10.1090/S0002-9939-08-09693-7.  Google Scholar

[11]

H. Miura and O. Sawada, On the regularizing rate estimates of Koch-Tataru's solution to the Navier-Stokes equations, Asymptot. Anal., 49 (2006), 1-15.  Google Scholar

[12]

H. C. Pak and Y. J. Park, Existence of solution for the Euler equations in a critical Besov space $B^1_{\infty,1} (\mathbbR^n)$, Comm. Partial Differential Equations, 29 (2004), 1149-1166. doi: 10.1081/PDE-200033764.  Google Scholar

[13]

O. Sawada and R. Takada, On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity, J. Funct. Anal., 260 (2011), 2148-2162. doi: 10.1016/j.jfa.2010.12.011.  Google Scholar

[14]

H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in $\mathbbR^3$, Trans. Amer. Math. Soc., 157 (1971), 373-397.  Google Scholar

[15]

R. Takada, Local existence and blow-up criterion for the Euler equations in Besov spaces of weak type, J. Evol. Equ., 8 (2008), 693-725. doi: 10.1007/s00028-008-0403-6.  Google Scholar

[16]

Y. Taniuchi, T. Tashiro and T. Yoneda, On the two-dimensional Euler equations with spatially almost periodic initial data, J. Math. Fluid Mech., 12 (2010), 594-612. doi: 10.1007/s00021-009-0304-7.  Google Scholar

[17]

H. Triebel, "Theory of Function Spaces," Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar

[18]

M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.  Google Scholar

[19]

V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066. Google Scholar

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