Analytic rates of solutions to the Euler equations

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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

Abstract
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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 and 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.
[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.
[3]	J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles, Ann. École Norm. Sup. (4), 26 (1993), 517-542.
[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.
[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.
[6]	N. M. Günther, Über ein Hauptproblem der Hydrodynamik, Math. Z., 24 (1926), 448-499. doi: 10.1007/BF01216794.
[7]	C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 33 (1969), 386-405.
[8]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	H. Triebel, "Theory of Function Spaces," Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[18]	M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.
[19]	V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066.

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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.
[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.
[3]	J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles, Ann. École Norm. Sup. (4), 26 (1993), 517-542.
[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.
[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.
[6]	N. M. Günther, Über ein Hauptproblem der Hydrodynamik, Math. Z., 24 (1926), 448-499. doi: 10.1007/BF01216794.
[7]	C. Kahane, On the spatial analyticity of solutions of the Navier-Stokes equations, Arch. Ration. Mech. Anal., 33 (1969), 386-405.
[8]	T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$, J. Funct. Anal., 9 (1972), 296-305.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[17]	H. Triebel, "Theory of Function Spaces," Monogr. Math., 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.
[18]	M. Vishik, Hydrodynamics in Besov spaces, Arch. Ration. Mech. Anal., 145 (1998), 197-214. doi: 10.1007/s002050050128.
[19]	V. Yudovich, Nonstationary flow of an ideal incompressible liquid, Zh. Vych. Mat., 3 (1963), 1032-1066.

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