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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

 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.
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. 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. [3] J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles,, Ann. École Norm. Sup. (4), 26 (1993), 517. [4] Y. Giga, A. Mahalov and B. Nicolaenko, The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data,, in, 163 (2007), 213. [5] Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem,, in, 1, 2, (2003), 549. [6] N. M. Günther, Über ein Hauptproblem der Hydrodynamik,, Math. Z., 24 (1926), 448. 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. [8] T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$,, J. Funct. Anal., 9 (1972), 296. [9] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. 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. 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. [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. 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. 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. [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. 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. doi: 10.1007/s00021-009-0304-7. [17] H. Triebel, "Theory of Function Spaces,", Monogr. Math., 78 (1983). doi: 10.1007/978-3-0346-0416-1. [18] M. Vishik, Hydrodynamics in Besov spaces,, Arch. Ration. Mech. Anal., 145 (1998), 197. doi: 10.1007/s002050050128. [19] V. Yudovich, Nonstationary flow of an ideal incompressible liquid,, Zh. Vych. Mat., 3 (1963), 1032.

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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. 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. [3] J.-Y. Chemin, Persistance de structures géométriques dans les fluides incompressibles bidimensionneles,, Ann. École Norm. Sup. (4), 26 (1993), 517. [4] Y. Giga, A. Mahalov and B. Nicolaenko, The Cauchy problem for the Navier-Stokes equations with spatially almost periodic initial data,, in, 163 (2007), 213. [5] Y. Giga and O. Sawada, On regularizing-decay rate estimates for solutions to the Navier-Stokes initial value problem,, in, 1, 2, (2003), 549. [6] N. M. Günther, Über ein Hauptproblem der Hydrodynamik,, Math. Z., 24 (1926), 448. 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. [8] T. Kato, Nonstationary flows of viscous and ideal fluids in $\mathbbR^3$,, J. Funct. Anal., 9 (1972), 296. [9] T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations,, Comm. Pure Appl. Math., 41 (1988), 891. 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. 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. [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. 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. 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. [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. 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. doi: 10.1007/s00021-009-0304-7. [17] H. Triebel, "Theory of Function Spaces,", Monogr. Math., 78 (1983). doi: 10.1007/978-3-0346-0416-1. [18] M. Vishik, Hydrodynamics in Besov spaces,, Arch. Ration. Mech. Anal., 145 (1998), 197. doi: 10.1007/s002050050128. [19] V. Yudovich, Nonstationary flow of an ideal incompressible liquid,, Zh. Vych. Mat., 3 (1963), 1032.
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