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Improvement of some anisotropic regularity criteria for the Navier--Stokes equations
Analytic rates of solutions to the Euler equations
1. | Department of Mathematical and Design Engineering, Gifu University, Yanagido 1-1, Gifu, 501-1193, Japan |
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. |
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. |
[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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