The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in R3

Keisuke KIKUCHI

doi:10.2969/jmsj/03840575

The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in R³

Keisuke KIKUCHI

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JOURNAL FREE ACCESS

1986 Volume 38 Issue 4 Pages 575-598

DOI https://doi.org/10.2969/jmsj/03840575

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Published: 1986 Received: September 28, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.
2) C. Bardos and U. Frisch, Finite-time regularity for bounded and unbounded ideal incompressible fluid using Hölder estimates, Turbulence and Navier-Stokes Equation, Lecture Notes in Math., 565, Springer, 1976, pp. 1-13.
3) O. V. Besov, V. P. II'in and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vol. II, Winston-Wiley, 1979.
4) J. P. Bourguignon and H. Brézis, Remark on the Euler equation, J. Funct. Anal., 15 (1974), 341-363.
5) M. Cantor, Perfect fluid flows over Rⁿ with asymptotic conditions, J. Funct. Anal., 18 (1975), 73-84.
6) M. Cantor, Spaces of functions with asymptotic conditions on Rⁿ, Indiana Univ. Math. J., 24 (1975), 897-902.
7) M. Cantor, Boundary value problems for asymptotically homogeneous elliptic second order operators, J. Differential Equations, 34 (1979), 102-113.
8) D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-153.
9) C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa Ser. 4, 5 (1978), 29-63.
10) D. Fujiwara and H. Morimoto, An L_r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
11) N. M. Günter, Potential Theory and its Applications to Basic Problems of Mathematical Physics, Frederick Unger Publ., 1967.
12) S. Ito, The existence and the uniqueness of regular solution of non-stationary Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 9 (1961), 103-140.
13) V. Judovic, Two-dimensional nonstationary problem of the flow of an ideal incompressible fluid through a given region, Mat. Sb. (N. S.), 64 (1964), 562-588 (Russian) =Amer. Math. Soc. Transl. (2), 57 (1966), 277-304.
14) T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rat. Mech. Anal., 25 (1967), 188-200.
15) T. Kato, Nonstationary flows of viscous and ideal fluids in R³, J. Funct. Anal., 9 (1972), 296-305.
16) T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.
17) K. Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 63-92.
18) C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss., 130, Springer, 1966.
19) L. Nirenberg and H. Walker, The null spaces of elliptic partial differential operator in Rⁿ, J. Math. Anal. Appl., 42 (1973), 271-301.
20) H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R₃, Trans. Amer. Math. Soc., 157 (1971), 373-397.
21) H. S. G. Swann, The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in R₃, Trans. Amer. Math. Soc., 179 (1973), 167-180.
22) R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43.

Right : [1] S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. Pure Appl. Math., 12 (1959), 623-727.
[2] C. Bardos and U. Frisch, Finite-time regularity for bounded and unbounded ideal incompressible fluid using Hölder estimates, Turbulence and Navier-Stokes Equation, Lecture Notes in Math., 565, Springer, 1976, pp. 1-13.
[3] O. V. Besov, V. P. II'in and S. M. Nikol'skii, Integral Representations of Functions and Imbedding Theorems, Vol. II, Winston-Wiley, 1979.
[4] J. P. Bourguignon and H. Brézis, Remark on the Euler equation, J. Funct. Anal., 15 (1974), 341-363.
[5] M. Cantor, Perfect fluid flows over Rⁿ with asymptotic conditions, J. Funct. Anal., 18 (1975), 73-84.
[6] M. Cantor, Spaces of functions with asymptotic conditions on Rⁿ, Indiana Univ. Math. J., 24 (1975), 897-902.
[7] M. Cantor, Boundary value problems for asymptotically homogeneous elliptic second order operators, J. Differential Equations, 34 (1979), 102-113.
[8] D. Ebin and J. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102-153.
[9] C. Foias and R. Temam, Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation, Ann. Scuola Norm. Sup. Pisa Ser. 4, 5 (1978), 29-63.
[10] D. Fujiwara and H. Morimoto, An L_r-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
[11] N. M. Günter, Potential Theory and its Applications to Basic Problems of Mathematical Physics, Frederick Unger Publ., 1967.
[12] S. Itô, The existence and the uniqueness of regular solution of non-stationary Navier-Stokes equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 9 (1961), 103-140.
[13] V. Judovic, Two-dimensional nonstationary problem of the flow of an ideal incompressible fluid through a given region, Mat. Sb. (N. S.), 64 (1964), 562-588 (Russian) = Amer. Math. Soc. Transl. (2), 57 (1966), 277-304.
[14] T. Kato, On classical solutions of the two-dimensional non-stationary Euler equation, Arch. Rat. Mech. Anal., 25 (1967), 188-200.
[15] T. Kato, Nonstationary flows of viscous and ideal fluids in R³, J. Funct. Anal., 9 (1972), 296-305.
[16] T. Kato and C. Y. Lai, Nonlinear evolution equations and the Euler flow, J. Funct. Anal., 56 (1984), 15-28.
[17] K. Kikuchi, Exterior problem for the two-dimensional Euler equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1983), 63-92.
[18] C. B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Grundlehren Math. Wiss., 130, Springer, 1966.
[19] L. Nirenberg and H. Walker, The null spaces of elliptic partial differential operator in Rⁿ, J. Math. Anal. Appl., 42 (1973), 271-301.
[20] H. S. G. Swann, The convergence with vanishing viscosity of nonstationary Navier-Stokes flow to ideal flow in R₃, Trans. Amer. Math. Soc., 157 (1971), 373-397.
[21] H. S. G. Swann, The existence and uniqueness of nonstationary ideal incompressible flow in bounded domains in R₃, Trans. Amer. Math. Soc., 179 (1973), 167-180.
[22] R. Temam, On the Euler equations of incompressible perfect fluids, J. Funct. Anal., 20 (1975), 32-43.

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