Weighted Lq-theory for the Stokes resolvent in exterior domains

Reinhard FARWIG; Hermann SOHR

doi:10.2969/jmsj/04920251

Weighted L^q-theory for the Stokes resolvent in exterior domains

Reinhard FARWIG, Hermann SOHR

著者情報

ジャーナルフリー

1997 年 49 巻 2 号 p. 251-288

DOI https://doi.org/10.2969/jmsj/04920251

詳細

Published: 1997 Received: 1995/03/24 Available on J-STAGE: 2006/10/20 Accepted: - Advance online publication: - Revised: -
訂正情報
訂正日: 2006/10/20 訂正理由: - 訂正箇所: 引用文献情報訂正内容: Wrong : 1) J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin, 1976.
2) W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in L_q-spaces, Math. Z., 196 (1987), 415-425.
3) P. Deuring, The resolvent problem for the Stokes system in exterior domains: An elementary approach, Math. Methods Appl. Sci., 13 (1990), 335-349.
4) R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
5) R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643.
6) A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
7) D. Fujiwara and H. Morimoto, An L_r-theorem of Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
8) J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985.
9) Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_r spaces, Math. Z., 178 (1981), 297-329.
10) Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130.
11) V. Girault, The divergence, curl and Stokes operator in exterior domains in R³, Recent developments in theoretical fluid mechanics, (eds. G. P. Galdi and J. Necas), Pitman Research Notes in Mathematics, 291, New York, 1993, pp. 34-77.
12) H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.
13) D. S. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 255 (1979), 343-362.
14) T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.
15) G. de Rham, Variétés différentiables, Hermann, Paris, 1960.
16) E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.
17) C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition in L_q-spaces for bounded and exterior domains, Advances in Math. for Appl. Sci., Vol. 11, World Scientific, Singapore, 1992.
18) V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8 (1977), 467-529.
19) M. Specovius-Neugebauer, Die Stokes-Gleichung in Cantor-Räumen and die Holomorphie der Stokes-Halbgruppe in gewichteten L^p-Räumen, Dissertation, Paderborn, 1984.
20) M. Specovius-Neugebauer, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.
21) M. Specovius-Neugebauer, The Helmholtz decomposition of weighted L^r-spaces, Comm. Partial Differential Equations, 15 (1990), 273-288.
22) A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, Orlando, 1986.
23) H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
24) W. von Wahl, Regularitätsfragen für instationären Navier-Stokes Gleichungen in höheren Dimensionen, J. Math. Soc. Japan, 32 (1980), 263-283.
25) W. von Wahl, Vorlesungen über das Außenraumproblem für die instationaren Gleichungen von Navier-Stokes, SFB 256, Vorlesungsreihe, no. 11, Univ. Bonn, 1989.

Right : [1] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Springer-Verlag, Berlin, 1976.
[2] W. Borchers and H. Sohr, On the semigroup of the Stokes operator for exterior domains in L_q-spaces, Math. Z., 196 (1987), 415-425.
[3] P. Deuring, The resolvent problem for the Stokes system in exterior domains: An elementary approach, Math. Methods Appl. Sci., 13 (1990), 335-349.
[4] R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Z., 211 (1992), 409-447.
[5] R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607-643.
[6] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[7] D. Fujiwara and H. Morimoto, An L_γ-theorem of Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 685-700.
[8] J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland, Amsterdam, 1985.
[9] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_r spaces, Math. Z., 178 (1981), 297-329.
[10] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130.
[11] V. Girault, The divergence, curl and Stokes operator in exterior domains in R³, Recent developments in theoretical fluid mechanics, (eds. G. P. Galdi and J. Necas), Pitman Research Notes in Mathematics, 291, New York, 1993, pp. 34-77.
[12] H. Komatsu, Fractional powers of operators, Pacific J. Math., 19 (1966), 285-346.
[13] D. S. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc., 255 (1979), 343-362.
[14] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.
[15] G. de Rham, Variétés différentiables, Hermann, Paris, 1960.
[16] E. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on euclidean and homogeneous spaces, Amer. J. Math., 114 (1992), 813-874.
[17] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition in L_q-spaces for bounded and exterior domains, Advances in Math. for Appl. Sci., Vol. 11, World Scientific, Singapore, 1992.
[18] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8 (1977), 467-529.
[19] M. Specovius-Neugebauer, Die Stokes-Gleichung in Cantor-Räumen und die Holomorphie der Stokes-Halbgruppe in gewichteten L^p-Räumen, Dissertation, Paderborn, 1984.
[20] M. Specovius-Neugebauer, Exterior Stokes problems and decay at infinity, Math. Methods Appl. Sci., 8 (1986), 351-367.
[21] M. Specovius-Neugebauer, The Helmholtz decomposition of weighted L^γ-spaces, Comm. Partial Differential Equations, 15 (1990), 273-288.
[22] A. Torchinsky, Real-variable methods in harmonic analysis, Academic Press, Orlando, 1986.
[23] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
[24] W. von Wahl, Regularitätsfragen für instationären Navier-Stokes Gleichungen in höheren Dimensionen, J. Math. Soc. Japan, 32 (1980), 263-283.
[25] W. von Wahl, Vorlesungen über das Außenraumproblem für die instationären Gleichungen von Navier-Stokes, SFB 256, Vorlesungsreihe, no. 11, Univ. Bonn, 1989.

訂正日: 2006/10/20 訂正理由: - 訂正箇所: PDFファイル訂正内容: -

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