Generalized resolvent estimates for the Stokes system in bounded and unbounded domains

Reinhard FARWIG; Hermann SOHR

doi:10.2969/jmsj/04640607

Reinhard FARWIG, Hermann SOHR

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JOURNAL FREE ACCESS

1994 Volume 46 Issue 4 Pages 607-643

DOI https://doi.org/10.2969/jmsj/04640607

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Published: 1994 Received: April 12, 1993 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. Amrouche and V. Girault, On the Existence and Regularity of the Solution of Stokes Problem in Arbitrary Dimension, Proc. Japan Acad. Ser. A, 67 (1991), 171-175.
3) M. E. Bogovskii, Solution of the First Boundary Value Problem for the Equation of Continuity of an Incompressible Medium, Soviet Math. Dokl., 20 (1979), 1094-1098.
4) M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Seminar S. L. Sobolev, No. 1, 80, Akademia Nauk SSR, Sibirskoe Otdelenie Matematiki, Nowosibirsk, 1980, pp. 5-40.
5) W. Borchers and T. Miyakawa, L²-Decay for the Navier-Stokes Flow in Halfspaces, Math. Ann., 282 (1988), 139-155.
6) W. Borchers and T. Miyakawa, On some coercive estimates for the Stokes problem in unbounded domains, Lecture Notes in Math., 1530, 1992, pp. 71-84.
7) 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.
9) L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
10) G. de Rham, Variétés différentiables, Paris, Hermann, 1960.
11) P. Deuring, The resolvent problem for the Stokes system in exterior domains, An elementary approach, Math. Methods Appl. Sci., 13 (1990), 335-349.
12) R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in L^q-spaces, Lecture Notes in Math., 1530, 1992, pp. 97-110.
13) R. Farwig, C. G. Simader and H. Sohr, An L^q-Theory for Weak Solutions of the Stokes System in Exterior Domains, Math. Methods Appl. Sci., 16 (1993), 707-723.
14) A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
15) D. Fujiwara and H. Morimoto, An L_γ-theorem of Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA, 24 (1977), 685-700.
16) G. P. Galdi and C. G. Simader, Existence, uniqueness and L^q-estimates for the Stokes problem in an exterior domain, Arch. Rational Mech. Anal., 112 (1990), 291-318.
17) G. P. Galdi, C. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl.
18) Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_γ spaces, Math. Z., 178 (1981), 297-329.
19) Y. Giga and H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA, 36 (1989), 103-130.
20) J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math., 136 (1976), 61-102.
21) M. McCracken, The resolvent problem for the Stokes equations on halfspace in L_p, SIAM J. Math. Anal., 12 (1981), 221-228.
22) T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.
23) M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, New York-San Francisco-London, Academic Press, 1975.
24) C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neuman problem in L_q-spaces for bounded and exterior domains, Series on Advances in Mathematics for Applied Sciences, Vol. 11, Singapore, World Scientific, 1992, pp. 1-35.
25) V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8 (1977), 467-529.
26) R. Temam, Navier-Stokes equations, Amsterdam-New York-Oxford, North-Holland, 1977.
27) H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Amsterdam -New York-Oxford, North-Holland, 1978.
28) S. Ukai, A solution formula for the Stokes equation in Rⁿ₊, Comm. Pure Appl. Math., 40 (1987), 611-621.
29) W. von Wahl, Regularitätsfragen für die instationären Navier-Stokesschen Gleichungen in höheren Dimensionen, J. Math. Soc. Japan, 32 (1980), 263-283.
30) W. von Wahl, Vorlesungen über das Auf3enraumproblem für die instationären Gleichungen von Navier-Stokes, SFB 256 Nichtlineare partielle Differentialglei-chungen, Vorlesungsreihe Nr. 11, Universität Bonn, 1989.

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. Amrouche and V. Girault, On the Existence and Regularity of the Solution of Stokes Problem in Arbitrary Dimension, Proc. Japan Acad. Ser. A, 67 (1991), 171-175.
[3] M. E. Bogovskii, Solution of the First Boundary Value Problem for the Equation of Continuity of an Incompressible Medium, Soviet Math. Dokl., 20 (1979), 1094-1098.
[4] M. E. Bogovskii, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Seminar S. L. Sobolev, No. 1, 80, Akademia Nauk SSR, Sibirskoe Otdelenie Matematiki, Nowosibirsk, 1980, pp. 5-40.
[5] W. Borchers and T. Miyakawa, L²-Decay for the Navier-Stokes Flow in Halfspaces, Math. Ann., 282 (1988), 139-155.
[6] W. Borchers and T. Miyakawa, On some coercive estimates for the Stokes problem in unbounded domains, Lecture Notes in Math., 1530, 1992, pp. 71-84.
[7] 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.
[8] W. Borchers and H. Sohr, On the equations rot v=g and div u=f with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67-87.
[9] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Sem. Mat. Univ. Padova, 31 (1961), 308-340.
[10] G. de Rham, Variétés différentiables, Paris, Hermann, 1960.
[11] P. Deuring, The resolvent problem for the Stokes system in exterior domains, An elementary approach, Math. Methods Appl. Sci., 13 (1990), 335-349.
[12] R. Farwig and H. Sohr, An approach to resolvent estimates for the Stokes equations in L^q-spaces, Lecture Notes in Math., 1530, 1992, pp. 97-110.
[13] R. Farwig, C. G. Simader and H. Sohr, An L^q-Theory for Weak Solutions of the Stokes System in Exterior Domains, Math. Methods Appl. Sci., 16 (1993), 707-723.
[14] A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
[15] D. Fujiwara and H. Morimoto, An L_γ-theorem of Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA, 24 (1977), 685-700.
[16] G. P. Galdi and C. G. Simader, Existence, uniqueness and L^q-estimates for the Stokes problem in an exterior domain, Arch. Rational Mech. Anal., 112 (1990), 291-318.
[17] G. P. Galdi, C. G. Simader and H. Sohr, On the Stokes problem in Lipschitz domains, Ann. Mat. Pura Appl.
[18] Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L_γ spaces, Math. Z., 178 (1981), 297-329.
[19] Y. Giga and H. Sohr, On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA, 36 (1989), 103-130.
[20] J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math., 136 (1976), 61-102.
[21] M. McCracken, The resolvent problem for the Stokes equations on halfspace in L_p, SIAM J. Math. Anal., 12 (1981), 221-228.
[22] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.
[23] M. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. II, New York-San Francisco-London, Academic Press, 1975.
[24] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition and the Neuman problem in L_q-spaces for bounded and exterior domains, Series on Advances in Mathematics for Applied Sciences, Vol. 11, Singapore, World Scientific, 1992, pp. 1-35.
[25] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8 (1977), 467-529.
[26] R. Temam, Navier-Stokes equations, Amsterdam-New York-Oxford, North-Holland, 1977.
[27] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Amsterdam -New York-Oxford, North-Holland, 1978.
[28] S. Ukai, A solution formula for the Stokes equation in Rⁿ₊, Comm. Pure Appl. Math., 40 (1987), 611-621.
[29] W. von Wahl, Regularitätsfragen für die instationären Navier-Stokesschen Gleichungen in höheren Dimensionen, J. Math. Soc. Japan, 32 (1980), 263-283.
[30] W. von Wahl, Vorlesungen über das Auf3enraumproblem für die instationären Gleichungen von Navier-Stokes, SFB 256 Nichtlineare partielle Differentialglei-chungen, Vorlesungsreihe Nr. 11, Universität Bonn, 1989.

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