Density properties for solenoidal vector fields, with applications to the Navier-Stokes equations in exterior domains

Hideo KOZONO; Hermann SOHR

doi:10.2969/jmsj/04420307

Hideo KOZONO, Hermann SOHR

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1992 Volume 44 Issue 2 Pages 307-330

DOI https://doi.org/10.2969/jmsj/04420307

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Published: 1992 Received: April 17, 1991 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: SUBTITLE Details: Wrong : Dedicated to Professor Dr. Reimund Rautmann on his 60th birthday

Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. A. Adams, Sobolev spaces, New York-San Francisco-London, Academic Press, 1977.
2) N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation method., Ann. Math. Pure Appl., 68 (1965), 51-117.
3) J. Bergh and J. Löfström, Interpolation spaces, Berlin-Heidelberg-New York, Springer 1976.
4) M. E. Bogovski, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Sov. Math. Dokl., 20 (1979), 1094-1098.
5) M. E. Bogovski, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Seminar S. L. Sobolev, No. 1, 80, Akademia Nauk SSSR, Sibirskoe Otdelenie Matematiki, Nowosibirsk, 5-40 (1980).
6) 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.
7) L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova 31 (1961), 308-340.
8) A. Friedman, Partial differential equations, New York. Holt Rinehart & Winston, 1969.
9) H. Fujita, On the existence and regularity of steady state solutions of the Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo, Sec. IA, 9 (1961), 59-102.
10) D. Fujiwara and H. Morimoto, An L_r theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sec. IA, 24 (1977), 685-700.
11) Y. Giga, Domains of fractional powers of the Stokes operator in L_r spaces. Arch. Rational Mech. Anal., 89 (1985), 251-265.
12) Y. Giga, Solutions for semilinear parabolic equations in L^p and regularity of weak solutions of the Navier-Stokes system, J. Differential Eq. 62 (1986), 182-212.
13) Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sec. IA, 36 (1989), 103-130.
14) J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math., 136 (1976), 61-102.
15) H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J., 42 (1991), 1-28. Indiana Univ. Math. J., 40 (1991), 1-27.
16) O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York, Gordon & Breach, 1969.
17) K. Masuda, Weak solutions of the Navier-Stokes equations, Tohoku Math. J., 36 (1984), 623-646.
18) T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in exterior domains, Hiroshima Math. J., 12 (1982), 115-140.
19) M. Reed and B. Simon, Method of Modern Mathematical Physics II, New York-San Francisco-London, Academic Press, 1975.
20) J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Problems, R. Langer ed., Madison, The University of Wisconsin Press, 1963, pp. 69-98.
21) C. G. Simader, Private communication, 1988.
22) C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition in L^q-spaces for bounded and exterior domains, to appear in Mathematical Problems Relating to the Navier-Stokes Equations, Series on Advanced in Mathematics for Applied Sciences, 11 World Scientific.
23) H. Sohr and W. von Wahl, On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations, Manuscripta Math., 49 (1984), 27-59.
24) R. Temam, Navier-Stokes equations, Amsterdam, North-Holland, 1977.
25) H. Triebel, Interpolation theory, function spaces, differential operators, Amsterdam, North-Holland, 1978.
26) K. Yosida, Functional analysis, Berlin-Heidelberg-New York., Springer, 1965.

Right : [1] R. A. Adams, Sobolev spaces, New York-San Francisco-London, Academic Press, 1977.
[2] N. Aronszajn and E. Gagliardo, Interpolation spaces and interpolation method., Ann. Math. Pure Appl., 68 (1965), 51-117.
[3] J. Bergh and J. Löfström, Interpolation spaces, Berlin-Heidelberg-New York, Springer 1976.
[4] M. E. Bogovski, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Sov. Math. Dokl., 20 (1979), 1094-1098.
[5] M. E. Bogovski, Solution of some vector analysis problems connected with operators div and grad (in Russian), Trudy Seminar S. L. Sobolev, No. 1, 80, Akademia Nauk SSSR, Sibirskoe Otdelenie Matematiki, Nowosibirsk, 5-40 (1980).
[6] 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.
[7] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Mat. Sem. Univ. Padova 31 (1961), 308-340.
[8] A. Friedman, Partial differential equations, New York. Holt Rinehart & Winston, 1969.
[9] H. Fujita, On the existence and regularity of steady state solutions of the Navier-Stokes equations, J. Fac. Sci. Univ. Tokyo, Sec. IA, 9 (1961), 59-102.
[10] D. Fujiwara and H. Morimoto, An L_r theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sec. IA, 24 (1977), 685-700.
[11] Y. Giga, Domains of fractional powers of the Stokes operator in L_r spaces. Arch. Rational Mech. Anal., 89 (1985), 251-265.
[12] Y. Giga, Solutions for semilinear parabolic equations in L^p and regularity of weak solutions of the Navier-Stokes system, J. Differential Eq. 62 (1986), 182-212.
[13] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sec. IA, 36 (1989), 103-130.
[14] J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math., 136 (1976), 61-102.
[15] H. Kozono and H. Sohr, New a priori estimates for the Stokes equations in exterior domains, Indiana Univ. Math. J., 42 (1991), 1-28. Indiana Univ. Math. J., 40 (1991), 1-27.
[16] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York, Gordon & Breach, 1969.
[17] K. Masuda, Weak solutions of the Navier-Stokes equations, Tôhoku Math. J., 36 (1984), 623-646.
[18] T. Miyakawa, On nonstationary solutions of the Navier-Stokes equations in exterior domains, Hiroshima Math. J., 12 (1982), 115-140.
[19] M. Reed and B. Simon, Method of Modern Mathematical Physics II, New York-San Francisco-London, Academic Press, 1975.
[20] J. Serrin, The initial value problem for the Navier-Stokes equations, in Nonlinear Problems, R. Langer ed., Madison, The University of Wisconsin Press, 1963, pp. 69-98.
[21] C. G. Simader, Private communication, 1988.
[22] C. G. Simader and H. Sohr, A new approach to the Helmholtz decomposition in L^q-spaces for bounded and exterior domains, to appear in Mathematical Problems Relating to the Navier-Stokes Equations, Series on Advanced in Mathematics for Applied Sciences, 11 World Scientific.
[23] H. Sohr and W. von Wahl, On the singular set and the uniqueness of weak solutions of the Navier-Stokes equations, Manuscripta Math., 49 (1984), 27-59.
[24] R. Temam, Navier-Stokes equations, Amsterdam, North-Holland, 1977.
[25] H. Triebel, Interpolation theory, function spaces, differential operators, Amsterdam, North-Holland, 1978.
[26] K. Yosida, Functional analysis, Berlin-Heidelberg-New York., Springer, 1965.

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