Acta Mathematica
- Acta Math.
- Volume 165 (1990), 189-227.
Algebraic L2 decay for Navier-Stokes flows in exterior domains
Wolfgang Borchers and Tetsuro Miyakawa
Full-text: Open access
Article information
Source
Acta Math., Volume 165 (1990), 189-227.
Dates
Received: 2 May 1989
First available in Project Euclid: 31 January 2017
Permanent link to this document
https://projecteuclid.org/euclid.acta/1485890620
Digital Object Identifier
doi:10.1007/BF02391905
Mathematical Reviews number (MathSciNet)
MR1075041
Zentralblatt MATH identifier
0722.35014
Rights
1990 © Almqvist & Wiksell
Citation
Borchers, Wolfgang; Miyakawa, Tetsuro. Algebraic L 2 decay for Navier-Stokes flows in exterior domains. Acta Math. 165 (1990), 189--227. doi:10.1007/BF02391905. https://projecteuclid.org/euclid.acta/1485890620
References
- Agmon, S., Douglis, A. & Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., 17 (1964), 35–92.
- Bogovski, M. E., Solutions of the first boundary value problem for the equations of continuity of an incompressible medium. Soviet Math. Dokl., 20 (1979), 1094–1098.
- Borchers, W. & Miyakawa, T., L2 decay for the Navier-Stokes flow in half-spaces. Math. Ann., 282 (1988), 139–155.Digital Object Identifier: doi:10.1007/BF01457017
Mathematical Reviews (MathSciNet): MR960838
Zentralblatt MATH: 0627.35076 - Borchers, W. & Sohr, H., On the semigroup of the Stokes operator for exterior domains in Lq spaces. Math. Z., 196 (1987), 415–425.Digital Object Identifier: doi:10.1007/BF01200362
Mathematical Reviews (MathSciNet): MR913666
Zentralblatt MATH: 0636.76027 - —, The equations div u=f and rot v=g with homogeneous Dirichlet boundary condition. Hokkaido Math. J., 19 (1990), 67–87.
- Cattabriga, L., Su un problema al contorno relativo al sistema di equazioni di Stokes. Rend. Sem. Mat. Univ. Padova, 31 (1961), 308–340.
- Chang, I-Dee & Finn, R., On the solutions of a class of equations occurring in continuum mechanics with application to the Stokes paradox. Arch. Rational Mech. Anal., 7 (1961), 388–401.Digital Object Identifier: doi:10.1007/BF00250771
Mathematical Reviews (MathSciNet): MR123803
Zentralblatt MATH: 0104.42401 - Friedman, A., Partial Differential Equations. Holt, Rinehard & Winston, New York, 1969.Zentralblatt MATH: 0224.35002
- Fujiwara, D. & Morimoto, H., An Lr-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 24 (1977), 685–700.
- Galdi, G. & Maremonti, P., Monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the Navier-Stokes equations in exterior domains. Arch. Rational Mech. Anal., 94 (1986), 253–266.Digital Object Identifier: doi:10.1007/BF00279866
Mathematical Reviews (MathSciNet): MR846064
Zentralblatt MATH: 0617.35108 - Giga, Y., Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z., 178 (1981), 297–329.Digital Object Identifier: doi:10.1007/BF01214869
Zentralblatt MATH: 0473.35064
Mathematical Reviews (MathSciNet): MR635201 - —, Domains of fractional powers of the Stokes operator in Lr spaces. Arch. Rational Mech. Anal., 89 (1985), 251–265.Digital Object Identifier: doi:10.1007/BF00276874
Zentralblatt MATH: 0584.76037
Mathematical Reviews (MathSciNet): MR786549 - Giga, Y. & Sohr, H., On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo, Sect. IA Math., 36 (1989), 103–130.Mathematical Reviews (MathSciNet): MR991022
- Heywood, J. G., On uniqueness questions in the theory of viscous flow. Acta Math., 138 (1976), 61–102.Mathematical Reviews (MathSciNet): MR425390
- —, The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J., 29 (1980), 639–681.Digital Object Identifier: doi:10.1512/iumj.1980.29.29048
Zentralblatt MATH: 0494.35077
Mathematical Reviews (MathSciNet): MR589434 - Hopf, E., Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr., 4 (1951), 213–231.
- Iwashita, H., Lq−Lr estimates for solutions of nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problem in Lq spaces. Math. Ann., 285 (1989), 265–288.Digital Object Identifier: doi:10.1007/BF01443518
Zentralblatt MATH: 0659.35081
Mathematical Reviews (MathSciNet): MR1016094 - Kajikiya, R. & Miyakawa, T., On L2 decay of weak solutions of the Navier—Stokes equations in Rn. Math. Z., 192 (1986), 135–148.Digital Object Identifier: doi:10.1007/BF01162027
Mathematical Reviews (MathSciNet): MR835398
Zentralblatt MATH: 0607.35072 - Kato, T., Perturbation Theory for Linear Operators. 2nd ed., Springer-Verlag, Berlin, 1976.Zentralblatt MATH: 0342.47009
- —, Strong Lp-solutions of the Navier—Stokes equation in Rm with applications to weak solutions. Math. Z., 187 (1984), 471–480.Digital Object Identifier: doi:10.1007/BF01174182
Zentralblatt MATH: 0545.35073
Mathematical Reviews (MathSciNet): MR760047 - Komatsu, H., Fractional powers of operators. Pacific J. Math., 19 (1966), 285–346.
- Krein, S., Linear Differential Equations in Banach Spaces. Amer. Math. Soc., Providence, 1972.
- Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow. Gordon & Breach, New York, 1969.Zentralblatt MATH: 0184.52603
- Leray, J., Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math., 63 (1934), 193–248.
- Maremonti, P., On the asymptotic behavior of the L2 norm of suitable weak solutions to the Navier—Stokes equations in three-dimensional exterior domains. Comm. Math. Phys., 118 (1988), 385–400.Digital Object Identifier: doi:10.1007/BF01466723
Zentralblatt MATH: 0682.35088
Mathematical Reviews (MathSciNet): MR958803 - Martinez, C., Sanz, M. & Marco, L., Fractional powers of operators. J. Math. Soc. Japan, 40 (1988), 331–347.Mathematical Reviews (MathSciNet): MR930604
Digital Object Identifier: doi:10.2969/jmsj/04020331
Zentralblatt MATH: 0628.47006 - Masuda, K., Weak solutions of the Navier—Stokes equations. Tôhoku Math. J., 36 (1984), 623–646.
- Miyakawa, T., On nonstationary solutions of the Navier—Stokes equations in an exterior domain. Hiroshima Math. J., 12 (1982), 115–140.
- Miyakawa, T. & Sohr, H., On energy inequality, smoothness and large time behavior in L2 for weak solutions of the Navier—Stokes equazions in exterior domains. Math. Z., 199 (1988) 455–478.Digital Object Identifier: doi:10.1007/BF01161636
Mathematical Reviews (MathSciNet): MR968313
Zentralblatt MATH: 0642.35067 - Prodi, G., Un teorema di unicità per le equazioni di Navier—Stokes. Annali di Mat., 48 (1959), 173–182.
- Reed, M. & Simon, B., Methods of Modern Mathematical Physics, vol. II; Fourier analysis, self-adjointness. Academic Press, New York, 1975.
- De Rham, G., Differentiable Manifolds. Springer-Verlag, Berlin, 1984.Zentralblatt MATH: 0534.58003
- Schonbek, M.E., L2 decay for weak solutions of the Navier—Stokes equations. Arch. Rational Mech. Anal., 88 (1986), 209–222.
- —, Large time behaviour of solutions of the Navier—Stokes equations. Comm. Partial Differential Equations, 11 (1986), 733–763.
- Serrin, J., The initial value problem for the Navier—Stokes equations, in Nonlinear Problems, R. Langer ed. The University of Wisconsin Press, Madison, 1963, pp. 69–98.
- Simader, C. G., On Dirichlet's Boundary Value Problem. Lecture Notes in Math., no. 268, Springer-Verlag, Berlin, 1972.Zentralblatt MATH: 0242.35027
- Solonnikov, V. A., Estimates for solutions of nonstationary Navier—Stokes equations. J. Soviet Math., 8 (1977), 467–529.
- Sohr, H., von Wahl, W. & Wiegner, M., Zur Asymptotik der Gleichungen von Navier—Stokes. Nachr. Akad. Wiss. Göttingen, Math.-Phys. Kl. II, 3 (1986), 45–59.
- Specovius, M., Die Stokes Gleichungen in Cantor Räumen und die Analytizität der Stokes-Halbgruppe in gewichteten Lp-Räumen. Dissertation, Paderborn, 1984.
- Stein, E., Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton, 1970.Zentralblatt MATH: 0207.13501
- Triebel, H., Interpolation Theory, Function Spaces, Differential Operators. North-Holland Publ. Co., Amsterdam, 1978.
- Westphal, U., Ein Kalkül für gebrochene Potenzen infinitesimaler Erzeuger von Halbgruppen und Gruppen von Operatoren; Teil I: Halbgruppenerzeuger. Compositio Math., 22 (1970), 67–103.
- Wiegner, M., Decay results for weak solutions of the Navier—Stokes equations in Rn. J. London Math. Soc., 35 (1987), 303–313.
- Yosida, K., Functional Analysis. Springer-Verlag, Berlin, 1965.Zentralblatt MATH: 0126.11504
- Kozono, H. & Sohr, H., Lq-regularity theory for the Stokes operator in exterior domains. Preprint.
- You have access to this content.
- You have partial access to this content.
- You do not have access to this content.
More like this
- On the Stokes and Navier-Stokes equations in a perturbed half-space
Kubo, Takayuki and Shibata, Yoshihiro, Advances in Differential Equations, 2005 - On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains
KOBA, Hajime, Journal of the Mathematical Society of Japan, 2017 - The Stokes and Navier-Stokes equations in an aperture domain
KUBO, Takayuki, Journal of the Mathematical Society of Japan, 2007
- On the Stokes and Navier-Stokes equations in a perturbed half-space
Kubo, Takayuki and Shibata, Yoshihiro, Advances in Differential Equations, 2005 - On stability of Leray's stationary solutions of the Navier–Stokes system in exterior domains
KOBA, Hajime, Journal of the Mathematical Society of Japan, 2017 - The Stokes and Navier-Stokes equations in an aperture domain
KUBO, Takayuki, Journal of the Mathematical Society of Japan, 2007 - $L^p$-Weighted Theory for Navier-Stokes Equations in
Exterior Domains
Amrouche, Chérif and Hoang, Nguyen H., Communications in Mathematical Analysis, 2010 - Spectral properties of the semigroup for the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain
Aoyama, Reika and Kagei, Yoshiyuki, Advances in Differential Equations, 2016 - On the Lq-Lr estimates of the Stokes semigroup in a two dimensional exterior domain
DAN, Wakako and SHIBATA, Yoshihiro, Journal of the Mathematical Society of Japan, 1999 - Note on decay of solutions of steady Navier-Stokes equations in $3$-D exterior
domains
Novotný, A. and Padula, M., Differential and Integral Equations, 1995 - Asymptotic behavior of the energy and pointwise
estimates for solutions to the Navier-Stokes equations
Brandolese, Lorenzo, Revista Matemática Iberoamericana, 2004 - Free Boundary Value Problem for the One-Dimensional Compressible Navier-Stokes Equations with a Nonconstant Exterior Pressure
Lian, Ruxu and Hu, Liping, Journal of Applied Mathematics, 2014 - On the $\mathcal{R}$-Boundedness of Solution Operators for the Stokes Equations
with Free Boundary Condition
Shibata, Yoshihiro, Differential and Integral Equations, 2014