Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On an exterior initial-boundary value problem for Navier-Stokes equations


Author: Yoshihiro Shibata
Journal: Quart. Appl. Math. 57 (1999), 117-155
MSC: Primary 35Q30; Secondary 76D05, 76E99
DOI: https://doi.org/10.1090/qam/1672187
MathSciNet review: MR1672187
Full-text PDF Free Access

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • [1] K. I. Babenko, On stationary solutions of the problem of flow past a body of a viscous incompressible fluid, Math. USSR Sb. 20, 1-25 (1973)
  • [2] J. Bemelmans, Eine Außenraumaufgabe für die instationären Navier-Stokes-Gleichungen, Math. Zeit. 162, 145-173 (1978) MR 506580
  • [3] M. E. Bogovskii, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20, 1094-1098 (1979) MR 553920
  • [4] M. E. Bogovskii, Solution for some vector analysis problems connected with operators div and grad, Theory of cubature formulas and application of functional analysis to problems of mathematical physics, Trudy Sem. S. L. Sobolev, #1, 80, Novosibirsk: Acad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., 1980, pp. 5-40 MR 631691
  • [5] W. Borchers and T. Miyakawa, On stability of exterior stationary Navier--Stokes flows, Acta Math. 174, 311-382 (1995) MR 1351321
  • [6] L. Cattabriga, Su un problema al contorno relativo al sistema di equazioni di Stokes, Rend. Semin. Mat. Univ. Padova 31, 308-340 (1961) MR 0138894
  • [7] Z. M. Chen, Solutions of the stationary and nonstationary Navier--Stokes equations in exterior domains, Pacific J. Math. 159, 227-240 (1993) MR 1214071
  • [8] R. Farwig, A variational approach in weighted Sobolev spaces to the operator $ - \Delta + \partial /\partial {x_1}$ in exterior domains of $ {\mathbb{R}^{3}}$, Math. Zeit. 210, 449-464 (1992) MR 1171183
  • [9] R. Farwig, The stationary exterior 3 D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces, Math. Zeit. 211, 409-447 (1992) MR 1190220
  • [10] R. Farwig and H. Sohr, The stationary and non-stationary Stokes system in exterior domains with non-zero divergence and non-zero boundary values, Math. Meth. Appl. Sci. 17, 269-291 (1994) MR 1265181
  • [11] H. Faxén, Fredholm'shce Integraleichungen zu der Hydrodynamik zäher Flüssigkeiten I, Ark. Mat. Astr. Fys. 21 A 14, 1-20 (1928/29)
  • [12] R. Finn, On steady-state solutions of the Navier-Stokes partial differential equations, Arch. Rational Mech. Anal. 3, 139-151 (1959) MR 0107442
  • [13] R. Finn, Estimates at infinity for stationary solutions of the Navier-Stokes equations, Bull. Math. dela Soc. Sci. Math. Phys. de la R. P. Roumaine 3 (51), 387-418 (1959) MR 0166495
  • [14] R. Finn, An energy theorem for viscous fluid motions, Arch. Rational Mech. Anal. 6, 371-381 (1960) MR 0166497
  • [15] R. Finn, On the steady-state solutions of the Navier--Stokes equations, III, Acta Math. 105, 197-244 (1961) MR 0166498
  • [16] R. Finn, On the exterior stationary problem for the Naviei--Stokes equations and associated perturbation problems, Arch. Rational Mech. Anal. 19, 363-406 (1965) MR 0182816
  • [17] R. Finn, Stationary solutions of the Navier-Stokes equations, Proc. Sympos. Appl. Math. 19, 121-153 (1965)
  • [18] D. Fujiwara and H. Morimoto, An $ {L_r}$-theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo, Sec., 1 24, 685-700 (1977) MR 0492980
  • [19] G. P. Galdi, An introduction to the mathematical theory of the Naviei--Stokes equations, Vol. I, Linearized Steady Problems; Vol. II, Nonlinear Steady Problems, Springer Tracts in Natural Philosophy Vol. 38, 39, Springer--Verlag, New York at al, 1994
  • [20] 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, 291-318 (1990) MR 1077262
  • [21] Y. Giga and T. Miyakawa, Solutions in $ {L_r}$ to the Navier-Stokes initial value problem, ibid 89, 267-281 (1985) MR 786550
  • [22] Y. Giga and H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sec., IA. Math. 36, 103-130 (1989) MR 991022
  • [23] J. G. Heywood, On stationary solutions of the Navier-Stokes equations as limits of non-stationary solutions, Arch. Rational Mech. Anal. 37, 48-60 (1970) MR 0412639
  • [24] J. G. Heywood, The exterior nonstationary problem for the Navier--Stokes equations, Acta Math. 129, 11-34 (1972) MR 0609550
  • [25] J. G. Heywood, The Navier--Stokes equations : On the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29, 639-681 (1980) MR 589434
  • [26] E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachr. 4, 213-231 (1950-51) MR 0050423
  • [27] L. Hörmander, The analysis of linear partial differential operators I, Grund. Math. Wiss. 256, Springer--Verlag, Berlin, 1983
  • [28] H. Iwashita, $ {L_q} - {L_r}$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier--Stokes initial value problems in $ {L_q}$ spaces, Math. Ann. 285, 265-288 (1989) MR 1016094
  • [29] T. Kato, Strong $ {L^p}$-solutions of the Navier-Stokes equation in $ {R^m}$ with applications to weak solutions, Math. Zeit. 187, 471-480 (1984) MR 760047
  • [30] T. Kobayashi and Y. Shibata, On the Oseen equation in exterior domains, Math. Ann. 310, 1-45 (1998) MR 1600022
  • [31] H. Kozono and M. Yamazaki, Navier-Stokes equations in exterior domains, Preprint in 1994
  • [32] J. Leray, Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l'hydrodynamique, J. Math. Pures Appl. IX. Sér. 12, 1-82 (1933)
  • [33] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63, 193-248 (1934)
  • [34] P. Maremonti, Stabilità asintotica in media per moti fluidi viscosi in domini esterni, Ann. Mat. Pura Appl. 97, 57-75 (1985)
  • [35] K. Masuda, On the stability of incompressible viscous fluid motions past objects, J. Math. Soc. Japan 27, 294-327 (1975) MR 0440224
  • [36] T. Miyakawa, On nonstationary solutions of the Navier--Stokes equations in an exterior domain, Hiroshima Math. J. 12, 115-140 (1982) MR 647234
  • [37] A. Novotny and M. Padula, Physically reasonable solutions to steady compressible Navier-Stokes equations in 3 D-exterior domains $ \left( {v_\infty } \ne 0 \right)$, Math. Ann. 308, 439-489 (1997) MR 1457741
  • [38] C. W. Oseen, Neuere Methoden und Ergebniss in der Hydrodynamik, Academische Verlagsgesellschaft m.b.H., Leipnig, 1927
  • [39] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Appl. Math. Sci. 44, Springer--Verlag, New York, 1983 MR 710486
  • [40] V. A. Solonikov, General boundary value problems for Douglis--Nirenberg elliptic systems which are elliptic in the sense of Douglis--Nirenberg I, Amer. Math. Soc. Transl. (2)56, 193-232 (1966), Izv. Acad. Nauk SSSR Ser. Mat. 28, 665-706 (1964); II, Russian, Proc. Steklov Inst. Math. 92, 233-297 (1966)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q30, 76D05, 76E99

Retrieve articles in all journals with MSC: 35Q30, 76D05, 76E99


Additional Information

DOI: https://doi.org/10.1090/qam/1672187
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society