References
V. N. Maslennikova and M. A. Timoshin, “To theL p-theory of the flow problem for the Stokes system,” in: Application of New Methods of Analysis to Differential Equations [in Russian], Voronezh Univ., Voronezh, 1989, pp. 63–77.
M. A. Timoshin, “On the dimension of the kernel of the differential operator in the first boundary value problem for the Stokes system inL p-spaces in the case of unbounded domains with compact boundary,” in: Boundary Value Problems and Spaces of Differentiable Functions [in Russian], Izdat. Univ. Druzhby Narodov, Moscow, 1989, pp. 124–147.
V. N. Maslennikova and M. A. Timoshin, “OnL p -theory for the Stokes problem in unbounded domains with compact boundary,” Dokl. Akad. Nauk SSSR,313, No. 6, 1341–1345 (1990).
G. P. Galdi and C. G. Simader, “Existence, uniqueness andL q-estimates for the Stokes problem in an exterior domain,” Arch. Rational Mech. Anal.,112, 291–318 (1990).
V. N. Maslennikova and M. E. Bogovskii, “Density of compactly-supported solenoidal vector fields,” Sibirsk. Mat. Zh.,19, No. 5, 1092–1108 (1978).
G. de Rham, Differentiable Manifolds [Russian translation], Izdat. Inostr. Lit., Moscow (1956).
R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis [Russian translation], Mir, Moscow (1981).
M. E. Bogovskii, “Solution of some vector analysis problems connected with the operators div and grad,” in: The Theory of Cubature Formulas and Applications of Functional Analysis to Equations of Mathematical Physics [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk, No. 1, 1980, pp. 5–40.
N. Dunford and J. T. Schwartz, Linear Operators. Vol. 1: General Theory [Russian translation], Izdat. Inostr. Lit., Moscow (1962).
G. Lamb, Hydrodynamics [Russian translation], Gostekhizdat, Moscow (1947).
J. G. Heywood, “On uniqueness questions in the theory of viscous flow,” Acta Math.,136, No. 1, 61–102 (1976).
N. E. Kochin, I. A. Kibel', and N. V. Roze, Theoretical Hydrodynamics. Vol. 2 [in Russian], Fizmatgiz, Moscow (1962).
E. Stein and G. Weiss, Introduction to Harmonic Analysis on Euclidean Spaces [Russian translation], Mir, Moscow (1974).
W. Rudin, Functional Analysis [Russian translation], Mir, Moscow (1975).
V. N. Maslennikova and M. E. Bogovskii, “Approximation of potential and solenoidal vector fields,” Sibirsk. Mat. Zh.,24, No. 5, 149–171 (1983).
V. N. Maslennikova and M. E. Bogovskii, “Approximation of solenoidal and potential vector fields in Sobolev spaces and problems of mathematical physics,” in: Partial Differential Equations [in Russian], Nauka, Novosibirsk, 1986, pp. 129–137.
M. E. Bogovskii, OnLp-Theory for the Navier-Stokes System in Unbounded Domains with Non-compact Boundary [in Russian], Diss. Kand. Fiz.-Mat. Nauk, Moscow (1984).
V. I. Burenkov, Function Spaces.L p -Spaces [in Russian], Izdat. Univ. Druzhby Narodov, Moscow (1987).
L. Cattabriga, “Su un problema al contorno relativo al sistema di equazioni di Stokes,” Rend. Sem. Math. Univ. Padova, No. 31, 308–340 (1961).
S. L. Sobolev, “Density of compactly-supported functions in the spaceL (m) p (E n ),” Sibirsk. Mat. Zh.,4, No. 3, 673–682 (1963).
Author information
Authors and Affiliations
Additional information
The research was financially supported by the Russian Foundation for Fundamental Research (code 93-011-1771).
Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 1, pp. 135–162, January–February, 1994.
The authors express profound gratitude to M. E. Bogovskii for useful discussions.
Rights and permissions
About this article
Cite this article
Maslennikova, V.N., Timoshin, M.A. Generalized solutions with first-order derivatives inL p to the flow problem for the stokes system. Sib Math J 35, 123–149 (1994). https://doi.org/10.1007/BF02104954
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02104954