Existence of weak solutions for the Navier-Stokes equations with initial data in
Author:
Calixto P. Calderón
Journal:
Trans. Amer. Math. Soc. 318 (1990), 179-200
MSC:
Primary 35Q10; Secondary 35D05, 76D05
DOI:
https://doi.org/10.1090/S0002-9947-1990-0968416-0
Addendum:
Trans. Amer. Math. Soc. 318 (1990), 201-207.
MathSciNet review:
968416
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: The existence of weak solutions for the Navier-Stokes equations for the infinite cylinder with initial data in is considered in this paper. We study the case of initial data in
,
, and
. An existence theorem is proved covering these important cases and therefore, the "gap" between the Hopf-Leray theory
and that of Fabes-Jones-Riviere
is bridged. The existence theorem gives a new method of constructing global solutions. The cases
are treated at the end of the paper.
- 1.
l. A. Benedek and R. Panzone, The space
with mixed norm, Duke Math. J. 28 (1961), 301-324. MR 0126155 (23:A3451)
- [2] E. B. Fabes and N. M. Riviere, Singular integrals with mixed homogeneity, Studia Math. 27 (1966), 19-38. MR 0209787 (35:683)
- [3]
E. B. Fabes, B. F. Jones, and N. M. Riviere, The initial value problem for the Navier-Stokes equations with data in
, Arch. Rational Mech. Anal. 45 (1972), 222-240. MR 0316915 (47:5463)
- [4] E. B. Fabes, J. E. Lewis, and N. M. Riviere, Singular integrals and hydrodynamical potentials, Amer. J. Math. 99 (1977), 601-625. MR 0454745 (56:12993)
- [5] -, Boundary value problems for the Navier-Stokes equations, Amer. J. Math. 99 (1977), 626-668. MR 0460928 (57:919)
- [6] E. Hopf, Ueber die Anfangsaufgabe für die hydrodinamischen Grundgleichungen, Math. Nachr. 4 (1951). MR 0050423 (14:327b)
- [7] O. A. Ladyzhenskaja, The mathematical theory of viscous incompressible flow, Gordon and Breach, 1969. MR 0254401 (40:7610)
- [8] J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. MR 1555394
- [9] C. W. Oseen, Neuere Methoden und Ergebnisse in der Hydrodinamik, Akademie Verlagsgessellsschaft, Leipzig, m.b.h. 1927, p. 68.
- [10] G. Prodi, Un teorema di unicitá per le equazioni di Navier Stokes, Ann. Mat. Pura Appl. 48 (1959), 173-182. MR 0126088 (23:A3384)
- [11] J. Serrin, The initial value problem for the Navier Stokes equations, Nonlinear Problems (R. E. Langer, ed.), Univ. of Wisconsin Press, 1963, pp. 69-83. MR 0150444 (27:442)
- [12] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, N.J., 1970. MR 0290095 (44:7280)
- [13]
F. B. Weissler, Initial value problem in
for the Navier Stokes fluids, Arch. Rational Mech. Anal. 74 (1980), 219-229. MR 591222 (83k:35071)
Retrieve articles in Transactions of the American Mathematical Society with MSC: 35Q10, 35D05, 76D05
Retrieve articles in all journals with MSC: 35Q10, 35D05, 76D05
Additional Information
DOI:
https://doi.org/10.1090/S0002-9947-1990-0968416-0
Article copyright:
© Copyright 1990
American Mathematical Society