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The Cauchy problem for Navier-Stokes equations. (English) Zbl 0880.35091
The 3D Cauchy problem for the Navier-Stokes system is considered \[ \frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla )u +\nabla p=f,\qquad \text{div} u=0\quad \text{in}\quad \mathbb{R}^3\times (0,T) \]
\[ u(x,0)=a(x),\quad x\in \mathbb{R}^3, \quad u(x,t)\to 0\quad \text{as}\quad |x|\to\infty \] It is proved that the strong generalized solution (from \(H^1\)) of the problem exists and is unique for sufficiently small \(T\). The author proves that if the data of the problem \(a\) and \(f\) are sufficiently small then the solution is global, i.e. \(T=+\infty\). The smallness conditions in the present paper are weaker than the similar conditions in H. Beirão da Veiga [Indiana Univ. Math. J. 36, 149-166 (1987; Zbl 0601.35093)].

MSC:
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
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[1] Adams, R. A., Sobolev Spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Beirão da Veiga, H., Existence and asymptotic behavior for strong solution of the navier – stokes equations on the whole space, Indiana Univ. Math. J., 36, 149-166, (1987) · Zbl 0601.35093
[3] Chorin, A. J.; Marsden, J. E., A Mathematical Introduction to Fluid Mechanics, (1990), Springer-Verlag New York/Berlin · Zbl 0712.76008
[4] Fabes, B. E.; Jones, B. F.; Rivere, N. M., The initial value problem for the navier – stokes equations with data inL^{p}, Arch. Rational Mech. Anal., 45, 222-240, (1972) · Zbl 0254.35097
[5] Giga, Y., Solutions for semilinear parabolic equations inL^{p}and regularity of weak solutions of the navier – stokes system, J. Differential Equations, 61, 186-212, (1986) · Zbl 0577.35058
[6] Hopf, E., Über die anfangswertaufgabe für die hydrodynamischen grundgleichungen, Math. Nachr., 4, 213-231, (1951) · Zbl 0042.10604
[7] Kato, T., StrongL^{p}-solutions of the navier – stokes equations inR^{n}, with application to weak solutions, Math. Z., 187, 471-480, (1984) · Zbl 0545.35073
[8] Kato, T.; Ponce, G., Commutator estimates and the Euler and navier – stokes equations, Comm. Pure. Appl. Math., 41, 891-907, (1988) · Zbl 0671.35066
[9] Ladyzhenskaya, O. A., The Mathematical Theory of Viscous Incompressible Flow, (1969), Gordon & Breach New York · Zbl 0184.52603
[10] Leray, J., Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63, 193-248, (1934) · JFM 60.0726.05
[11] Lions, J. L., Problemes aux limites dans les Équations aux Dérivées partielles, (1967), Presses Univ. Montreal Montreal
[12] Lions, J. L., Quelques methodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[13] Prodi, G., Un teorema di unicità per le equazioni di navier – stokes, Ann. Mat. Pura Appl., 48, 173-182, (1959) · Zbl 0148.08202
[14] Serrin, J., The initial value problem for the navier – stokes equations, (Langer, R. E., Nonlinear Problems, (1963), Univ. of Wisconsin Press Madison), 69-98
[15] Serrin, J., On the interior regularity of weak solutions of the navier – stokes equations, Arch. Rational Mech. Anal., 9, 187-195, (1962) · Zbl 0106.18302
[16] Schonbek, M. E., L^{2}, Arch. Rational Mech. Anal., 88, 209-222, (1985) · Zbl 0602.76031
[17] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
[18] Temam, R., Navier-Stokes Equations, (1977), North-Holland Amsterdam · Zbl 0335.35077
[19] Wiegner, M., Decay results for weak solutions of the navier – stokes equations onR^{n}, J. London Math. Soc. (2), 35, 303-313, (1987) · Zbl 0652.35095
[20] Wiegner, M., Decay and stability inL^{p}for strong solutions of the Cauchy problem for the navier – stokes equations, Lecture Notes in Math., 1431, (1988), Springer-Verlag New York/Berlin, p. 95-99
[21] Kato, T., Liapunov functions and monotonicity in the navier – stokes equation, Lecture Notes in Math., 1450, (1989), Springer-Verlag New York/Berlin, p. 53-64
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