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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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References:
 [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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