Ponce, G.; Racke, R.; Sideris, T. C.; Titi, E. S. Global stability of large solutions to the 3D Navier-Stokes equations. (English) Zbl 0795.35082 Commun. Math. Phys. 159, No. 2, 329-341 (1994). The authors consider the Navier-Stokes equations in a domain \(\Omega\subset \mathbb{R}^ 3\), i.e. \[ u_ t- \nu\Delta u+ (u\cdot\nabla) u+\nabla p=f,\;\text{div } u=0 \text{ in } \Omega\times [0,\infty), \;u=0 \text{ on } \partial\Omega \times [0,\infty), \;u(0)= u_ 0 \text{ in } \Omega. \tag{*} \] Let \((u,p)\) be a global strong solution of (*) which satisfies \[ \int_ 0^ \infty \|\nabla u(t)\|_{L^ 2(\Omega)}^ 4 dt<\infty. \tag{**} \] Then it is shown that for any data which are sufficiently close to \((u_ 0,f)\) (*) has a global strong solution as well.Combining this result with known existence theorems for global strong solutions having axial, rotational or helical symmetry and satisfying (**) the authors prove global existence theorems for large data.As a further application they obtain the stability of unforced two- dimensional flow under three-dimensional perturbations. Reviewer: K.Deckelnick (Freiburg i.Br.) Cited in 2 ReviewsCited in 72 Documents MSC: 35Q30 Navier-Stokes equations 35B35 Stability in context of PDEs Keywords:Navier-Stokes equations; stability; strong solutions; large data × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Constantin, P., Foias, C.: Navier-Stokes equations. Lectures in Mathematics. Chicago: The University of Chicago Press, 1988 · Zbl 0687.35071 [2] Giga, Y.: Regularity criteria for weak solutions of the Navier-Stokes system. Proc. Symposia in Pure Math.45, Providence, RI: AMS 1986, pp. 449–453 · Zbl 0598.35094 [3] Grauer, R., Sideris, T.C.: Numerical computation of 3D incompressible ideal fluids with swirl. Phys. Rev. Lett.67, 3511–3514 (1991) · doi:10.1103/PhysRevLett.67.3511 [4] Heywood, J.G.: The Navier-Stokes equations: On the existence, regularity and decay of solutions. Indiana Univ. Math. J.29, 639–681 (1980) · Zbl 0494.35077 · doi:10.1512/iumj.1980.29.29048 [5] Hopf, E.: Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr.4, 213–231 (1951) · Zbl 0042.10604 [6] Kajikiya, R., Miyakawa, T.: OnL 2 decay of weak solutions of the Navier-Stokes equations in \(\mathbb{R}\) n . Math. Z.192, 135–148 (1986) · Zbl 0607.35072 · doi:10.1007/BF01162027 [7] Kato, T.: StrongL p -solutions of the Navier-Stokes equation in \(\mathbb{R}\) m , with applications to weak solutions. Math. Z.187, 471–480 (1984) · Zbl 0545.35073 · doi:10.1007/BF01174182 [8] Ladyzhenskaya, O. A.: The mathematical theory of viscous incompressible flow. New York: Gordon and Breach Science Publishers, 1969 · Zbl 0184.52603 [9] Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math.63, 193–248 (1934) · JFM 60.0726.05 · doi:10.1007/BF02547354 [10] Leray, J.: Étude de diverses équations intégrales non linéares et de quelques problèmes que pose l’Hydrodynamique. J. Math. Pure Appl.12, 1–82 (1933) · Zbl 0006.16702 [11] Lions, J. L.: Sur la regularité et l’unicité des solutions turbulentes des équations de Navier Stokes. Rend. Sem. Mat. Padova,30, 16–23 (1960) · Zbl 0098.17205 [12] Mahalov, A., Titi, E. S., Leibovich, S.: Invariant helical subspaces for the Navier-Stokes equations. Arch. Rat. Mech. Anal.112, 193–222 (1990) · Zbl 0708.76044 · doi:10.1007/BF00381234 [13] Prodi, G.: Un teorema di unicita per le equazioni di Navier-Stokes. Annali di Mat.48, 173–182 (1959) · Zbl 0148.08202 · doi:10.1007/BF02410664 [14] Pumir, A., Siggia, E.: Development of singular solutions to the axisymmetric Euler equations. Phys. Fluids A4, 1472–1491 (1992) · Zbl 0825.76121 · doi:10.1063/1.858422 [15] Schonbek, M. E.: Large time behaviour of solutions to the Navier-Stokes equations. Comm. P. D. E.11, 733–763 (1986) · Zbl 0607.35071 · doi:10.1080/03605308608820443 [16] Serrin, J.: The initial value problem for the Navier-Stokes equations. Nonlinear Problems, R. E. Langer, (ed.), Madison, Wis.: University of Wisconsin Press, 1963, pp. 69–98 · Zbl 0115.08502 [17] Sohr, H.: Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z.184, 359–375 (1983) · Zbl 0506.35084 · doi:10.1007/BF01163510 [18] Stein, E. M.: Singular integrals and differentiability properties of functions. Princeton NJ: Princeton University Press 1970 · Zbl 0207.13501 [19] Temam, R.: The Navier-Stokes equations. Theory and numerical analysis. Amsterdam: North-Holland 1979 · Zbl 0426.35003 [20] Ukhovskii, M. R., Iudovich, V. I.: Axially symmetric flows of ideal and viscous fluids filling the whole space. J. Appl. Math. Mech.32, 52–62 (1968) · doi:10.1016/0021-8928(68)90147-0 [21] von Wahl, W.: The equations of Navier-Stokes and abstract parabolic equations. Aspects of Mathematics, E8. Braunschweig/Wiesbaden: Friedr. Vieweg & Sohn 1985 · Zbl 0575.35074 [22] Wiegner, M.: Decay and stability inL p for strong solutions of Cauchy problem for the Navier-Stokes equations. The Navier-Stokes equations. Theory and numerical methods. Proceedings Oberwolfach (1988), eds. J.G. Heywood, et al, Lect. Notes Math.1431, Berlin, Heidelberg, New York: Springer 1990, pp. 95–99 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.