A note on the $3$-D Navier-Stokes equations

Jan W. Cholewa, Tomasz Dłotko


We consider the Navier-Stokes model in a bounded smooth domain $\Omega\subset \mathbb R^3$. Assuming a smallness condition on the external force $f$, which does not necessitate smallness of $\| f\|_{[L^2(\Omega)]^3}$-norm, we show that for any smooth divergence free initial data $u_0$ there exists ${\mathcal T}={\mathcal T}(\|u_0\|_{[L^2(\Omega)]^3})$ satisfying $$ {\mathcal T} \to 0 \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to 0 \quad \text{and} \quad {\mathcal T} \to \infty \quad \text{as } \|u_0\|_{[L^2(\Omega)]^3}\to \infty, $$% and such that either a corresponding regular solution ceases to exist until $\mathcal T$ or, otherwise, it is globally defined and approaches a maximal compact invariant set $\mathbb A$. The latter set $\mathbb A$ is a global attractor for the semigroup restricted to initial velocities $u_0$ in a certain ball of fractional power space $X^{1/4}$ associated with the Stokes operator, which in turn does not necessitate smallness of the gradient norm $\|\nabla u_0\|_{[L^2(\Omega)]^3}$. Moreover, $\mathbb A$ attracts orbits of bounded sets in $X$ through Leray-Hopf type solutions obtained as limits of viscous parabolic approximations.


Navier-Stokes equations; global solutions; small data; asymptotic behavior

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H. Amann, Linear and Quasilinear Parabolic Problems, Volume I, Abstract Linear Theory, Birkhäuser, Basel, 1995.

H. Amann, On the strong solvability of the Navier–Stokes equations, J. Math. Fluid Mech. 2 (2000), 16–98.

J. Arrieta and A.N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier–Stokes and heat equations, Trans. Amer. Math. Soc. 352 (1999), 285–310.

J. Avrin, Singular initial data and uniform global bounds for the hyper-viscous Navier–Stokes equation with periodic boundary conditions, J. Differential Equations 190 (2003), 330–351.

M. Cannone, A generalization of a theorem by Kato on Navier–Stokes equations, Rev.Mat. Iberoam. 13 (1997), 515–541.

M. Cannone and G. Karch, About the regularized Navier–Stokes equations, J. Math. Fluid Mech. 7 (2005), 1–28.

M. Cannone, F. Planchon and M. Schonbek, Strong solutions to the incompressible Navier–Stokes equations in the half-space, Comm. Partial Differential Equations 25 (2000), 903–924.

J.W. Cholewa and T. Dlotko, Local attractor for n-D Navier–Stokes system, Hiroshima Math. J. 28 (1998), 309–319.

J.W. Cholewa and T. Dlotko, Global Attractors in Abstract Parabolic Problems, Cambridge University Press, Cambridge, 2000.

J.W. Cholewa and T. Dlotko, Parabolic equations with critical nonlinearities, Topol. Methods Nonlinear Anal. 21 (2003), 311–324.

J.W. Cholewa and T. Dlotko, Fractional Navier–Stokes equations, Discrete Contin. Dyn. Syst. Ser. B, doi:10.3934/dcdsb.2017149.

T. Dlotko, Navier–Stokes equation and its fractional approximations, Appl. Math. Optim. 77 (2018), 99–128.

D. Fujiwara and H. Morimoto, An Lr -theorem of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo 24 (1977), 685–700.

Y. Giga, Analyticity of the semigroup generated by the Stokes operator in Lr spaces, Math. Z. 178 (1981), 297–329.

Y. Giga, Domains of fractional powers of the Stokes operator in Lr spaces, Arch. Rational Mech. Anal. 89 (1985), 251–265.

Y. Giga, Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier–Stokes system, J. Differential Equations 61 (1986), 186–212.

Y. Giga and T. Miyakawa, Solutions in Lr of the Navier–Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267–281.

T. Kato, Strong Lp -solutions of the Navier–Stokes equation in Rm , with applications to weak solutions, Math. Z. 187 (1984), 471–480.

T. Kato and H. Fujita, On the nonstationary Navier–Stokes system, Rend. Sem. Math. Univ. Padova 32 (1962), 243–260.

H. Koch and D. Tataru, Well-posedness for the Navier–Stokes equations, Adv. Math. 157 (2001), 22–35.

S.G. Krein, Linear Equations in Banach Spaces, Birkhäuser, Boston, 1982.

O.A. Ladyzhenskaya, On some gaps in two of my papers on the Navier–Stokes equations and the way of closing them, J. Math. Sci. 115 (2003), 2789–2891.

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier–Villars, Paris, 1969.

G. Lukaszewicz and P. Kalita, Navier–Stokes Equations. An Introduction with Applications, Springer, Berlin, 2016.

J. Renclawowicz and W. Zajaczkowski, Nonstationary flow for the Navier–Stokes equations in a cylindrical pipe, Math. Meth. Appl. Sci. 35 (2012), 1434–1455.

M.-H. Ri, P. Zhang and Z. Zhang, Global well-posedness for Navier–Stokes equations with small initial value in Bn,∞ (Ω), J. Math. Fluid Mech. 18 (2016), 103–131.

J.C. Robinson, Infinite-Dimensional Dynamical Systems. An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge University Press, Cambridge, 2001.

H. Sohr, The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001.

W. von Wahl, Equations of Navier–Stokes and Abstract Parabolic Equations, Vieweg, Braunschweig/Wiesbaden, 1985.


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