Topological Methods in Nonlinear Analysis

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

Jan W. Cholewa and Tomasz Dłotko

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Abstract

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.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 52, Number 1 (2018), 195-212.

Dates
First available in Project Euclid: 24 April 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1524535232

Digital Object Identifier
doi:10.12775/TMNA.2017.049

Mathematical Reviews number (MathSciNet)
MR3867985

Zentralblatt MATH identifier
07029867

Citation

Cholewa, Jan W.; Dłotko, Tomasz. A note on the $3$-D Navier-Stokes equations. Topol. Methods Nonlinear Anal. 52 (2018), no. 1, 195--212. doi:10.12775/TMNA.2017.049. https://projecteuclid.org/euclid.tmna/1524535232


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References

  • 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 $L_r$-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 $L_r$ spaces, Math. Z. 178 (1981), 297–329.
  • Y. Giga, Domains of fractional powers of the Stokes operator in $L_r$ spaces, Arch. Rational Mech. Anal. 89 (1985), 251–265.
  • Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solution of the Navier–Stokes system, J. Differential Equations 61 (1986), 186–212.
  • Y. Giga and T. Miyakawa, Solutions in $L_r$ of the Navier–Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267–281.
  • T. Kato, Strong $L^p$-solutions of the Navier–Stokes equation in $R^m$, 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. Łukaszewicz and P. Kalita, Navier–Stokes Equations. An Introduction with Applications, Springer, Berlin, 2016.
  • J. Rencławowicz 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 $B^0_{n,\infty}(\Omega)$, 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.