Topological Methods in Nonlinear Analysis

Hopf-bifurcation theorem and stability for the magneto-hydrodynamics equations

Weiping Yan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This paper is devoted to the study of the dynamical behavior for the 3D viscous Magneto-hydrodynamics equations. We first prove that this system under smooth external forces possesses time dependent periodic solutions, bifurcating from a steady solution. If the time periodic solution is smooth, then the linear stability of the time periodic solution implies nonlinear stability is obtained in $\textbf{L}^p$ for all $p\in(3,\infty)$.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 46, Number 1 (2015), 471-493.

Dates
First available in Project Euclid: 30 March 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1459343903

Digital Object Identifier
doi:10.12775/TMNA.2015.055

Mathematical Reviews number (MathSciNet)
MR3443696

Zentralblatt MATH identifier
1362.76066

Citation

Yan, Weiping. Hopf-bifurcation theorem and stability for the magneto-hydrodynamics equations. Topol. Methods Nonlinear Anal. 46 (2015), no. 1, 471--493. doi:10.12775/TMNA.2015.055. https://projecteuclid.org/euclid.tmna/1459343903


Export citation

References

  • T. Brand, M. Kunze, G. Schneider and T. Seelbach, Hopf bifurcation and exchange of stability in diffusive media, Arch. Rational Mech. Anal. 171 (2004), 263–296.
  • Z.M. Chen and W.G. Price, it Time dependent periodic Navier–Stokes flows on a two-dimensional torus, Comm. Math. Phys 179 (1996), 577–597.
  • B. Climent-Ezquerra, F. Guillen-Gonzalez and M.A. Rojas-Medar, Time-periodic solutions for a generalized Boussinesq model with Neumann boundary conditions for temperature, Proc. Roy. Soc. Sect. A 463 (2007), 2153–2164.
  • M.G. Crandall and P. Rabinowitz, The Hopf Bifurcation theorem in infinite dimensions, Arch. Rational Mech. Anal. 67 (1977), 53–72.
  • S. Friendlander, N. Pavlović and R. Shvydkoy, Nonlinear instability for the Navier–Stokes equations, Commun. Math. Phys. 264 (2006), 335–347.
  • S. Friendlander, W. Strauss and M. Vishik, Nonlinear instability in an ideal fluids, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 187–209.
  • Y. Giga, Analyticity of the semigroup generated by the Stokes operator in $\textbf{L}_r$ spaces, Math. Z. 178 (1981), 297–329.
  • ––––, The Stokes operator in $\textbf{L}_r$ spaces, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 85–89.
  • R. Glowinski and G. Guidoboni, Hopf bifurcation in viscous incompressible flow down an inclined plane: a numerical approach, J. Math. Fluid Mech. 10 (2008), 434–454.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer–Verlag, New York, 1981.
  • G. Iooss, Bifurcation des solutions périodiques de certains problèmes dévolution, C.R. Acad. Sci. Paris Sér. A 273 (1971), 624–627.
  • ––––, Bifurcation of a periodic solution of the Navier–Stokes equations into an invariant torus, Arch. Rational Mech. Anal. 58 (1975), 35–56.
  • G. Iooss and A. Mielke, Bifurcating time-periodic solutions of Navier–Stokes equations in infinite cylinders, J. Nonlinear Sci. 1 (1991), 107–146.
  • V.I. Iudovich, Apprearance of auto-oscillations in a fluid, Prikl. Mat. Mekh. 35 (1971), 638–655.
  • J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193–248.
  • J.E. Marsden and M. Mccracken, The Hopf Bifurcation and its Applications, Springer, Berlin, New York, 1976.
  • A. Melcher, G. Schneider and H. Uecker, A Hopf-bifurcation theorem for the vorticity formulation of the Navier–Stokes equations in $\mathbb{R}^3$, Commun. Partial Differential Equations 33 (2008), 772–783.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York, 1983.
  • D.H. Sattinger, Bifurcation of periodic solutions of the Navier–Stokes equations, Arch. Rational Mech. Anal. 41 (1971), 66–80.
  • V.A. Solonnikov, Some stationary boundary value problems of magnetohydrodynamic, Trudy Mat. Inst. Steklov. 59 (1960), 174–187.
  • M. Vishik and S. Friendlander, Nonlinear instability in two dimensional ideal fluids: the case of a dominant eigenvalue, Commun. Math. Phys. 243 (2003), 261–273.
  • V. I. Yudovich, The Linearization Method in Hydrodynamical Stability Theory, Transl. Math. Monogr. vol. 74, Providence, RI, Amer. Math. Soc. 1989.
  • R. Temam, Some Developments on Navier–Stokes Equations in the Second Half of the $20$th Century. Development of Mathematics 1950–2000, Basel, Birkhäuser, 2000, 1049–1106.