Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation.

*(English)*Zbl 0937.35135The paper considers two well-known models,
\[
u_t = -(1+\nabla^2)^2 u- \varepsilon^2 u^3,
\]

\[ u_t = -(1+\nabla^2)^2 u - \varepsilon^2 |u|^2 u, \] for the real and complex wave fields, respectively, with a positive supercriticality parameter \(\varepsilon^2\). It is well known that both equations have a continuous family of rolls, i.e., quasi-one-dimensional stationary periodic solutions \(u(x)\). The rolls are known in an exact form in the case of the complex equation. In the real equation, their analytical form can only be found as an expansion in powers of \(\varepsilon\); however, the existence of the rolls has been proved rigorously. A part of the solutions (within the bounds of the so-called Busse balloon) are dynamically stable against small perturbations; the solutions lying exactly at the borders of the balloon are neutrally stable in the linear approximation, but nonlinearly stable.

In the present work, it is proved that a sufficiently small perturbation added to the dynamically stable roll solution decays in time according to the diffusive law, i.e., in a suitable Banach space, the norm of the deviation of the perturbed solution from the exact one does not decay slower than \(t^{-3/2}\). The proof is performed by means of the renomalization-group technique, i.e., evolution of the solution is at first considered on a finite interval, and then the transformation of the perturbation, corresponding to the finite-interval evolution, is iterated many times in combination with a rescaling of the perturbation at each step.

\[ u_t = -(1+\nabla^2)^2 u - \varepsilon^2 |u|^2 u, \] for the real and complex wave fields, respectively, with a positive supercriticality parameter \(\varepsilon^2\). It is well known that both equations have a continuous family of rolls, i.e., quasi-one-dimensional stationary periodic solutions \(u(x)\). The rolls are known in an exact form in the case of the complex equation. In the real equation, their analytical form can only be found as an expansion in powers of \(\varepsilon\); however, the existence of the rolls has been proved rigorously. A part of the solutions (within the bounds of the so-called Busse balloon) are dynamically stable against small perturbations; the solutions lying exactly at the borders of the balloon are neutrally stable in the linear approximation, but nonlinearly stable.

In the present work, it is proved that a sufficiently small perturbation added to the dynamically stable roll solution decays in time according to the diffusive law, i.e., in a suitable Banach space, the norm of the deviation of the perturbed solution from the exact one does not decay slower than \(t^{-3/2}\). The proof is performed by means of the renomalization-group technique, i.e., evolution of the solution is at first considered on a finite interval, and then the transformation of the perturbation, corresponding to the finite-interval evolution, is iterated many times in combination with a rescaling of the perturbation at each step.

Reviewer: B.A.Malomed (Tel Aviv)

##### MSC:

35Q35 | PDEs in connection with fluid mechanics |

76E30 | Nonlinear effects in hydrodynamic stability |

37K45 | Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems |

##### Keywords:

diffusive convergence; quasi-one-dimensional stationary periodic solutions; perturbation; dynamically stable roll solution; renormalization-group technique
PDF
BibTeX
XML
Cite

\textit{H. Uecker}, Commun. Partial Differ. Equations 24, No. 11--12, 2109--2146 (1999; Zbl 0937.35135)

Full Text:
DOI

**OpenURL**

##### References:

[1] | DOI: 10.1007/BF02096573 · Zbl 0765.35052 |

[2] | DOI: 10.1002/cpa.3160470606 · Zbl 0806.35067 |

[3] | Collet P., Instabilities and Fronts in Extended Systems (1990) · Zbl 0732.35074 |

[4] | DOI: 10.1007/s002200050238 · Zbl 0912.35140 |

[5] | DOI: 10.1088/0951-7715/7/3/003 · Zbl 0801.35046 |

[6] | Manneville Paul, Dissipative Structures and Weak Turbulence (1992) |

[7] | Mielke Alexander, Nonlinear dynamics and pattern formation in the natural environment pp 206– (1995) |

[8] | DOI: 10.1007/s002200050230 · Zbl 0897.76037 |

[9] | DOI: 10.1007/BF02679126 · Zbl 0871.76028 |

[10] | Reed M., Methods of Modern Mathematical Physics IV (1978) · Zbl 0401.47001 |

[11] | DOI: 10.1007/BF00945930 · Zbl 0805.35125 |

[12] | DOI: 10.1007/BF02108820 · Zbl 0861.35107 |

[13] | Schneider Guido, Submitted to Arch. Rat. Mech. Anal. (1997) |

[14] | Schneider Guido, Proceedings International Conference on Asyrnptotics of Nonlinear Diflusive Systems, Sendai 28 pp 159– (1998) |

[15] | DOI: 10.1007/BF02429847 · Zbl 0795.35112 |

[16] | DOI: 10.1007/s002050050042 · Zbl 0882.35061 |

[17] | DOI: 10.1007/BF02761845 · Zbl 0476.35043 |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.