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Diffusive stability of rolls in the two-dimensional real and complex Swift-Hohenberg equation. (English) Zbl 0937.35135
The 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.

MSC:
35Q35 PDEs in connection with fluid mechanics
76E30 Nonlinear effects in hydrodynamic stability
37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems
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