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Correction to: Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

  • M. D. Korzec
  • P. Nayar
  • P. RybkaEmail author
Correction
  • 53 Downloads

1 Correction to: J Dyn Diff Equat (2016) 28:49–67  https://doi.org/10.1007/s10884-015-9510-6

The original version of this article, unfortunately, contained an error.

In [1], we studied
$$\begin{aligned} \begin{array}{ll} h_t = \frac{\delta }{2}| \nabla h|^2 + \Delta (\Delta ^2 h - \Delta \mathrm{div}\,D_F W(\nabla h))&{} \hbox {in }\Omega \times \mathbb {R}_+,\\ h(x, 0) = h_0(x)&{} \hbox {for } x \in \Omega , \end{array} \end{aligned}$$
(1)
for \(\Omega = (0,L)^d\), \(d=1\) or \(d=2\) with periodic boundary conditions. The nonlinearity had the following form,
$$\begin{aligned} \begin{array}{ll} W(F) = \frac{1}{4}(F^2-1)^2, &{} d=1,\\ W(F_1,F_2)= \frac{\alpha }{12}(F_1^4 + F_2^4) + \frac{\beta }{2} F_1^2 F_2^2 - \frac{1}{2}(F_1^2 + F_2^2) + A, &{} d=2, \end{array} \end{aligned}$$
where \(\alpha , \beta >0\) are anisotropy coefficients.
The way to obtain long-time results was through the study of the differentiated system (1), \(u=\nabla h\), i.e. we differentiated (1) with respect to x. Here is the resulting problem,
$$\begin{aligned} \begin{array}{ll} u_t = \frac{\delta }{2} \nabla |u|^2 + \Delta ^3 u - \nabla \Delta \mathrm{div}\,D_u W(u) &{} \hbox {in } \Omega \times \mathbb {R}_+,\\ u(x,0) = u_0(x)&{} \hbox {for }x\in \Omega , \end{array} \end{aligned}$$
(2)
where \(u = (u_1, u_2) =(h_x, h_y)\) (resp. \(u= h_x\)), if \(d=2\), (resp. \(d=1\)).

We proved in [1] the following result about (2).

Theorem 1

([1, Theorem 4], [1, Theorem 5]) Let us consider \(\Omega =(0,L)^d\) with \(d=1,2\) and \(L>0\) arbitrary. The semigroup \(S(t):\dot{H}^2_{per}(\Omega ) \rightarrow \dot{H}^2_{per}(\Omega ), u_0 \mapsto S(t)u_0 = u(t) \) generated by equation (2) with periodic boundary conditions has a global attractor.

We also claimed that the following result holds true.

Theorem 2

([1, Theorem 6]) The semigroup generated by equation (1) has a global attractor in \(H^3_{per}\) for \(d=1\) and \(d=2\).

However, this claim is not valid, because if h is solution to (1), then due to [1, Lemma 13] we know that \(\nabla h\in L^2(0,T; \dot{H}^5_{per})\) and integration of (1) over \(\Omega \) yields,
$$\begin{aligned} \frac{d}{dt} \int _\Omega h (x,t)\,dx = \int _\Omega \frac{\partial h}{\partial t} (x,t)\,dx = \int _\Omega \delta |\nabla h|^2\,dx \ge 0. \end{aligned}$$
However, \(\frac{d}{dt} \int _\Omega h (x,t)\,dx = 0\) if and only if \(h\equiv const.\) Moreover, \(h = const.\) is a steady state of (1). As a result, if h is not a constant steady state, then
$$\begin{aligned} 0< \frac{d}{dt} \int _\Omega h (x,t)\,dx. \end{aligned}$$
This fact was overlooked in [1], making the claim in Theorem 2 invalid.

Notes

Reference

  1. 1.
    Korzec, M.D., Nayar, P., Rybka, P.: Global attractors of sixth order PDEs describing the faceting of growing surfaces. J. Dyn. Differ. Equ. 28, 49–67 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsTechnical University BerlinBerlinGermany
  2. 2.Institute of MathematicsThe University of WarsawWarszawaPoland
  3. 3.Institute of Applied Mathematics and MechanicsThe University of WarsawWarszawaPoland

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