1 / 18

On the longtime behavior of solutions to a model for epitaxial growth

On the longtime behavior of solutions to a model for epitaxial growth. Atsushi Yagi Hideaki Fujimura Jian Luca Mola Maurizio Grasselli. Model Equation. Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr, Phys. Rev. Lett. (1994). Surface Diffusion. Simplification due to W.W. Mullins (1957).

shannon-adkins
Download Presentation

On the longtime behavior of solutions to a model for epitaxial growth

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. On the longtime behavior of solutions to a model forepitaxial growth Atsushi Yagi Hideaki Fujimura Jian Luca Mola Maurizio Grasselli

  2. Model Equation Johnson-Orme-Hunt-Graff-Sudijono-Sauder-Orr, Phys. Rev. Lett.(1994)

  3. Surface Diffusion Simplification due to W.W. Mullins (1957)

  4. Schwoebel Effect Schwoebel障壁 テラス 原子 ステップ Negative diffusion due to M.D.Johnson et al. (1994)

  5. Epitaxial Growth Model Our problem is:

  6. Analytical Results • Fundamentals • Convergence as t   • Finite dimensional Attractors • Homogeneous Stationary Solutions

  7. Abstract Formulation Abstract parabolic semi-linear evolution equation:

  8. Global Solutions and Dynamical System Global existence:

  9. Lyapunov Function Therefore, (u) becomes aLyapunov function

  10. S(t)u0 ū as t  Simon-Łojasiewicz theory:

  11. Finite Dimensional Attractors Eden, Foias, Nicolaenko,Temam (1994) introduced: B1 h B2

  12. Fractal Dimension Mañé-Hölder type embedding:

  13. Exponential attractors for (E) (S(t),Hm1(Ω),L2,m(Ω)) has exponential attractors.

  14. Stationary Solutions Linearized principle:

  15. Stability-Instability of 0 We have the estimate dF(M) N.

  16. Summary • (EGM) generates a dynamical system (S(t),H1(),L2()). • u0H1(), S(t)u0 ū as t , where ū is a stationary solution of (EGM). • The set {stationary solutions}  M (finite dimensional attractor). • If m < a1, the homogeneous stationary solutions are stable; otherwise, unstable.

  17. Problems • How is the structure of stationary solutions? • What is a profile of stationary solution?

  18. Thank You for Your Kind Attention

More Related