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Nonlinear convective instability of fronts: a case study

Ghazaryan, Anna R.
http://rave.ohiolink.edu/etdc/view?acc_num=osu1117552079

Abstract Details

2005, Doctor of Philosophy, Ohio State University, Mathematics.
We consider a model system, consisting of two nonlinearly coupled partial differential equations, to investigate nonlinear convective instabilities of travelling waves. The system exhibits front solutions which are travelling waves that asymptotically connect two different spatially homogeneous rest states. In the coordinate frame that moves with the front, the rest state ahead of the front is asymptotically stable, while the rest state behind the front experiences a Turing instability: upon increasing a bifurcation parameter, spatially periodic patterns arise. We show that the front is nonlinearly convectively unstable with respect to perturbations that are exponentially localized at positive infinity. More precisely, in the co-moving coordinate frame, the corresponding solution converges pointwise to a translate of the front, and the perturbation to the front is transported toward negative infinity. The proof of this stability result (which appears to be the first of this kind for dissipative systems) is based on the interplay of various norms, including norms for ordinary, uniformly local, and exponentially weighted Sobolev spaces. Among the methods employed are energy and semigroup estimates derived using multiplier theory in uniformly local spaces.
Bjorn Sandstede (Advisor)
176 p.
Mathematics
stability; travelling wave; front; convective instability; nonlinear stability; reaction diffusion equation

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Citations

  • Ghazaryan, A. R. (2005). Nonlinear convective instability of fronts: a case study [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1117552079

    APA Style (7th edition)

  • Ghazaryan, Anna. Nonlinear convective instability of fronts: a case study. 2005. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1117552079.

    MLA Style (8th edition)

  • Ghazaryan, Anna. "Nonlinear convective instability of fronts: a case study." Doctoral dissertation, Ohio State University, 2005. http://rave.ohiolink.edu/etdc/view?acc_num=osu1117552079

    Chicago Manual of Style (17th edition)

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© 2005, all rights reserved.
This open access ETD is published by The Ohio State University and OhioLINK.