Long time behavior and stability of special solutions of nonlinear partial differential equations
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- Tez No: 400092
- Danışmanlar: MİLENA STANİSLAVOVA
- Tez Türü: Doktora
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 2011
- Dil: İngilizce
- Üniversite: University of Kansas
- Enstitü: Yurtdışı Enstitü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 117
Özet
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Özet (Çeviri)
This dissertation deals with a variety of problems concerning solutions of a large classof partial differential equations (PDEs) of mathematical physics, which can be viewedas dynamical systems on an infinite-dimensional space. Many PDEs support coherentstructures like solitary waves (both ground states and bound states), as well as travelingwave solutions. These coherent structures are very important objects when modelingphysical processes and their stability is essential in practical applications. Stable statesof the system are key because they attract all nearby configurations, while the loss ofstability or being able to control it is of practical importance as well. In this dissertation,I apply spectral and variational methods, evolution semigroups, as well as thetechniques of Fourier analysis, to study some outstanding open problems in the theoryof stability and long time behavior for solutions of nonlinear PDEs. The point of viewis that of infinite-dimensional dynamical systems which takes advantage of the analogybetween PDEs and ODEs by looking at systems whose time evolution occurs on ap-propriately defined infinite-dimensional function spaces. In general, the main difficultyin the study of long time behavior of the solutions occurs in higher dimensional spacesand on unbounded domains. To overcome this difficulty, either the modified equationshave been studied, or the initial data and the domain have been restricted. In the studyof stability, one of the most interesting problems is the relation between the linear stability/instability and the nonlinear stability/instability. This question is more or lessresolved in the ODE case, but it is much more complicated in the case of PDEs whereinfinite-dimensional function spaces and unbounded operators are needed to describethe situation. Based on the linear results, the challenge is to establish nonlinear stability/instability and complete invariant manifolds description for these equations. Mycontribution described in this dissertation can be divided in two parts.In the first part, I study the long-time behavior of the solutions of the Kuramoto-Sivashinsky (KS) equation and the Burgers-Sivashinsky equation. KS equation hasbeen widely studied and many results have been obtained for bounded domains in dimensionone. However when the dimension is higher, the problem becomes much morechallenging due to the nonlinear term. Previous results for dimension two have beenobtained either for restricted initial data and a thin domain, or for a modified version ofthe KS equation. I work on a two-dimensional modified Kuramoto-Sivashinsky equationand prove the existence of a global attractor on a bounded domain. Next, I studythe long-time behavior of the solutions of the one-dimensional Burgers-Sivashinskyequation for general initial data as opposed to the usually considered odd initial data.My main contribution is in the study of radially symmetric solutions of the KS equationin dimension two and higher. More precisely, I study the long-time behavior ofradially symmetric solutions of the KS equation in a shell domain in three-dimensionsand prove the existence of a time independent bound for the L2 norm of the solution. Ialso show that similar results hold in any dimension n as long as we have the domain,which excludes the origin. We utilize various techniques from analysis and PDE suchas energy estimates, coercivity and evolution semigroups .In the second part, we deal with the conditional stability of radial steady state solutionsfor the one-dimensional Klein-Gordon equation. It is known that these solutionsare linearly unstable and it has been proved that they are also nonlinearly unstable.Our results complement these. I consider the one-dimensional case and constructthe infinite-dimensional invariant manifolds explicitly. The result is a precise centerstablemanifold theorem, which includes the co-dimension of the manifolds and thedecay rates. I use spectral theory, dynamical systems methods, functional analysis andStrichartz estimates to obtain this. The main difficulty in dimension one compared tohigher dimensions is that the required decay of the Klein-Gordon semigroup does notfollow from Strichartz estimates alone. Thus I apply additional weighted decay estimatesin order to close the argument. In this part of my dissertation, the goal is todevelop a systematic approach to study the fine properties of the solutions in the vicinityof the center-stable manifold and to apply the conditional stability results to controlthe perturbations in order to keep the stable configurations.
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