Change search
ReferencesLink to record
Permanent link

Direct link
Convection-diffusion equation in unbounded cylinder and related homogenization problems
Luleå tekniska universitet.
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

The thesis consists of two closely related papers (A and B). Paper A is concerned with the study of the behaviour at infinity of solutions to second order elliptic equation with first order terms stated in a half-cylinder. The coefficients of the equation are assumed to be measurable and bounded; Neumann boundary condition is imposed on the lateral boundary of the cylinder, while on the base we assign the Dirichlet boundary condition. Under the assumption that the coefficients of the equation stabilize to a periodic regime exponentially, and the functions on the right-hand side decay sufficiently fast at infinity, we prove the existence and the uniqueness of a bounded solution and its stabilization to a constant at the exponential rate at infinity. Also we provide a necessary and sufficient condition for the uniqueness of a bounded solution. Our approach relies on the results from local qualitative elliptic theory, such as Harnack's inequality, Nash and De Giorgi estimates, the maximum principle, positive operator theory and a number of nontrivial a priori estimates. The problems of this type have many interesting applications in physics and mechanics and also appear while constructing the asymptotic expansions of solutions to equations describing different phenomena in highly inhomogeneous media. In particular, these results allow one to construct boundary layer correctors. Paper B is devoted to the homogenization of a stationary convectiondiffusion equation in a thin cylinder being a union of two nonintersecting rods with a junction at the origin. It is assumed that each of these cylinders has a periodic microstructure, and that the microstructure period is of the same order as the cylinder diameter. Under some natural assumptions on the data we construct and justify the asymptotic expansion of a solution which consists of the interior expansion and the boundary layer correctors, arising both in the vicinity of the rod ends and the vicinity of the junction. In contrast to the divergence form operators, in the case of convectiondiffusion equation the asymptotic behaviour of solutions depends crucially on the direction of the so-called effective convection (effective axial drift). In the present work we only consider the case when in each of the two cylinders (being the constituents of the rod) the effective convection is directed from the end of the cylinder towards the junction.

Place, publisher, year, edition, pages
Luleå: Luleå tekniska universitet, 2009.
Licentiate thesis / Luleå University of Technology, ISSN 1402-1757
Research subject
URN: urn:nbn:se:ltu:diva-17074Local ID: 16d9d720-fe8a-11dd-95be-000ea68e967bISBN: 978-91-86233-16-7OAI: diva2:990068
Godkänd; 2009; 20090219 (johfab); LICENTIATSEMINARIUM Ämnesområde: Matematik Examinator: Professor Lars-Erik Persson, Luleå tekniska universitet Tid: Onsdag den 1 april 2009 kl 10.00 Plats: D770, Luleå tekniska universitetAvailable from: 2016-09-29 Created: 2016-09-29Bibliographically approved

Open Access in DiVA

fulltext(584 kB)0 downloads
File information
File name FULLTEXT01.pdfFile size 584 kBChecksum SHA-512
Type fulltextMimetype application/pdf

Search outside of DiVA

GoogleGoogle Scholar
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

ReferencesLink to record
Permanent link

Direct link