Besov Regularity of Stochastic Partial Differential Equations on Bounded Lipschitz Domains - Softcover

Cioica, Petru

Softcover

ISBN 10: 3832539204 ISBN 13: 9783832539207

Publisher: Logos Verlag Berlin GmbH, 2015

Synopsis

Stochastic partial differential equations (SPDEs, for short) are the mathematical models of choice for space time evolutions corrupted by noise. Although in many settings it is known that the resulting SPDEs have a unique solution, in general, this solution is not given explicitly. Thus, in order to make those mathematical models ready to use for real life applications, appropriate numerical algorithms are needed. To increase efficiency, it would be tempting to design suitable adaptive schemes based, e.g., on wavelets. However, it is not a priori clear whether such adaptive strategies can outperform well-established uniform alternatives. Their theoretical justification requires a rigorous regularity analysis in so-called non-linear approximation scales of Besov spaces. In this thesis the regularity of (semi-)linear second order SPDEs of Ito type on general bounded Lipschitz domains is analysed. The non-linear approximation scales of Besov spaces are used to measure the regularity with respect to the space variable, the time regularity being measured first in terms of integrability and afterwards in terms of Hlder norms. In particular, it is shown that in specific situations the spatial Besov regularity of the solution in the non-linear approximation scales is generically higher than its corresponding classical Sobolev regularity. This indicates that it is worth developing spatially adaptive wavelet methods for solving SPDEs instead of using uniform alternatives.

Edition details

• Publisher: Logos Verlag Berlin GmbH
• Publication date: 2015
• ISBN 10: 3832539204
• ISBN 13: 9783832539207
• Binding: Paperback
• Language: English
• Number of pages: 162