Generalized solutions to the porous medium equation

Research output: ThesisDoctoral ThesisCollection of Articles

Researchers

  • Pekka Lehtelä

Research units

Abstract

In this thesis we study various notions of generalized solutions to the porous medium equation. The central question throughout the work is finding classifications for the different notions of solutions and supersolutions, as well as clarifying the connections between these classes.  The solutions we are considering are defined by weak formulations of the equation. Our interest lies in the different regularity assumptions, and we prove that these notions are in fact equivalent. Furthermore, we study weak supersolutions defined in a similar manner via weak formulations, as well as supercaloric functions defined in terms of a parabolic comparison principle. First, we address the question on the equivalence of these classes of supersolutions by proving that in the locally bounded case the equivalence holds. Secondly, we show that unbounded supercaloric functions can be divided into two mutually exclusive classes with significantly different properties. We give several characterizations for these classes.  In order to treat the various notions of supersolutions, we develop several tools, which turn out to be interesting in their own right as well. As one of the main tools, we prove a weak Harnack type estimate for weak supersolutions. Finally, we consider the obstacle problem for the porous medium equation. In particular, we prove the existence of weak solutions satisfying useful regularity properties.

Details

Translated title of the contributionHuokoisen aineen yhtälön yleistetyt ratkaisut
Original languageEnglish
QualificationDoctor's degree
Awarding Institution
Supervisors/Advisors
Publisher
  • Aalto University
Print ISBNs978-952-60-7977-6
Electronic ISBNs978-952-60-7978-3
Publication statusPublished - 2018
MoE publication typeG5 Doctoral dissertation (article)

    Research areas

  • porous medium equation, weak solution, weak supersolution, supercaloric function, comparison principle, weak Harnack estimate, obstacle problem

ID: 30203382