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.
|Translated title of the contribution||Huokoisen aineen yhtälön yleistetyt ratkaisut|
|Publication status||Published - 2018|
|MoE publication type||G5 Doctoral dissertation (article)|
- porous medium equation, weak solution, weak supersolution, supercaloric function, comparison principle, weak Harnack estimate, obstacle problem