# Generalized solutions to the porous medium equation

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**Generalized solutions to the porous medium equation.** / Lehtelä, Pekka.

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*Generalized solutions to the porous medium equation*. Aalto University.

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TY - THES

T1 - Generalized solutions to the porous medium equation

AU - Lehtelä, Pekka

PY - 2018

Y1 - 2018

N2 - 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.

AB - 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.

KW - porous medium equation

KW - weak solution

KW - weak supersolution

KW - supercaloric function

KW - comparison principle

KW - weak Harnack estimate

KW - obstacle problem

KW - huokoisen aineen yhtälö

KW - heikko ratkaisu

KW - heikko superratkaisu

KW - superkalorinen funktio

KW - vertailuperiaate

KW - heikko Harnack-estimaatti

KW - esteongelma

KW - porous medium equation

KW - weak solution

KW - weak supersolution

KW - supercaloric function

KW - comparison principle

KW - weak Harnack estimate

KW - obstacle problem

M3 - Doctoral Thesis

SN - 978-952-60-7977-6

T3 - Aalto University publication series DOCTORAL DISSERTATIONS

PB - Aalto University

ER -

ID: 30203382