Porous medium equation and fast diffusion equation as gradient systems

Samuel Littig; Jürgen Voigt

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 869-889
  • ISSN: 0011-4642

Abstract

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We show that the Porous Medium Equation and the Fast Diffusion Equation, u ˙ - Δ u m = f , with m ∈ ( 0 , ∞ ) , can be modeled as a gradient system in the Hilbert space H - 1 ( Ω ) , and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets Ω ⊆ ℝ n and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.

How to cite

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Littig, Samuel, and Voigt, Jürgen. "Porous medium equation and fast diffusion equation as gradient systems." Czechoslovak Mathematical Journal 65.4 (2015): 869-889. <http://eudml.org/doc/276116>.

@article{Littig2015,
abstract = {We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot\{u\}-\Delta u^m=f$, with $m\in (0,\infty )$, can be modeled as a gradient system in the Hilbert space $H^\{-1\}(\Omega )$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $\Omega \subseteq \mathbb \{R\}^n$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.},
author = {Littig, Samuel, Voigt, Jürgen},
journal = {Czechoslovak Mathematical Journal},
keywords = {porous medium equation; gradient system; fast diffusion; asymptotic behaviour; order preservation},
language = {eng},
number = {4},
pages = {869-889},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Porous medium equation and fast diffusion equation as gradient systems},
url = {http://eudml.org/doc/276116},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Littig, Samuel
AU - Voigt, Jürgen
TI - Porous medium equation and fast diffusion equation as gradient systems
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 869
EP - 889
AB - We show that the Porous Medium Equation and the Fast Diffusion Equation, $\dot{u}-\Delta u^m=f$, with $m\in (0,\infty )$, can be modeled as a gradient system in the Hilbert space $H^{-1}(\Omega )$, and we obtain existence and uniqueness of solutions in this framework. We deal with bounded and certain unbounded open sets $\Omega \subseteq \mathbb {R}^n$ and do not require any boundary regularity. Moreover, the approach is used to discuss the asymptotic behaviour and order preservation of solutions.
LA - eng
KW - porous medium equation; gradient system; fast diffusion; asymptotic behaviour; order preservation
UR - http://eudml.org/doc/276116
ER -

References

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  10. Souplet, P., 10.1080/03605309908821454, Commun. Partial Differ. Equations 24 (1999), 951-973. (1999) Zbl0926.35064MR1680893DOI10.1080/03605309908821454
  11. Vázquez, J. L., The Porous Medium Equation, Mathematical Theory, Oxford Mathematical Monographs; Oxford Science Publications Oxford University Press (2007). (2007) Zbl1107.35003MR2286292
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