Porous medium equation and fast diffusion equation as gradient systems

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• 作者：Samuel Littig ; Jürgen Voigt
• 关键词：porous medium equation ; gradient system ; fast diffusion ; asymptotic behaviour ; order preservation ; 35G25 ; 47J35 ; 47H99 ; 34G20
• 刊名：Czechoslovak Mathematical Journal
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：65
• 期：4
• 页码：869-889
• 全文大小：208 KB
• 参考文献：[1] V. Barbu: Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer Monographs in Mathematics, Berlin, Springer, 2010.MATH CrossRef
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  [5] R. Chill, E. Fašangová: Gradient Systems-13th International Internet Seminar. Matfyzpress, Charles University in Prague, 2010.
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  [8] F. Otto: The geometry of dissipative evolution equations: the porous medium equation. Commun. Partial Differ. Equations 26 (2001), 101–174.MATH CrossRef
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  [11] J. L. Vázquez: The Porous Medium Equation, Mathematical Theory. Oxford Mathematical Monographs; Oxford Science Publications, Oxford University Press, 2007.
  [12] W. P. Ziemer: Weakly Differentiable Functions. Sobolev Spaces and Functions of Bounded Variation. Graduate Texts in Mathematics 120, Springer, 1989.
  
• 作者单位：Samuel Littig (1)
  Jürgen Voigt (1)
  
  1. Fachrichtung Mathematik, TU Dresden, Helmholtzstrasse 10, D-01062, Dresden, Germany
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Analysis
  Convex and Discrete Geometry
  Ordinary Differential Equations
  Mathematical Modeling and IndustrialMathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9141

文摘

We show that the Porous Medium Equation and the Fast Diffusion Equation, \(\dot u - \Delta {u^m} = f\), 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. Keywords porous medium equation gradient system fast diffusion asymptotic behaviour order preservation}