scholarly journals The higher integrability of weak solutions of porous medium systems

2018 ◽  
Vol 8 (1) ◽  
pp. 1004-1034 ◽  
Author(s):  
Verena Bögelein ◽  
Frank Duzaar ◽  
Riikka Korte ◽  
Christoph Scheven
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Type Systems ◽  
The Self ◽  
Higher Integrability ◽  
Medium Type

Abstract In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

scholarly journals Higher integrability for obstacle problem related to the singular porous medium equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Qifan Li
Keyword(s):  
Porous Medium ◽  
Weak Solution ◽  
Weak Solutions ◽  
Obstacle Problem ◽  
Porous Medium Equation ◽  
The Self ◽  
Spatial Gradient ◽  
Higher Integrability ◽  
Medium Equation ◽  
Funct Anal

Abstract In this paper we study the self-improving property of the obstacle problem related to the singular porous medium equation by using the method developed by Gianazza and Schwarzacher (J. Funct. Anal. 277(12):1–57, 2019). We establish a local higher integrability result for the spatial gradient of the mth power of nonnegative weak solutions, under some suitable regularity assumptions on the obstacle function. In comparison to the work by Cho and Scheven (J. Math. Anal. Appl. 491(2):1–44, 2020), our approach provides some new aspects in the estimations of the nonnegative weak solution of the obstacle problem.

scholarly journals Global higher integrability of weak solutions of porous medium systems

2020 ◽  
Vol 19 (3) ◽  
pp. 1697-1745 ◽  
Author(s):  
Kristian Moring ◽  
◽  
Christoph Scheven ◽  
Sebastian Schwarzacher ◽  
Thomas Singer ◽  
...  
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Higher Integrability ◽  
Global Higher Integrability

scholarly journals Hölder regularity for porous medium systems

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Naian Liao
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Parabolic Systems ◽  
Type Result ◽  
Local Boundedness ◽  
Liouville Type ◽  
Medium Type ◽  
Hölder Estimates

AbstractWe establish Hölder continuity for locally bounded weak solutions to certain parabolic systems of porous medium type, i.e. $$\begin{aligned} \partial _t \mathbf{u}-\mathrm{div}(m|\mathbf{u}|^{m-1}D\mathbf{u})=0,\quad m>0. \end{aligned}$$ ∂ t u - div ( m | u | m - 1 D u ) = 0 , m > 0 . As a consequence of our local Hölder estimates, a Liouville type result for bounded global solutions is also established. In addition, sufficient conditions are given to ensure local boundedness of local weak solutions.

Hölder Regularity for Singular Parabolic Obstacle Problems of Porous Medium Type

2018 ◽  
Vol 2020 (6) ◽  
pp. 1671-1717 ◽  
Author(s):  
Yumi Cho ◽  
Christoph Scheven
Keyword(s):  
Porous Medium ◽  
Weak Solution ◽  
Nonlinear Equation ◽  
Weak Solutions ◽  
Obstacle Problems ◽  
Hölder Continuous ◽  
Parabolic Obstacle Problems ◽  
Regularity Of Weak Solutions ◽  
Medium Type ◽  
Bounded Weak Solutions

Abstract We study the regularity of weak solutions to parabolic obstacle problems related to equations of singular porous medium type that are modeled after the nonlinear equation $$\partial_{t} u - \Delta u^{m} = 0.$$For the range of exponents 0 < m < 1, we prove that locally bounded weak solutions are locally Hölder continuous, provided the obstacle function is. Moreover, in the case $\frac{(n-2)_{+}}{n+2} < m < 1$ we show that every weak solution is locally bounded and therefore Hölder continuous.

Numerical modelling of the self-heating process of a wet porous medium

2011 ◽  
Vol 54 (25-26) ◽  
pp. 5200-5206 ◽  
Author(s):  
A. Ejlali ◽  
D.J. Mee ◽  
K. Hooman ◽  
B.B. Beamish
Keyword(s):  
Porous Medium ◽  
Numerical Modelling ◽  
The Self ◽  
Heating Process ◽  
Self Heating
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