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.

scholarly journals Nonexistence of Global Solutions to Higher-Order Time-Fractional Evolution Inequalities with Subcritical Degeneracy

Mathematics ◽  
2021 ◽  
Vol 9 (21) ◽  
pp. 2765
Author(s):  
Ravi P. Agarwal ◽  
Soha Mohammad Alhumayan ◽  
Mohamed Jleli ◽  
Bessem Samet
Keyword(s):  
Weak Solutions ◽  
Function Method ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Global Weak Solutions ◽  
Test Function ◽  
Test Function Method

In this paper, we study the nonexistence of global weak solutions to higher-order time-fractional evolution inequalities with subcritical degeneracy. Using the test function method and some integral estimates, we establish sufficient conditions depending on the parameters of the problems so that global weak solutions cannot exist globally.

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.

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.

scholarly journals Nonexistence Results for Higher Order Fractional Differential Inequalities with Nonlinearities Involving Caputo Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1866
Author(s):  
Mohamed Jleli ◽  
Bessem Samet ◽  
Calogero Vetro
Keyword(s):  
Fractional Derivative ◽  
Caputo Fractional Derivative ◽  
A Priori Estimates ◽  
Sufficient Conditions ◽  
Global Solutions ◽  
Higher Order ◽  
Differential Inequalities ◽  
Structure Of Solutions ◽  
Nonlinear Capacity ◽  
Fractional Differential

Higher order fractional differential equations are important tools to deal with precise models of materials with hereditary and memory effects. Moreover, fractional differential inequalities are useful to establish the properties of solutions of different problems in biomathematics and flow phenomena. In the present work, we are concerned with the nonexistence of global solutions to a higher order fractional differential inequality with a nonlinearity involving Caputo fractional derivative. Namely, using nonlinear capacity estimates, we obtain sufficient conditions for which we have no global solutions. The a priori estimates of the structure of solutions are obtained by a precise analysis of the integral form of the inequality with appropriate choice of test function.

scholarly journals Magneto-rotatory compressible couple-stress fluid heated from below in porous medium

2016 ◽  
Vol 38 (1) ◽  
pp. 55-63
Author(s):  
Chander Bhan Mehta
Keyword(s):  
Magnetic Field ◽  
Porous Medium ◽  
Couple Stress ◽  
Normal Modes ◽  
Sufficient Conditions ◽  
Couple Stress Fluid ◽  
Onset Of Convection ◽  
Medium Permeability ◽  
Effects Of Rotation ◽  
The Magnetic Field

Abstract The study is aimed at analysing thermal convection in a compressible couple stress fluid in a porous medium in the presence of rotation and magnetic field. After linearizing the relevant equations, the perturbation equations are analysed in terms of normal modes. A dispersion relation governing the effects of rotation, magnetic field, couple stress parameter and medium permeability have been examined. For a stationary convection, the rotation postpones the onset of convection in a couple stress fluid heated from below in a porous medium in the presence of a magnetic field. Whereas, the magnetic field and couple stress postpones and hastens the onset of convection in the presence of rotation and the medium permeability hastens and postpones the onset of convection with conditions on Taylor number. Further the oscillatory modes are introduced due to the presence of rotation and the magnetic field which were non-existent in their absence, and hence the principle of exchange stands valid. The sufficient conditions for nonexistence of over stability are also obtained.

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