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.

REGULARITY OF WEAK SOLUTIONS FOR THE NAVIER–STOKES EQUATIONS IN THE CLASS L∞(BMO-1)

2012 ◽  
Vol 14 (03) ◽  
pp. 1250020 ◽  
Author(s):  
WENDONG WANG ◽  
ZHIFEI ZHANG
Keyword(s):  
Weak Solution ◽  
Weak Solutions ◽  
Stokes Equations ◽  
Removable Singularity ◽  
Navier Stokes ◽  
Singularity Theorem ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Mild Assumption

We study the regularity of weak solution for the Navier–Stokes equations in the class L∞( BMO-1). It is proved that the weak solution in L∞( BMO-1) is regular if it satisfies a mild assumption on the vorticity direction, or it is axisymmetric. A removable singularity theorem in ∈ L∞( VMO-1) is also proved.

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 Regularity of Weak Solutions to the Inhomogeneous Stationary Navier–Stokes Equations

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1336
Author(s):  
Alfonsina Tartaglione
Keyword(s):  
Bounded Domain ◽  
Weak Solutions ◽  
Analogous Result ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Hölder Continuous ◽  
Navier Stokes Equations ◽  
Regularity Of Weak Solutions ◽  
Constant Viscosity ◽  
Stationary Navier Stokes Equations

One of the most intriguing issues in the mathematical theory of the stationary Navier–Stokes equations is the regularity of weak solutions. This problem has been deeply investigated for homogeneous fluids. In this paper, the regularity of the solutions in the case of not constant viscosity is analyzed. Precisely, it is proved that for a bounded domain Ω⊂R2, a weak solution u∈W1,q(Ω) is locally Hölder continuous if q=2, and Hölder continuous around x, if q∈(1,2) and |μ(x)−μ0| is suitably small, with μ0 positive constant; an analogous result holds true for a bounded domain Ω⊂Rn(n>2) and weak solutions in W1,n/2(Ω).

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.

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