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 Higher Integrability of Weak Solutions to a Class of Double Obstacle Systems

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Zhenhua Hu ◽  
Shuqing Zhou
Keyword(s):  
Differential Equation ◽  
Weak Solutions ◽  
Second Order ◽  
Obstacle Problems ◽  
Elliptic Differential Equation ◽  
Higher Integrability ◽  
Global Higher Integrability ◽  
Quasilinear Elliptic

We first introduce double obstacle systems associated with the second-order quasilinear elliptic differential equationdiv(A(x,∇u))=div f(x,u), whereA(x,∇u),f(x,u)are twon×Nmatrices satisfying certain conditions presented in the context, then investigate the local and global higher integrability of weak solutions to the double obstacle systems, and finally generalize the results of the double obstacle problems to the double obstacle systems.

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 Higher integrability for weak solutions to a degenerate parabolic system with singular coefficients

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yan Dong ◽  
Guangwei Du ◽  
Kelei Zhang
Keyword(s):  
Parabolic System ◽  
Weak Solutions ◽  
Vector Fields ◽  
Degenerate Parabolic System ◽  
Higher Integrability ◽  
Singular Coefficients ◽  
Degenerate Parabolic ◽  
Alpha Beta ◽  
Hörmander’S Condition ◽  
Reverse Hölder

Abstract In this paper, we study the degenerate parabolic system $$ u_{t}^{i} + X_{\alpha }^{*} \bigl(a_{ij}^{\alpha \beta }(z){X_{\beta }} {u^{j}}\bigr) = {g_{i}}(z,u,Xu) + X_{\alpha }^{*} f_{i}^{\alpha }(z,u,Xu), $$ u t i + X α ∗ ( a i j α β ( z ) X β u j ) = g i ( z , u , X u ) + X α ∗ f i α ( z , u , X u ) , where $X=\{X_{1},\ldots,X_{m} \}$ X = { X 1 , … , X m } is a system of smooth real vector fields satisfying Hörmander’s condition and the coefficients $a_{ij}^{\alpha \beta }$ a i j α β are measurable functions and their skew-symmetric part can be unbounded. After proving the $L^{2}$ L 2 estimates for the weak solutions, the higher integrability is proved by establishing a reverse Hölder inequality for weak solutions.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

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