scholarly journals Higher integrability of weak solution of a nonlinear problem arising in the electrorheological fluids

2016 ◽  
Vol 15 (4) ◽  
pp. 1335-1350 ◽  
Author(s):  
Zhong Tan ◽  
Jianfeng Zhou
Keyword(s):  
Weak Solution ◽  
Nonlinear Problem ◽  
Electrorheological Fluids ◽  
Higher Integrability

scholarly journals On regularity properties of a surface growth model

2021 ◽  
pp. 1-24
Author(s):  
Jan Burczak ◽  
Wojciech S. Ożański ◽  
Gregory Seregin
Keyword(s):  
Weak Solution ◽  
Growth Model ◽  
Surface Growth ◽  
Suitable Weak Solution ◽  
Scale Invariant ◽  
Local Smoothness ◽  
Higher Integrability ◽  
Regularity Properties ◽  
Smoothness Of Solutions

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

scholarly journals Flow of electrorheological fluid under conditions of slip on the boundary

2006 ◽  
Vol 2006 ◽  
pp. 1-14 ◽  
Author(s):  
R. H. W. Hoppe ◽  
M. Y. Kuzmin ◽  
W. G. Litvinov ◽  
V. G. Zvyagin
Keyword(s):  
Mathematical Model ◽  
Weak Solution ◽  
Electrorheological Fluid ◽  
Electrorheological Fluids ◽  
Topological Method

We study a mathematical model describing flows of electrorheological fluids. A theorem of existence of a weak solution is proved. For this purpose the approximating-topological method is used.

scholarly journals Higher integrability near the initial boundary for nonhomogeneous parabolic systems of 𝑝-Laplacian type

2020 ◽  
Vol 32 (6) ◽  
pp. 1539-1559
Author(s):  
Sun-Sig Byun ◽  
Wontae Kim ◽  
Minkyu Lim
Keyword(s):  
Weak Solution ◽  
Type System ◽  
Regularity Theory ◽  
Structural Data ◽  
Parabolic Systems ◽  
Initial Boundary ◽  
Optimal Regularity ◽  
Higher Integrability ◽  
Regularity Results ◽  
Higher Regularity

AbstractWe establish a sharp higher integrability near the initial boundary for a weak solution to the following p-Laplacian type system:\left\{\begin{aligned} \displaystyle u_{t}-\operatorname{div}\mathcal{A}(x,t,% \nabla u)&\displaystyle=\operatorname{div}\lvert F\rvert^{p-2}F+f&&% \displaystyle\phantom{}\text{in}\ \Omega_{T},\\ \displaystyle u&\displaystyle=u_{0}&&\displaystyle\phantom{}\text{on}\ \Omega% \times\{0\},\end{aligned}\right.by proving that, for given {\delta\in(0,1)}, there exists {\varepsilon>0} depending on δ and the structural data such that\lvert\nabla u_{0}\rvert^{p+\varepsilon}\in L^{1}_{\operatorname{loc}}(\Omega)% \quad\text{and}\quad\lvert F\rvert^{p+\varepsilon},\lvert f\rvert^{(\frac{% \delta p(n+2)}{n})^{\prime}+\varepsilon}\in L^{1}(0,T;L^{1}_{\operatorname{loc% }}(\Omega))\implies\lvert\nabla u\rvert^{p+\varepsilon}\in L^{1}(0,T;L^{1}_{% \operatorname{loc}}(\Omega)).Our regularity results complement established higher regularity theories near the initial boundary for such a nonhomogeneous problem with {f\not\equiv 0} and we provide an optimal regularity theory in the literature.

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.

On a Nonlinear System of Reaction-Diffusion Equations

2006 ◽  
Vol 11 (2) ◽  
pp. 115-121 ◽  
Author(s):  
G. A. Afrouzi ◽  
S. H. Rasouli
Keyword(s):  
Weak Solution ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Dirichlet Boundary Conditions ◽  
Nonlinear Elliptic System ◽  
Positive Parameter ◽  
Nonlinear Elliptic ◽  
Positive Weak Solution

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n,   x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n,   x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn,   x ∈ Ω,ui = 0,   x ∈ ∂Ω,   i = 1, 2, ..., n,  where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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