scholarly journals Structural stability for the Forchheimer equations interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^{3}$

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jincheng Shi ◽  
Yan Liu
Keyword(s):  
Structural Stability ◽  
Continuous Dependence ◽  
A Priori ◽  
Bounded Region ◽  
A Priori Bounds ◽  
Darcy Equations ◽  
Nonlinear Fluid ◽  
Continuous Dependence Results

AbstractThe structural stability for the Forchheimer fluid interfacing with a Darcy fluid in a bounded region in $\mathbb{R}^{3}$ R 3 was studied. We assumed that the nonlinear fluid was governed by the Forchheimer equations in $\Omega _{1}$ Ω 1 , while in $\Omega _{2}$ Ω 2 , we supposed that the flow satisfies the Darcy equations. With the aid of some useful a priori bounds, we were able to demonstrate the continuous dependence results for the Forchheimer coefficient λ.

scholarly journals Structural stability for the Boussinesq equations interfacing with Darcy equations in a bounded domain

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yuanfei Li ◽  
Shuanghu Zhang ◽  
Changhao Lin
Keyword(s):  
Bounded Domain ◽  
Structural Stability ◽  
Differential Inequality ◽  
Continuous Dependence ◽  
A Priori ◽  
Boussinesq Equations ◽  
Interface Boundary ◽  
A Priori Bounds ◽  
First Order ◽  
Darcy Equations

AbstractA priori bounds were derived for the flow in a bounded domain for the viscous-porous interfacing fluids. We assumed that the viscous fluid was slow in $\Omega _{1}$ Ω 1 , which was governed by the Boussinesq equations. For a porous medium in $\Omega _{2}$ Ω 2 , we supposed that the flow satisfied the Darcy equations. With the aid of these a priori bounds we were able to demonstrate the result of the continuous dependence type for the Boussinesq coefficient λ. Following the method of a first-order differential inequality, we can further obtain the result that the solution depends continuously on the interface boundary coefficient α. These results showed that the structural stability is valid for the interfacing problem.

scholarly journals Convergence Results for the Double-Diffusion Perturbation Equations

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 67
Author(s):  
Jincheng Shi ◽  
Shiguang Luo
Keyword(s):  
Boundary Conditions ◽  
Structural Stability ◽  
A Priori ◽  
Double Diffusion ◽  
A Priori Bounds ◽  
Poincaré Inequalities ◽  
Convergence Results ◽  
Poincare Inequalities ◽  
Diffusion Perturbation ◽  
Perturbation Equations

We study the structural stability for the double-diffusion perturbation equations. Using the a priori bounds, the convergence results on the reaction boundary coefficients k1, k2 and the Lewis coefficient Le could be obtained with the aid of some Poincare´ inequalities. The results showed that the structural stability is valid for the the double-diffusion perturbation equations with reaction boundary conditions. Our results can be seen as a version of symmetry in inequality for studying the structural stability.

scholarly journals A priori bounds of the solution of a one point IBVP for a singular fractional evolution equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Said Mesloub ◽  
Hassan Eltayeb Gadain
Keyword(s):  
Differential Equations ◽  
Evolution Equation ◽  
A Priori Estimates ◽  
Boundary Value ◽  
A Priori ◽  
Initial Boundary ◽  
A Priori Bounds ◽  
Fractional Evolution Equation ◽  
Priori Estimates ◽  
Initial Boundary Value

Abstract A priori bounds constitute a crucial and powerful tool in the investigation of initial boundary value problems for linear and nonlinear fractional and integer order differential equations in bounded domains. We present herein a collection of a priori estimates of the solution for an initial boundary value problem for a singular fractional evolution equation (generalized time-fractional wave equation) with mass absorption. The Riemann–Liouville derivative is employed. Results of uniqueness and dependence of the solution upon the data were obtained in two cases, the damped and the undamped case. The uniqueness and continuous dependence (stability of solution) of the solution follows from the obtained a priori estimates in fractional Sobolev spaces. These spaces give what are called weak solutions to our partial differential equations (they are based on the notion of the weak derivatives). The method of energy inequalities is used to obtain different a priori estimates.

scholarly journals Periodic solution for ϕ-Laplacian neutral differential equation

2019 ◽  
Vol 17 (1) ◽  
pp. 172-190 ◽  
Author(s):  
Shaowen Yao ◽  
Zhibo Cheng
Keyword(s):  
Differential Equation ◽  
Periodic Solution ◽  
Periodic Solutions ◽  
A Priori ◽  
Neutral Differential Equation ◽  
A Priori Bounds ◽  
T Tau

Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

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