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 Convergence Results on the Boundary Conditions for 2D Large-Scale Primitive Equations in Oceanic Dynamics

2020 ◽  
Vol 2020 ◽  
pp. 1-18
Author(s):  
Yuanfei Li
Keyword(s):  
Boundary Conditions ◽  
Large Scale ◽  
Weather Prediction ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergence Result ◽  
Primitive Equations ◽  
Convergence Results ◽  
Initial Boundary Value

In this paper, the initial boundary value problem for the two-dimensional large-scale primitive equations of large-scale oceanic motion in geophysics is considered, which are fundamental models for weather prediction. By establishing rigorous a priori bounds with coefficients and deriving some useful inequalities, the convergence result for the boundary conditions is obtained.

scholarly journals Analysis of some heterogeneous catalysis models with fast sorption and fast surface chemistry

2021 ◽  
Author(s):  
Björn Augner ◽  
Dieter Bothe
Keyword(s):  
Heterogeneous Catalysis ◽  
Boundary Conditions ◽  
Surface Chemistry ◽  
A Priori ◽  
Strong Solutions ◽  
Mass Conservation ◽  
Chemical Reactor ◽  
Classical Solutions ◽  
A Priori Bounds ◽  
Reaction Diffusion Systems

AbstractWe investigate limit models resulting from a dimensional analysis of quite general heterogeneous catalysis models with fast sorption (i.e. exchange of mass between the bulk phase and the catalytic surface of a reactor) and fast surface chemistry for a prototypical chemical reactor. For the resulting reaction–diffusion systems with linear boundary conditions on the normal mass fluxes, but at the same time nonlinear boundary conditions on the concentrations itself, we provide analytic properties such as local-in-time well-posedness, positivity, a priori bounds and comment on steps towards global existence of strong solutions in the class $$\mathrm {W}^{(1,2)}_p(J \times \Omega ; {{\,\mathrm{{\mathbb {R}}}\,}}^N)$$ W p ( 1 , 2 ) ( J × Ω ; R N ) , and of classical solutions in the Hölder class $$\mathrm {C}^{(1+\alpha , 2 + 2\alpha )}({\overline{J}} \times {\overline{\Omega }}; {{\,\mathrm{{\mathbb {R}}}\,}}^N)$$ C ( 1 + α , 2 + 2 α ) ( J ¯ × Ω ¯ ; R N ) . Exploiting that the model is based on thermodynamic principles, we further show a priori bounds related to mass conservation and the entropy principle.

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.

Second Order Boundary Value Problems with Nonlinear Two-Point Boundary Conditions

2000 ◽  
Vol 7 (4) ◽  
pp. 677-688 ◽  
Author(s):  
P. Kelevedjiev ◽  
N. Popivanov
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
A Priori ◽  
Existence Results ◽  
Second Order ◽  
Point Boundary ◽  
A Priori Bounds ◽  
Barrier Strips

Abstract The existence results obtained for the Dirichlet and mixed BVPs for the equation x ″ = f (t, x, x′) are extended to BVPs with full nonlinear conditions. The proofs are based on the theorem of Granas, Guenther and Lee, while barrier strips are used to obtain a priori bounds for solutions.

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