Search

The system of Brusselator-type reaction-diffusion equations (RDs) on open bounded convex domains 풟 ⊂ ℝ d (d ≤ 3) with Robin boundary conditions (Rbcs) has been mathematically analyzed. The Faedo–Galerkin approach is used to demonstrate the global existence and uniqueness of a weak solution to the system. The weak solution's higher regularity findings are constructed under more regular conditions on the initial data. In addition, continuous dependence on the initial conditions has been proved. [ABSTRACT FROM AUTHOR]

In this article, we investigate the influence of the vertical throughflow Reynolds number on the instability of Poiseuille flow in a bidisperse porous medium. The Brinkman model was employed to describe fluid flow in the porous medium with large pores, referred to as "macropores," while the Darcy model was utilized for fluid flow in the porous medium with small pores, referred to as "micropores". The resulting eigenvalue system was solved using the Chebyshev collocation method (CCM), renowned for its accuracy and flexibility, rendering it one of the most reliable methods available. Regardless of its direction, the impact of the vertical throughflow Reynolds number on system instability is not uniform; it exhibits a dual nature, acting as a destabilizing factor at specific values while serving as a stabilizing influence at others. In the case of the permeability ratio, porous parameter, and interaction parameter, our observations indicate that elevating these parameters results in an enhancement of system stability. [ABSTRACT FROM AUTHOR]

A mathematical analysis is performed for a system consisting of two coupled Cahn-Hilliard equations. These equations incorporate a diffusional mobility that depends on concentration. This modeling approach is often used to describe the process of phase separation in a thin layer of a binary liquid mixture covering a substrate, particularly when one of the components wets the substrate. The analysis establishes the existence of a weak formulation for this problem, which is supported by the use of a Lyapunov functional. Additionally, the analysis provides insights into the regularity properties of the weak formulation. [ABSTRACT FROM AUTHOR]

A model for double-diffusive convection in anisotropic and inhomogeneous porous media has been analysed. In particular, the effect of variable permeability and thermal diffusivity has been studied using the Brinkman model. Moreover, we analyse the effect of slip boundary conditions on the stability of the model. Due to numerous applications in micro-electro-mechanical-systems (MEMS) and other microfluidic devices, such a study is essential to have. Both linear instability analysis and nonlinear stability analysis are employed. We accurately analyse when stability and instability will commence and determine the critical Rayleigh number as a function of the slip coefficient.

We conducted a mathematical investigation on a system of interconnected Cahn-Hilliard equations featuring a logarithmic potential, nondegenerate mobility, and homogeneous Neumann boundary conditions. This system emerges from a model depicting the phase separation of a binary liquid mixture in a thin film. Assuming certain conditions on the initial data, we successfully established the existence, uniqueness, and stability estimates for the weak solution. Our approach involved initially replacing the logarithmic potential with a smooth counterpart, resulting in the regularization of the original problem (Q) into a regularized problem (Qϵ). Utilizing the Faedo-Galerkin method and compactness arguments, we demonstrated the existence and uniqueness of a solution for (Qϵ). Subsequently, by letting ε approach zero, we attained the existence of a solution for the original problem (Q). Additionally, we addressed higher regularity aspects of the weak solutions for both (Q) and (Qϵ). Employing the standard regularity theory for elliptic problems and introducing additional assumptions regarding the domain's boundary and the initial data, we established that the weak solutions belong to higher-order Sobolev spaces. [ABSTRACT FROM AUTHOR]

This paper examines a finite element method for the chemotaxis-growth model with indirect attractant production and logistic source. To begin, we propose a regularized problem of the truncated problem. We then obtain some a priori estimates of regularized solutions that are independent of the regularization parameter using a well-defined entropy inequality for the regularized problem. Additionally, we offer an efficient fully discrete finite element approximation of the regularized problem. A fixed point theorem is then used to prove that the approximate solutions exist. A discrete entropy inequality is also proposed for fully discrete finite element problems, as well as some stability bounds. We also investigate the convergence of the fully discrete problem. [ABSTRACT FROM AUTHOR]