On the global existence of weak solution for a multiphasic incompressible fluid model with Korteweg stress

Abstract

In this paper, we study a multiphasic incompressible fluid model, called the Kazhikhov–Smagulov model, with a particular viscous stress tensor, introduced by Bresch and co‐authors, and a specific diffusive interface term introduced for the first time by Korteweg in 1901. We prove that this model is globally well posed in a 3D bounded domain. Copyright © 2016 John Wiley & Sons, Ltd.

Nonmonotone slip problem for miscible liquids

“…where ∂j denotes the generalized gradient of a prescibed locally Lipschitz function j. The boundary friction law (8) has been considered for the Navier-Stokes problems in [16,13,17,20]. Finally, the initial conditions for the velocity and concentration are prescibed…”

“…The boundary friction law (8) has been considered for the Navier-Stokes problems in [16,13,17,20]. Finally, the initial conditions for the velocity and concentration are prescibed…”

“…Results on the unique weak solvability of models describing miscible liquids can be found [1,15] where the problem was studied in two dimensional case, in the absence of external and source forces, with no subdifferential boundary conditions, and in [2,3] who treated the three dimensional case with the homogeneous Dirichlet boundary condition on the whole boundary. A result on existence of the global weak solution for a multiphasic incompressible fluid model with Korteweg stress can be found in [8] where the Galerkin method combined with a fixed point argument have been employed.…”

Existence of global weak solutions of inhomogeneous incompressible Navier-Stokes equations with mass diffusion

“…al. [6] derived versions of (1.4) without θ 2 -nonlinearity by specific choice of µ and θ. Cook-Dimotakis [7] investigated (1.2) with the constant viscosity and (1.3) to understand Rayleigh-Taylor instability between miscible fluids. Goudon-Vasseur [10] re-analyzed Kazhikhov-Smagulov type modeling of mixture flows from a more microscopic viewpoint to propose a generalized model.…”

$$L^\infty $$ -Stability of IMEX-BDF2 Finite Volume Scheme for Convection-Diffusion Equation

“…This model was firstly studied in [14,1] (see also references therein). The mathematical analysis in a three-dimensional domain of KazhikhovSmagulov type models was carried out in recent works [5,8], where the authors study the Kazhikhov-Smagulov models with a specific Korteweg stress tensor. The numerical study of a Kazhikhov-Smagulov model for the two-dimensional case can be found in [4], where the authors propose an hybrid finite volume-finite element method combined with the backward Euler method in time.…”