Using two models that incorporate a nonlinear forward-backward heat equation, we demonstrate the existence of well-deened weak solutions containing shocks for diiusive problems. Occurrence of shocks is connected to multivalued inverse solutions and non-monotone potential functions. Unique viscous solutions are determined from perturbation theory by matching to a shock layer condition. Results o...

We study a diffuse interface model for two-phase flows of similar densities in superposed free flow and porous media. The consists the Navier–Stokes–Cahn–Hilliard system Darcy–Cahn–Hilliard media coupled through set domain boundary conditions. These conditions include nonlinear Lions condition linear Beavers–Joseph–Saffman–Jones condition. establish global existence weak solutions three dimensi...

We are concerned with the Cahn–Hilliard/Navier–Stokes equations for stationary compressible flows in a three-dimensional bounded domain. The governing consist of Navier–Stokes describing fluid and Cahn–Hilliard-type diffuse equation mass concentration difference. prove existence weak solutions when adiabatic exponent $$\gamma $$ satisfies >\frac{4}{3}$$ . proof is based on weighted total energy...

We study a Cahn-Hilliard-Hele-Shaw (or Cahn-Hilliard-Darcy) system for an incompressible mixture of two fluids. The relative concentration difference $\varphi$ is governed by convective nonlocal Cahn-Hilliard equation with degenerate mobility and logarithmic potential. volume averaged fluid velocity $\mathbf{u}$ obeys Darcy's law depending on the so-called Korteweg force $\mu\nabla \varphi$, wh...

In this paper, we present a novel second order in time mixed finite element scheme for the Cahn-Hilliard-Navier-Stokes equations with matched densities. The scheme combines a standard second order Crank-Nicholson method for the Navier-Stokes equations and a modification to the Crank-Nicholson method for the Cahn-Hilliard equation. In particular, a second order Adams-Bashforth extrapolation and ...

In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of stabilized finite element method based on multiscale enrichment spatial discretization semi-implicit scheme temporal discretization. With reasonable stability conditions, it is shown that proposed are energy stable. Furthermore, by defining a new projection operator...