We consider the equation $u_t\mathrm{-}i\mathrm{\Lambda }u=0$, where $\mathrm{\Lambda }=\lambda \left(D_x\right)$ is a first order pseudo-differential operator with real symbol $\lambda \left(\xi \right)$. Under a suitable convexity assumption on $\lambda$ we find the decay properties for $u\left(t,x\right)$. These can be applied to the linear Maxwell system in anisotropic media and to the nonlinear Cauchy Problem $u_t\mathrm{-}i\mathrm{\Lambda }u=f\left(u\right)$, $u\left(0,x\right)=g\left(x\right)$. If $f\left(u\right)$ is a smooth function which satisfies $f\left(u\right)\simeq {\left|u\right|}^p$ near $u=0$, and $g$ is small in suitably Sobolev norm, we prove global existence theorems provided $p$ is greater than a critical exponent.

A classical model for three-phase capillary immiscible flows in a porous medium is considered. Capillarity pressure functions are found, with a corresponding diffusion-capillarity tensor being triangular. The model is reduced to a degenerate quasilinear parabolic system. A global existence theorem is proved under some hypotheses on the model data.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies$$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$where $\Omega$ is an open set of ${ℝ}^2$ and $E$ is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on $\varphi$.

In this article we are interested in the following problem: to find a map $u:\Omega \to {ℝ}^2$ that satisfies $$\left\{\begin{array}{cc}Du\in E\hfill & \mathit{\text{a.e.}}\text{in}\Omega \hfill \\ u\left(x\right)=\varphi \left(x\right)\hfill & x\in \partial \Omega \hfill \end{array}\right.$$ where Ω is an open set of ${ℝ}^2$ and E is a compact isotropic set of ${ℝ}^{2\times 2}$. We will show an existence theorem under suitable hypotheses on φ.

For a two phase incompressible flow we consider a diffuse interface model aimed at addressing the movement of three-phase (fluid-fluid-solid) contact lines. The model consists of the Cahn Hilliard Navier Stokes system with a variant of the Navier slip boundary conditions. We show that this model possesses a natural energy law. For this system, a new numerical technique based on operator splitting and fractional time-stepping is proposed. The method is shown to be unconditionally stable. We present...

We study a singular perturbation problem arising in the scalar two-phase field model. Given a sequence of functions with a uniform bound on the surface energy, assume the Sobolev norms $W^{1,p}$ of the associated chemical potential fields are bounded uniformly, where $p\>\frac{n}{2}$ and $n$ is the dimension of the domain. We show that the limit interface as $\epsilon$ tends to zero is an integral varifold with a sharp integrability condition on the mean curvature.