We prove existence of weak solutions to doubly degenerate diffusion equations $$\dot{u}={\Delta }_p{u}^{m-1}+f(m,p\ge 2)$$ by Faedo-Galerkin approximation for general domains and general nonlinearities. More precisely, we discuss the equation in an abstract setting, which allows to choose function spaces corresponding to bounded or unbounded domains $\Omega \subset {ℝ}^n$ with Dirichlet or Neumann boundary conditions. The function $f$ can be an inhomogeneity or a nonlinearity involving terms of the form $f\left(u\right)$ or $div\left(F\right(u\left)\right)$. In the appendix, an introduction to weak differentiability...

For a class of degenerate diffusion processes of rank 2, i.e. when only Poisson brackets of order one are needed to span the whole space, we obtain a parametrix representation of McKean–Singer [J. Differential Geom.1 (1967) 43–69] type for the density. We therefrom derive an explicit gaussian upper bound and a partial lower bound that characterize the additional singularity induced by the degeneracy. This particular representation then allows to give a local limit theorem with the usual convergence...

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence...

We are interested in the feedback stabilization of a fluid flow over a flat plate, around a stationary solution, in the presence of perturbations. More precisely, we want to stabilize the laminar-to-turbulent transition location of a fluid flow over a flat plate. For that we study the Algebraic Riccati Equation (A.R.E.) of a control problem in which the state equation is a doubly degenerate linear parabolic equation. Because of the degenerate character of the state equation, the classical existence...

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and $L^1$ contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in $L^1$ and $L^{\infty }$, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive $L^1$ convergence without any convergence rate....

Finite element approximation for degenerate parabolic equations is considered. We propose a semidiscrete scheme provided with order-preserving and L1 contraction properties, making use of piecewise linear trial functions and the lumping mass technique. Those properties allow us to apply nonlinear semigroup theory, and the wellposedness and stability in L1 and L∞, respectively, of the scheme are established. Under certain hypotheses on the data, we also derive L1 convergence without any...

We consider a fully practical finite element approximation of the following degenerate system$$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$subject to an initial condition on the temperature, $u$, and boundary conditions on both $u$ and the electric potential, $\phi$. In the above $\rho \left(u\right)$ is the enthalpy incorporating the latent heat of melting, $\alpha \left(u\right)\>0$ is the temperature dependent heat conductivity, and $\sigma \left(u\right)\ge 0$ is the electrical conductivity. The latter is zero in the frozen zone, $u\le 0$, which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

We consider a fully practical finite element approximation of the following degenerate system $$\frac{\partial }{\partial t}\rho \left(u\right)-\nabla .\left(\alpha \left(u\right)\nabla u\right)\ni \sigma \left(u\right){\left|\nabla \phi \right|}^2,\nabla .\left(\sigma \left(u\right)\nabla \phi \right)=0$$ subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy...

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity...

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a general class of coupled FENE-type finitely extensible nonlinear elastic dumbbell models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ℝd, d = 2 or 3, for the velocity...

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ${ℝ}^d$,d= 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....

We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier–Stokes equations in a bounded domain Ω ⊂ ${ℝ}^d$, d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation....

Using the asymptotic a priori estimate method, we prove the existence of a pullback -attractor for a reaction-diffusion equation with an inverse-square potential in a bounded domain of ${ℝ}^N$ (N ≥ 3), with the nonlinearity of polynomial type and a suitable exponential growth of the external force. Then under some additional conditions, we show that the pullback -attractor has a finite fractal dimension and is upper semicontinuous with respect to the parameter in the potential.