DCDS

Several results from differential geometry of hypersurfaces in $\mathbb{R}^n$ are derived to form a tool box for the * direct mapping method*.
The latter technique has been widely employed to solve problems with moving interfaces, and to study the asymptotics of the induced semiflows.

EECT

The basic model for incompressible two-phase flows with phase transitions is derived from basic principles and shown to be thermodynamically consistent in the sense that the total energy is conserved and the total entropy is nondecreasing. The local well-posedness of such problems is proved by means of the technique of
maximal $L_p$-regularity in the case of equal densities. This way we obtain a local semiflow on a well-defined nonlinear state manifold. The equilibria of the system in absence of external forces are identified and it is shown that the negative total entropy is a strict Ljapunov functional for the system. If a solution does not develop singularities, it is proved that it exists globally in time, its orbit is relatively compact, and its limit set is nonempty and contained in the set of equilibria.

DCDS-B

A mathematical model for the dynamics of prion
proliferation is analyzed. The model involves a system of three
ordinary differential equations for the normal prion forms, the
abnormal prion forms, and polymers comprised of the abnormal forms.
The model is a special case of a more general model, which is also
applicable to other models of
infectious diseases. A theorem of threshold type is
derived for this general model. It is proved that below and at the threshold,
there is a unique steady state, the disease-free equilibrium, which is
globally asymptotically stable. Above the threshold, the disease-free
equilibrium is unstable, and there is another steady state,
the disease equilibrium, which is globally asymptotically stable.

DCDS-S

Isothermal incompressible multi-component two-phase flows with mass transfer, chemical reactions, and phase transition are modeled based on first principles. It is shown that the resulting system is thermodynamically consistent in the sense that the available energy is a strict Lyapunov functional, and the equilibria are identified. It is proved that the problem is well-posed in an $L_p$-setting, and generates a local semiflow in the proper state manifold. It is further shown that each non-degenerate equilibrium is dynamically stable in the natural state manifold. Finally, it is proved that a solution, which does not develop singularities, exists globally and converges to an equilibrium in the state manifold.

PROC

We show convergence of solutions to equilibria for
quasilinear and fully nonlinear
parabolic evolution equations in situations where the set of
equilibria is non-discrete, but forms a finite-dimensional
$C^1$-manifold which is normally stable.

DCDS

In this paper we study a temperature dependent phase field model
with memory. The case where both the equation for the temperature
and that for the order parameter is of fractional time order is
covered. Under physically reasonable conditions on the
nonlinearities we prove global well-posedness in the $L_p$ setting
and show that each solution converges to a steady state as time goes
to infinity.

DCDS-B

We study quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains. Our main results concern the asymptotic behavior of the solutions in the vicinity of an equilibrium. The local center, center–stable, and center–unstable manifolds are constructed and their dynamical properties are established using nonautonomous cutoff functions. Under natural conditions, we show that each solution starting close to the center manifold converges to a solution on the center manifold.

keywords:
exponential dichotomy
,
Parabolic system
,
anisotropic Slobodetskii spaces
,
invariant
manifold
,
extrapolation
,
Nemytskii
operators
,
center manifold reduction
,
initial--boundary value problem
,
stability
,
implicit function
theorem.
,
attractivity
,
maximal regularity
DCDS

We prove strong convergence to singular limits for a linearized
fully inhomogeneous Stefan problem subject to surface tension and kinetic
undercooling effects. Different combinations of $\sigma \to \sigma_0$
and $\delta\to\delta_0$, where $\sigma,\sigma_0\ge 0$ and
$\delta,\delta_0\ge 0$ denote surface tension and kinetic undercooling
coefficients respectively, altogether lead to five different types
of singular limits. Their strong convergence is based on uniform
maximal regularity estimates.

DCDS-S

Let $X$ be a complex Banach space and
$A:\,D(A) \to X$ a quasi-$m$-sectorial operator
in $X$. This paper is concerned with the
identification of diffusion coefficients
$\nu > 0$ in the initial-value problem:
\[
(d/dt)u(t) + {\nu}Au(t) = 0,
\quad t \in (0,T), \quad u(0) = x \in X,
\]
with additional condition $\|u(T)\| = \rho$,
where $\rho >0$ is known. Except for
the additional condition, the solution to the
initial-value problem is given by
$u(t) := e^{-t\,{\nu}A} x
\in C([0,T];X) \cap C^{1}((0,T];X)$.
Therefore, the identification of $\nu$ is reduced
to solving the equation
$\|e^{-{\nu}TA}x\| = \rho$.
It will be shown that the unique root
$\nu = \nu(x,\rho)$
depends on $(x,\rho)$ locally Lipschitz
continuously if the datum $(x,\rho)$ fulfills
the restriction $\|x\|> \rho$. This extends
those results in
Mola [6](2011).

EECT

Of concern is the motion of two fluids separated by a free interface in a porous medium,
where the velocities are given by Darcy's law.
We consider the case with and without phase transition.
It is shown that the resulting models can be understood as purely geometric evolution laws,
where the motion of the separating interface depends in a non-local way on the mean curvature.
It turns out that the models are volume preserving and surface area reducing,
the latter property giving rise to a Lyapunov function.
We show well-posedness of the models, characterize all equilibria,
and study the dynamic stability of the equilibria.
Lastly, we show that solutions which do not develop singularities exist globally
and converge exponentially fast to an equilibrium.