On the manifold of closed hypersurfaces in $\mathbb{R}^n$
Jan Prüss Gieri Simonett
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.
keywords: mean curvature surface gradient and surface divergence tubular neighborhood Principal curvature normal variation approximation of hypersurfaces. level function
On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities
Jan Prüss Yoshihiro Shibata Senjo Shimizu Gieri Simonett
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.
keywords: phase transitions time weights. entropy surface tension Two-phase Navier-Stokes equations well-posedness
Modeling and analysis of reactive multi-component two-phase flows with mass transfer and phase transition -- The isothermal incompressible case
Dieter Bothe Jan Prüss
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.
keywords: Two-phase flows phase transition mass transfer chemical reactions available energy quasilinear parabolic evolution equations maximal regularity generalized principle of linearized stability convergence to equilibria.
Analysis of a model for the dynamics of prions
Jan Prüss Laurent Pujo-Menjouet G.F. Webb Rico Zacher
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.
keywords: proliferation viral-host interaction. epidemics prions
On normal stability for nonlinear parabolic equations
Jan Prüss Gieri Simonett Rico Zacher
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.
keywords: fully nonlinear parabolic equations generalized principle of linearized stability normally stable Convergence towards equilibria center manifolds
Well-posedness and long-time behaviour for the non-isothermal Cahn-Hilliard equation with memory
Jan Prüss Vicente Vergara Rico Zacher
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.
keywords: Cahn-Hilliard equation with memory integro-differential equations phase field system with memory Łojasiewicz-Simon inequality Maximal regularity convergence to equilibrium.
Center manifolds and dynamics near equilibria of quasilinear parabolic systems with fully nonlinear boundary conditions
Yuri Latushkin Jan Prüss Ronald Schnaubelt
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
Singular limits for the two-phase Stefan problem
Jan Prüss Jürgen Saal Gieri Simonett
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.
keywords: Stefan problem singular limits maximal regularity. free boundary problem phase transition
Semigroup-theoretic approach to identification of linear diffusion coefficients
Gianluca Mola Noboru Okazawa Jan Prüss Tomomi Yokota
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).
keywords: linear evolution equations in Banach spaces linear parabolic equations Identification problems unknown constants well-posedness results.
On the Muskat problem
Jan Prüss Gieri Simonett
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.
keywords: normally hyperbolic. normally stable Darcy's law phase transition Lyapunov function Muskat problem porous medium free boundary problem

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