Local Well-Posedness for Relaxational Fluid Vesicle Dynamics

We prove the local well-posedness of a basic model for relaxational fluid vesicle dynamics by a contraction mapping argument. Our approach is based on the maximal $L_p$-regularity of the model's linearization.Comment: 29 page

Strong Well-Posedness for a Class of Dynamic Outflow Boundary Conditions for Incompressible Newtonian Flows

Based on energy considerations, we derive a class of dynamic outflow boundary conditions for the incompressible Navier-Stokes equations, containing the well-known convective boundary condition but incorporating also the stress at the outlet. As a key building block for the analysis of such problems, we consider the Stokes equations with such dynamic outflow boundary conditions in a halfspace and prove the existence of a strong solution in the appropriate Sobolev-Slobodeckij-setting with $L_p$ (in time and space) as the base space for the momentum balance. For non-vanishing stress contribution in the boundary condition, the problem is actually shown to have $L_p$-maximal regularity under the natural compatibility conditions. Aiming at an existence theory for problems in weakly singular domains, where different boundary conditions apply on different parts of the boundary such that these surfaces meet orthogonally, we also consider the prototype domain of a wedge with opening angle $\frac{\pi}{2}$ and different combinations of boundary conditions: Navier-Slip with Dirichlet and Navier-Slip with the dynamic outflow boundary condition. Again, maximal regularity of the problem is obtained in the appropriate functional analytic setting and with the natural compatibility conditions.Comment: 31 pages, 1 figur

A Kinematic Evolution Equation for the Dynamic Contact Angle and some Consequences

We investigate the moving contact line problem for two-phase incompressible flows with a kinematic approach. The key idea is to derive an evolution equation for the contact angle in terms of the transporting velocity field. It turns out that the resulting equation has a simple structure and expresses the time derivative of the contact angle in terms of the velocity gradient at the solid wall. Together with the additionally imposed boundary conditions for the velocity, it yields a more specific form of the contact angle evolution. Thus, the kinematic evolution equation is a tool to analyze the evolution of the contact angle. Since the transporting velocity field is required only on the moving interface, the kinematic evolution equation also applies when the interface moves with its own velocity independent of the fluid velocity. We apply the developed tool to a class of moving contact line models which employ the Navier slip boundary condition. We derive an explicit form of the contact angle evolution for sufficiently regular solutions, showing that such solutions are unphysical. Within the simplest model, this rigorously shows that the contact angle can only relax to equilibrium if some kind of singularity is present at the contact line. Moreover, we analyze more general models including surface tension gradients at the contact line, slip at the fluid-fluid interface and mass transfer across the fluid-fluid interface.Comment: 25 pages, 6 figures; accepted manuscript

On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows

We derive a class of energy preserving boundary conditions for incompressible Newtonian flows and prove local-in-time well-posedness of the resulting initial boundary value problems, i.e. the Navier-Stokes equations complemented by one of the derived boundary conditions, in an Lp-setting in domains, which are either bounded or unbounded with almost flat, sufficiently smooth boundary. The results are based on maximal regularity properties of the underlying linearisations, which are also established in the above setting.Comment: 53 page

Stability Analysis for a Class of Heterogeneous Catalysis Models

We prove stability for a class of heterogeneous catalysis models in the $L_p$-setting. We consider a setting in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. Under a reasonable condition on the involved parameters, we show that given equilibria are normally stable, i.e. solutions are attracted at an exponential rate. The potential incidence of instability is discussed as well.Comment: 14 page

Global Strong Solutions for a Class of Heterogeneous Catalysis Models

We consider a mathematical model for heterogeneous catalysis in a finite three-dimensional pore of cylinder-like geometry, with the lateral walls acting as a catalytic surface. The system under consideration consists of a diffusion-advection system inside the bulk phase and a reaction-diffusion-sorption system modeling the processes on the catalytic wall and the exchange between bulk and surface. We assume Fickian diffusion with constant coefficients, sorption kinetics with linear growth bound and a network of chemical reactions which possesses a certain triangular structure. Our main result gives sufficient conditions for the existence of a unique global strong $L^2$-solution to this model, thereby extending by now classical results on reaction-diffusion systems to the more complicated case of heterogeneous catalysis.Comment: 30 page

Dihydropyrimidine Dehydrogenase Testing prior to Treatment with 5-Fluorouracil, Capecitabine, and Tegafur: A Consensus Paper

Background: 5-Fluorouracil (FU) is one of the most commonly used cytostatic drugs in the systemic treatment of cancer. Treatment with FU may cause severe or life-threatening side effects and the treatment-related mortality rate is 0.2–1.0%. Summary: Among other risk factors associated with increased toxicity, a genetic deficiency in dihydropyrimidine dehydrogenase (DPD), an enzyme responsible for the metabolism of FU, is well known. This is due to variants in the DPD gene (DPYD). Up to 9% of European patients carry a DPD gene variant that decreases enzyme activity, and DPD is completely lacking in approximately 0.5% of patients. Here we describe the clinical and genetic background and summarize recommendations for the genetic testing and tailoring of treatment with 5-FU derivatives. The statement was developed as a consensus statement organized by the German Society for Hematology and Medical Oncology in cooperation with 13 medical associations from Austria, Germany, and Switzerland. Key Messages: (i) Patients should be tested for the 4 most common genetic DPYD variants before treatment with drugs containing FU. (ii) Testing forms the basis for a differentiated, risk-adapted algorithm with recommendations for treatment with FU-containing drugs. (iii) Testing may optionally be supplemented by therapeutic drug monitorin