Rigorous derivation of the Kuramoto-Sivashinsky equation in a 2D weakly nonlinear Stefan problem

In this paper we are interested in a rigorous derivation of the Kuramoto-Sivashinsky equation (K--S) in a Free Boundary Problem. As a paradigm, we consider a two-dimensional Stefan problem in a strip, a simplified version of a solid-liquid interface model. Near the instability threshold, we introduce a small parameter $\varepsilon$ and define rescaled variables accordingly. At fixed $\varepsilon$, our method is based on: definition of a suitable linear 1D operator, projection with respect to the longitudinal coordinate only, Lyapunov-Schmidt method. As a solvability condition, we derive a self-consistent parabolic equation for the front. We prove that, starting from the same configuration, the latter remains close to the solution of K--S on a fixed time interval, uniformly in $\varepsilon$ sufficiently small

Modeling fungal hypha tip growth via viscous sheet approximation

In this paper we present a new model for single-celled, non-branching hypha tip growth. The growth mechanism of hypha cells consists of transport of cell wall building material to the cell wall and subsequent incorporation of this material in the wall as it arrives. To model the transport of cell wall building material to the cell wall we follow Bartnicki-Garcia et al in assuming that the cell wall building material is transported in straight lines by an isotropic point source. To model the dynamics of the cell wall, including its growth by new material, we use the approach of Campas and Mahadevan, which assumes that the cell wall is a thin viscous sheet sustained by a pressure difference. Furthermore, we include a novel equation which models the hardening of the cell wall as it ages. We present numerical results which give evidence that our model can describe tip growth, and briefly discuss validation aspects

Understanding start-up problems in yeast glycolysis

Yeast glycolysis has been the focus of research for decades, yet a number of dynamical aspects of yeast glycolysis remain poorly understood at present. If nutrients are scarce, yeast will provide its catabolic and energetic needs with other pathways, but the enzymes catalysing upper glycolytic fluxes are still expressed. We conjecture that this overexpression facilitates the rapid transition to glycolysis in case of a sudden increase in nutrient concentration. However, if starved yeast is presented with abundant glucose, it can enter into an imbalanced state where glycolytic intermediates keep accumulating, leading to arrested growth and cell death. The bistability between regularly functioning and imbalanced phenotypes has been shown to depend on redox balance. We shed new light on these phenomena with a mathematical analysis of an ordinary differential equation model, including NADH to account for the redox balance. In order to gain qualitative insight, most of the analysis is parameter-free, i.e., without assigning a numerical value to any of the parameters. The model has a subtle bifurcation at the switch between an inviable equilibrium state and stable flux through glycolysis. This switch occurs if the ratio between the flux through upper glycolysis and ATP consumption rate of the cell exceeds a fixed threshold. If the enzymes of upper glycolysis would be barely expressed, our model predicts that there will be no glycolytic flux, even if external glucose would be at growth-permissable levels. The existence of the imbalanced state can be found for certain parameter conditions independent of the mentioned bifurcation. The parameter-free analysis proved too complex to directly gain insight into the imbalanced states, but the starting point of a branch of imbalanced states can be shown to exist in detail. Moreover, the analysis offers the key ingredients necessary for successful numerical continuation, which highlight the existence of this bistability and the influence of the redox balance

A Local Analysis of Similarity Solutions of the Thin Film Equation

We classify singularities of the equation (f n f 000 ) 0 = fijf 0 +fff . Solutions of this equation are similarity profiles of the equation u t + (u n u xxx ) x = 0, which arises in the description of thin film viscosity driven flow. The classification is based upon a transformation to a 4-dimensional quadratic system and a local critical point analysis. Key Words and Phrases. Similarity solutions, 4-dimensional quadratic dynamical systems, Poincar'e transformation, critical points. 1 Introduction In this paper we study similarity solutions of the the following 4th order degenerate diffusion equation. u t + (u n u xxx ) x = 0: (1.1) Here u = u(x; t) and n ? 0. This equation arises in the modelling of thin film slow viscous flows such as painting layers. Let us briefly go through the derivation of this model. We follow [10]. Consider a two-dimensional thin flow on a horizontal plate. The flow domain is described by f(x; y) : 0 y h(x; t)g. Assuming the inertia terms to..

Recent Results on Selfsimilar Solutions of Degenerate Nonlinear Diffusion Equations

this paper we restrict ourselves to m ? 1. This solution, which was first published by Zel'dovich and Kompanyeets [30], initiated the development of an extensive theory for equation (1.1) and its generalizations. Without going into to any detail, let us recall that [1]: the Cauchy problem for nonnegative integrable initial data is wellposed; the intermediate asymptotics of (weak) solutions are given by (1.11) with C determined b

Selfsimilar Solutions of Barenblatt's Model for Turbulence

. In this paper we consider Barenblatt's k \Gamma ffl model for turbulence. For the case of equal diffusion coefficients ff and fi Barenblatt found explicit compactly supported selfsimilar solutions. From these we obtain compactly supported solutions for ff 6= fi by transforming the equations into a four-dimensional quadratic system and verifying a transversality condition for a saddle point connection. This involves the Poincar'e transformation as well as classical properties of the hypergeometric equation and its solutions. AMS classification. 35K65. Running head. Barenblatt's model for turbulence. Keywords. Turbulence, compactly supported similarity solutions, quadratic systems, critical points at infinity, Poincar'e transformation, saddle point connections, transversality. y This work was supported by the Netherlands Organization for Scientific Research, NWO, and by EEC-grant SC1-0019-C-(TT). Introduction. In this paper we consider the system (KE) 8 ? ? ! ? ? : k t = ff i ..

Selfsimilar Solutions of the k-epsilon Model for Turbulence

. The k \Gamma ffl model for turbulence reads (KE) 8 ? ? ! ? ? : k t = ff i k 2 " k x j x \Gamma "; " t = fi i k 2 " " x j x \Gamma fl " 2 k : We look for compactly supported similarity solutions of the form k = 1 t 2¯ f(i); " = 1 t 2¯+1 g(i); i = x t 1\Gamma¯ ; where we restrict our attention to the case 0 ! ¯ ! 1. For ff = fi = 1; ¯ = fl 3(fl \Gamma 1) ; fl ? 3 2 ; there exists an explicit solution, namely f(i) = 1 6 (2 \Gamma )(1 \Gamma i 2 ) + ; g(i) = f(i); = 1 fl \Gamma 1 : We show that this solution can be used to obtain a compactly supported solution for ff 6= fi. This involves a transformation of the equations for f and g into a four-dimensional quadratic system, an analysis by Maple5 of the critical points at infinity of this system, and a transversality condition for a saddle point connection. The verification of this condition depends on classical properties of the hypergeometric equation and its solutions. Keywords. Turbulence, compac..