PDEs in Moving Time Dependent Domains

In this work we study partial differential equations defined in a domain that moves in time according to the flow of a given ordinary differential equation, starting out of a given initial domain. We first derive a formulation for a particular case of partial differential equations known as balance equations. For this kind of equations we find the equivalent partial differential equations in the initial domain and later we study some particular cases with and without diffusion. We also analyze general second order differential equations, not necessarily of balance type. The equations without diffusion are solved using the characteristics method. We also prove that the diffusion equations, endowed with Dirichlet boundary conditions and initial data, are well posed in the moving domain. For this we show that the principal part of the equivalent equation in the initial domain is uniformly elliptic. We then prove a version of the weak maximum principle for an equation in a moving domain. Finally we perform suitable energy estimates in the moving domain and give sufficient conditions for the solution to converge to zero as time goes to infinity.Comment: pp 559-577. Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics (2013) p. 36

A Class of Free Boundary Problems with Onset of a new Phase

A class of diffusion driven Free Boundary Problems is considered which is characterized by the initial onset of a phase and by an explicit kinematic condition for the evolution of the free boundary. By a domain fixing change of variables it naturally leads to coupled systems comprised of a singular parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even though the one dimensional case has been thoroughly investigated, results as basic as well-posedness and regularity have so far not been obtained for its higher dimensional counterpart. In this paper a recently developed regularity theory for abstract singular parabolic Cauchy problems is utilized to obtain the first well-posedness results for the Free Boundary Problems under consideration. The derivation of elliptic regularity results for the underlying static singular problems will play an important role

Positive solutions to indefinite Neumann problems when the weight has positive average

We deal with positive solutions for the Neumann boundary value problem associated with the scalar second order ODE $$u" + q(t)g(u) = 0, \quad t \in [0, T],$$ where $g: [0, +\infty[\, \to \mathbb{R}$ is positive on $\,]0, +\infty[\,$ and $q(t)$ is an indefinite weight. Complementary to previous investigations in the case $\int_0^T q(t) < 0$, we provide existence results for a suitable class of weights having (small) positive mean, when $g'(x) < 0$ at infinity. Our proof relies on a shooting argument for a suitable equivalent planar system of the type $$x' = y, \qquad y' = h(x)y^2 + q(t),$$ with $h(x)$ a continuous function defined on the whole real line.Comment: 17 pages, 3 figure

S-adenosyl-L-methionine: (S)-scoulerine 9-O-methyltransferase, a highly stereo- and regio-specific enzyme in tetrahydroprotoberberine biosynthesis

Suspension cultures of Berberis species are useful sources for the detection and isolation of a new enzyme which transfers the methyl group from S-adenosyl-L-methionine specifically to the 9-position of the (S)-enantiomer of scoulerine, producing (S)-tetrahydrocolumbamine. The enzyme was enriched 27-fold; it is not particle bound, has a pH optimum of 8.9, a molecular weight of 63 000 and shows a high degree of substrate specificity

Parking and the visual perception of space

Using measured data we demonstrate that there is an amazing correspondence among the statistical properties of spacings between parked cars and the distances between birds perching on a power line. We show that this observation is easily explained by the fact that birds and human use the same mechanism of distance estimation. We give a simple mathematical model of this phenomenon and prove its validity using measured data

Two-dimensional individual clustering model

10 pagesInternational audienceThis paper is devoted to study a model of individual clustering with two specific reproduction rates in two space dimensions. Given q > 2 and an initial condition in W 1,q (Ω), the local existence and uniqueness of solution have been shown in [6]. In this paper we give a detailed proof of existence of global solution

On well-posedness, stability, and bifurcation for the axisymmetric surface diffusion flow

In this article, we study the axisymmetric surface diffusion flow (ASD), a fourth-order geometric evolution law. In particular, we prove that ASD generates a real analytic semiflow in the space of (2 + \alpha)-little-H\"older regular surfaces of revolution embedded in R^3 and satisfying periodic boundary conditions. We also give conditions for global existence of solutions and prove that solutions are real analytic in time and space. Further, we investigate the geometric properties of solutions to ASD. Utilizing a connection to axisymmetric surfaces with constant mean curvature, we characterize the equilibria of ASD. Then, focusing on the family of cylinders, we establish results regarding stability, instability and bifurcation behavior, with the radius acting as a bifurcation parameter for the problem.Comment: 37 pages, 6 figures, To Appear in SIAM J. Math. Ana

Diffusion, Fragmentation and Coagulation Processes: Analytical and Numerical Results

We formulate dynamical rate equations for physical processes driven by a combination of diffusive growth, size fragmentation and fragment coagulation. Initially, we consider processes where coagulation is absent. In this case we solve the rate equation exactly leading to size distributions of Bessel type which fall off as $\exp(-x^{3/2})$ for large $x$-values. Moreover, we provide explicit formulas for the expansion coefficients in terms of Airy functions. Introducing the coagulation term, the full non-linear model is mapped exactly onto a Riccati equation that enables us to derive various asymptotic solutions for the distribution function. In particular, we find a standard exponential decay, $\exp(-x)$, for large $x$, and observe a crossover from the Bessel function for intermediate values of $x$. These findings are checked by numerical simulations and we find perfect agreement between the theoretical predictions and numerical results.Comment: (28 pages, 6 figures, v2+v3 minor corrections

The role of mathematical modeling in VOC analysis using isoprene as a prototypic example

Isoprene is one of the most abundant endogenous volatile organic compounds (VOCs) contained in human breath and is considered to be a potentially useful biomarker for diagnostic and monitoring purposes. However, neither the exact biochemical origin of isoprene nor its physiological role are understood in sufficient depth, thus hindering the validation of breath isoprene tests in clinical routine. Exhaled isoprene concentrations are reported to change under different clinical and physiological conditions, especially in response to enhanced cardiovascular and respiratory activity. Investigating isoprene exhalation kinetics under dynamical exercise helps to gather the relevant experimental information for understanding the gas exchange phenomena associated with this important VOC. A first model for isoprene in exhaled breath has been developed by our research group. In the present paper, we aim at giving a concise overview of this model and describe its role in providing supportive evidence for a peripheral (extrahepatic) source of isoprene. In this sense, the results presented here may enable a new perspective on the biochemical processes governing isoprene formation in the human body.Comment: 17 page