A Free Boundary Problem for an Elliptic–Parabolic System: Application to a Model of Tumor Growth

Abstract

In this article, we study a free boundary problem for a system of two partial differential equations, one parabolic and other elliptic. The system models the growth of a tumor with arbitrary initial shape. We establish the existence and uniqueness of a solution for some time interval. In the special case where we only have the elliptic equation, the problem coincides with the Hele–Shaw problem.     

Existence and Non-Existence Results for Quasilinear Elliptic Exterior Problems with Nonlinear Boundary Conditions

Existence and non-existence results are established for quasilinear elliptic problems with nonlinear boundary conditions and lack of compactness. The proofs combine variational methods with the geometrical feature, due to the competition between the different growths of the nonlinearities.     

A Class of Operator Equations in Nonlinear Mechanics

A nonlinear evolution equation in Hilbert space is introduced and is shown to govern a broad class of problems of nonlinear solid mechanics. The simplest oscillations of this equation are described by a “generalized Liénard equation.”

AMS(MOS) Classification Nos: 73D35, 73F15, 58D25, 35L35, 34G20, 34C15.     

Comparison of non‐reflective boundary conditions for a free‐rising turbulent axisymmetric plume

The stability and accuracy of radiation type non‐reflective outflow boundary conditions, as well as the standard Neumann boundary condition with zero normal derivative, have been compared for the numerical simulation of a turbulent axisymmetric plume with Reynolds number of 7700 and Prandtl number of 0.71. Comparison of the performance of the boundary conditions with respect to each other, and to the results obtained for an extended domain, shows that a one‐dimensional scheme in which advection and diffusion terms are included in the radiation equation is the optimum approach for the plume simulation. Copyright © 2013 John Wiley & Sons, Ltd.     

Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics.     

Prioritized Composition With Exclusion and Generation for the Interaction and Control of Discrete Event Systems

Interaction of multiple discrete event systems (DESs) represented as automata are carried out using composition operations. These operations on automata enforce concurrency, wherein an event exists in the composed automaton if it exists in the participating states of the interacting automata possessing the event in their event set. Heymann generalized this by introducing event priorities, wherein an event exists in the composed automaton if it exists in the participating state of the interacting automata having priority over the event. For two interacting automata P and Q, while prioritized composition can model the P, Q, AND, and OR boolean interactions, it cannot model boolean interactions which require exclusivity of participation, namely, “exclusive P”, “exclusive Q”, “exclusive P or exclusive Q”, “exclusive P and exclusive Q”. In order to also model these additional interactions we propose a generalization of prioritized composition by introducing an exclusivity set besides the existing priority sets. The resulting composition is called prioritized composition with exclusion. We also introduce prioritized composition with exclusion and generation that allows for all sixteen boolean modes of interaction possible when two automata interact. This is done by the further introduction of a nor-generative set. This event set together with the two priority sets and an exclusivity set makes it possible to model eight additional boolean interactions which do not require either of the interacting automata to participate for the event to be enabled in the composed automaton. The applicability of these interactions to decentralized supervisory decision fusion and in composing the rules based model of systems has been illustrated.     

Microwave-assisted Cannizzaro reaction—Optimisation of reaction conditions

The microwave-assisted Cannizzaro reaction was studied in order to develop fully reproducible synthetic protocols for transformation of aldehydes to carboxylic acid and alcohols. Optimised were the following process parameters: power, temperature, and time. Aromatic, heteroaromatic and aliphatic aldehydes were used in the studies. It was found that furfural, thiophene-2-carbaldehyde, pyridinecarboxaldehyde and aromatic aldehydes react under mild conditions, while 1-methyl-pyrrole-2-carboxaldehyde derivatives and aliphatic aldehydes require more drastic reaction conditions and a longer exposure time to microwave radiation.     

Density estimates for a random noise propagating through a chain of differential equations

We here provide two sided bounds for the density of the solution of a system of n differential equations of dimension d, the first one being forced by a non-degenerate random noise and the n−1 other ones being degenerate. The system formed by the n equations satisfies a suitable Hörmander condition: the second equation feels the noise plugged into the first equation, the third equation feels the noise transmitted from the first to the second equation and so on … , so that the noise propagates one way through the system. When the coefficients of the system are Lipschitz continuous, we show that the density of the solution satisfies Gaussian bounds with non-diffusive time scales. The proof relies on the interpretation of the density of the solution as the value function of some optimal stochastic control problem.     

非齐次边界条件泛定方程的代换选择

对具有非齐次边界条件的泛定方程齐次化过程中代换的选择进行研究和探讨.基于一些相关结论和齐次化的定义得出新的研究成果,即给出对三类非齐次边界条件齐次化都适用的代换W(x,t)=A(t)x<'3>+B(t).     

Kreiss symmetrizer and boundary conditions for the Euler-Korteweg system in a half space

The Euler-Korteweg system is a third order, dispersive system of PDEs, obtained from the standard Euler equations for compressible fluids by adding the so-called Korteweg stress tensor - encoding capillarity effects. Various results of well-posedness have been obtained recently for the Cauchy problem associated with the Euler-Korteweg system in the whole space. As to mixed problems, with initial and boundary value data, they are still mostly open. Here the linearized Euler-Korteweg system is studied in a half space by the use of normal mode analysis, which yields a generalized Kreiss-Lopatinski? condition that must be satisfied by the boundary conditions for the boundary value problem to be well-posed.Conversely, under the uniform Kreiss-Lopatinski? condition, generalized Kreiss symmetrizers are constructed in one space dimension for an extended system originally introduced for the Cauchy problem, which displays crucial quasi-homogeneity properties. A priori estimates without loss of derivatives are thus derived, and finally the well-posedness of the mixed problem is obtained by combining the estimates for the pure boundary value problem and trace results for solutions of the pure Cauchy problem.