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Catastrophes and partial differential equations

John Guckenheimer (1973)

Annales de l'institut Fourier

This paper outlines the manner in which Thom’s theory of catastrophes fits into the Hamilton-Jacobi theory of partial differential equations. The representation of solutions of a first order partial differential equation as lagrangian manifolds allows one to study the local structure of their singularities. The structure of generic singularities is closely related to Thom’s concept of the elementary catastrophe associated to a singularity. Three concepts of the stability of a singularity are discussed....

Comparison of Perron and Floquet Eigenvalues in Age Structured Cell Division Cycle Models

J. Clairambault, S. Gaubert, Th. Lepoutre (2009)

Mathematical Modelling of Natural Phenomena

We study the growth rate of a cell population that follows an age-structured PDE with time-periodic coefficients. Our motivation comes from the comparison between experimental tumor growth curves in mice endowed with intact or disrupted circadian clocks, known to exert their influence on the cell division cycle. We compare the growth rate of the model controlled by a time-periodic control on its coefficients with the growth rate of stationary models of the same nature, but with averaged coefficients....

Convergence of formal solutions of first order singular partial differential equations of nilpotent type

Masatake Miyake, Akira Shirai (2012)

Banach Center Publications

Let (x,y,z) ∈ ℂ³. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: P u ( x , y , z ) : = ( y x - z y ) u ( x , y , z ) = f ( x , y , z ) x , y , z , where P = y x - z y : x , y , z x , y , z . For this equation, our aim is to characterize the solvability on x , y , z by using the Im P, Coker P and Ker P, and we give the exact forms of these sets.

Differential operators of the first order with degenerate principal symbols

Rainer Felix (1992)

Banach Center Publications

Let there be given a differential operator on n of the form D = i , j = 1 n a i j · x j / x i + μ , where A = ( a i j ) is a real matrix and μ is a complex number. We study the following question: To what extent the mapping D : S ' ( n ) S ' ( n ) is surjective? We shall give some conditions on A and μ which assure the surjectivity of D.

Équations de transport dont les vitesses sont partiellement B V

Nicolas Lerner (2003/2004)

Séminaire Équations aux dérivées partielles

Nous démontrons l’unicité des solutions faibles pour une classe d’équations de transport dont les vitesses sont partiellement à variations bornées. Nous nous intéressons à des champs de vecteurs du type a 1 ( x 1 ) · x 1 + a 2 ( x 1 , x 2 ) · x 2 , a 1 B V ( x 1 N 1 ) , a 2 L x 1 1 B V ( x 2 N 2 ) , avec une borne sur la divergence de chacun des champs a 1 , a 2 . Ce modèle a été étudié récemment dans [LL] par C. Le Bris et P.-L. Lions avec une régularité W 1 , 1  ; nous montrons ici également que, dans le cas W 1 , 1 , le contrôle L de la divergence totale du champ est suffisant. Notre méthode consiste à démontrer...

Existence of solutions to nonlinear advection-diffusion equation applied to Burgers' equation using Sinc methods

Kamel Al-Khaled (2014)

Applications of Mathematics

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers' equation. The approximate solution...

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