J. B. McLeod

On the Uniqueness of Flow of a Navier-Stokes Fluid Due to a Stretching Boundary

When a sheet of polymer is extruded continuously from a die, it entrains the ambient fluid and a boundary layer develops. Such a boundary layer is markedly different from that in the Blasius flow past a flat plate in that the boundary layer grows in the direction of the motion of the sheet, starting at the die. Sakiadis [1]–[3] was the first to study such a boundary layer flow due to a rigid flat continuous surface moving in its own plane. Later, Erickson, Fan & Fox [4] studied the boundary layer due to the motion of a porous flat plate when the transverse velocity at the surface is non-zero.

Smooth static solutions of the Einstein/Yang-Mills equations

We consider the Einstein/Yang-Mills equations in 3+1 space time dimensions withSU(2) gauge group and prove rigorously the existence of a globally defined smooth static solution. We show that the associated Einstein metric is asymptotically flat and the total mass is finite. Thus, for non-abelian gauge fields the Yang-Mills repulsive force can balance the gravitational attractive force and prevent the formation of singularities in spacetime.

Classical methods in ordinary differential equations : with applications to boundary value problems

Introduction An introduction to shooting methods Some boundary value problems for the Painleve transcendents Periodic solutions of a higher order system A linear example Homoclinic orbits of the FitzHugh-Nagumo equations Singular perturbation problems--rigorous matching Asymptotics beyond all orders Some solutions of the Falkner-Skan equation Poiseuille flow: Perturbation and decay Bending of a tapered rod variational methods and shooting Uniqueness and multiplicity Shooting with more parameters Some problems of A. C. Lazer Chaotic motion of a pendulum Layers and spikes in reaction-diffusion equations, I Uniform expansions for a class of second order problems Layers and spikes in reaction-diffusion equations, II Three unsolved problems Bibliography Index

Bifurcation for non-differentiable operators with an application to elasticity

AbstractWe consider operator equations of the form

Two expressions for the distribution of eigenvalues

\left( {A_0 - \lambda _0 B_0 } \right)u = N\left( {\lambda ,u} \right)

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

, (1) where A0, B0 are linear operators between real Banach spaces and N(λ, u) is a nonlinear operator with the property that N(λ, 0)=0 for all real λ. Assuming that λ0, a specific value of λ, is an isolated eigenvalue of A0 − λB0 of multiplicity m, we study the phenomenon of bifurcation for equation (1), where it is merely assumed that N(λ, u) is Lipschitz continuous in u near u=0 with a small Lipschitz constant. It is shown that when (1) has a variational structure, for each suitable normalization of u, two non-zero solutions (λ, u) occur near (λ0,0) (m pairs occur if N is odd in u). Further results concern the existence of branches of solutions when m is odd and the asymptotic behavior of solutions in terms of the size of the Lipschitz constant. The motivation for the study and the main application of the results concerns buckling of a von Kármán plate resting on a foundation.

ON AN INFINITE SET OF NON-LINEAR DIFFERENTIAL EQUATIONS

Let us denote by α the set of n real numbers α 1 , …, α n , and by c k (α) and h k (α) the elementary and complete symmetric functions of degree k in α 1 , …, α n , and by c k (α) and h k (α) the elementary and complete symmetric functions of degree k in α l , …, α n , i.e. c k (α) is the sum of all possible products of k different α i and h k (α) is the sum of all possible products of k α i , where now in any product one or more α i may be repeated any number of times.

On a Functional Equation

It is a standard result that the eigenvalues associated with the equation ∇2 ψ + {λ — q(r)} ψ = 0, where q(r) is a function of r only, and tends to infinity as r→∞, are the roots of equations of the form ∫ R2 R1 {λ-q(r) - (l=1/2)2/r2}½ dr = (m + 1/2) π + εm, where l and m are integers, and εm is small when m is large. It has recently been proved that they are also the roots of equations of the form ∫p0 {λ - q(r) }½dr = (1/2l + m + 3/4) π + ηm, where ηm is also small when m is large. By a direct comparison of the integrals involved, this paper shows the two expressions to be consistent.