Well-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids

We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument

Bifurcation of singular solutions in reversible systems and applications to reaction - Diffusion equations

Dynamical systems that are reversible in the sense of Moser are investigated and bifurcation of trajectories connecting saddle points from stationary solutions is studied. As an application, reaction-diffusion models in one space dimension are considered. These equations are studied in the neighborhood of a point, where the set of spatially homogeneous solutions displays a Hopf bifurcation. It is shown that from such a point branches of solutions bifurcate, which can be described as waves travelling to or from a center. These waves may be exponentially damped at infinity or not. They can be regarded as one-dimensional analogues of “target patterns” or “spiral waves.

Transition from rotating waves to modulated rotating waves on the sphere

We study non-resonant and resonant Hopf bifurcation of a rotating wave in SO(3)-equivariant reaction-diffusion systems on a sphere. We obtained reduced differential equations on so(3), the characterization of modulated rotating waves obtained by Hopf bifurcation of a rotating wave, as well as results regarding the resonant case. Our main tools are the equivariant center manifold reduction and the theory of Lie groups and Lie algebras, especially for the group SO(3) of all rigid rotations on a sphere

On a class of quasilinear partial integrodifferential equations with singular kernels

AbstractWe prove local and global existence theorems for a model equation in nonlinear viscoelasticity. In contrast to previous studies, we allow the memory function to have a singularity. We approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions

A discrete systems approach to cardinal spline Hermite interpolation

AbstractA cardinal spline Hermite interpolation problem is posed by specifying values, and m−1 derivatives, m⩾1, at uniformly spaced knots tk; it may be solved by means of a generalized spline function w(t) (a standard spline function when m=1), piecewise a polynomial of degree n−1=2m+p−1, p⩾0, with w(j)(t) continuous across the knots for j=0,1,2,…,m+p−1. The problem is studied here for p>0 in the context of an (m+p)-dimensional system of linear recursion equations satisfied by the values of the m-th through m+p−1-st derivatives of w(t) at the knots, whose homogeneous term involves a p×p matrix A . In the case m=1 we relate the characteristic polynomial of A and certain controllability notions to the standard B-spline and we proceed to show how systems-theoretic ideas can be used to generate systems of basis splines for higher values of m

Role of inertia in two-dimensional deformation and breakup of a droplet

We investigate by Lattice Boltzmann methods the effect of inertia on the deformation and break-up of a two-dimensional fluid droplet surrounded by fluid of equal viscosity (in a confined geometry) whose shear rate is increased very slowly. We give evidence that in two dimensions inertia is {\em necessary} for break-up, so that at zero Reynolds number the droplet deforms indefinitely without breaking. We identify two different routes to breakup via two-lobed and three-lobed structures respectively, and give evidence for a sharp transition between these routes as parameters are varied.Comment: 4 pages, 4 figure

General decay of the solution for a viscoelastic wave equation with a time-varying delay term in the internal feedback

In this paper we consider a viscoelastic wave equation with a time-varying delay term, the coefficient of which is not necessarily positive. By introducing suitable energy and Lyapunov functionals, under suitable assumptions, we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases.Comment: 11 page

On Asymptotic Completeness of Scattering in the Nonlinear Lamb System, II

We establish the asymptotic completeness in the nonlinear Lamb system for hyperbolic stationary states. For the proof we construct a trajectory of a reduced equation (which is a nonlinear nonautonomous ODE) converging to a hyperbolic stationary point using the Inverse Function Theorem in a Banach space. We give the counterexamples showing nonexistence of such trajectories for nonhyperbolic stationary points

A model problem for the initial-boundary value formulation of Einstein's field equations

In many numerical implementations of the Cauchy formulation of Einstein's field equations one encounters artificial boundaries which raises the issue of specifying boundary conditions. Such conditions have to be chosen carefully. In particular, they should be compatible with the constraints, yield a well posed initial-boundary value formulation and incorporate some physically desirable properties like, for instance, minimizing reflections of gravitational radiation. Motivated by the problem in General Relativity, we analyze a model problem, consisting of a formulation of Maxwell's equations on a spatially compact region of spacetime with timelike boundaries. The form in which the equations are written is such that their structure is very similar to the Einstein-Christoffel symmetric hyperbolic formulations of Einstein's field equations. For this model problem, we specify a family of Sommerfeld-type constraint-preserving boundary conditions and show that the resulting initial-boundary value formulations are well posed. We expect that these results can be generalized to the Einstein-Christoffel formulations of General Relativity, at least in the case of linearizations about a stationary background.Comment: 25 page

Global-in-time solutions for the isothermal Matovich-Pearson equations

In this paper we study the Matovich-Pearson equations describing the process of glass fiber drawing. These equations may be viewed as a 1D-reduction of the incompressible Navier-Stokes equations including free boundary, valid for the drawing of a long and thin glass fiber. We concentrate on the isothermal case without surface tension. Then the Matovich-Pearson equations represent a nonlinearly coupled system of an elliptic equation for the axial velocity and a hyperbolic transport equation for the fluid cross-sectional area. We first prove existence of a local solution, and, after constructing appropriate barrier functions, we deduce that the fluid radius is always strictly positive and that the local solution remains in the same regularity class. To the best of our knowledge, this is the first global existence and uniqueness result for this important system of equations