Twin semigroups and delay equations

In the standard theory of delay equations, the fundamental solution does not 'live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with "retarded" equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations

The Formation, Structure, and Stability of a Shear Layer in a Fluid with Temperature-Dependent Viscosity

The presence of viscosity normally has a stabilizing effect on the flow of a fluid. However, experiments show that the flow of a fluid might form shear bands or shear layers, narrow bands in which the velocity of the fluid changes sharply. In general, adiabatic shear layers are observed not only in fluids but also in thermo-plastic materials subject to shear at a high-strain rate and in combustion. Therefore there is widespread interest in modeling the formation of shear layers. In this paper we investigate the basic system of conservation laws for a one-dimensional flow with temperature-dependent viscosity using a combination of analytical and numerical tools. We present results to substantiate the claim that the formation of shear layers is due to teh fact that viscosity decreases sufficiently quickly as temperature increases and analyze the structure and stability properties of the layers

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