Marshall Slemrod

Admissibility criteria for propagating phase boundaries in a van der Waals fluid

Abstract : This paper gives admissibility criteria for weak solutions to the partial differential equations governing isothermal motion of a van der Waals fluid. The main issue is that an admissibility criterion based on viscosity alone is too restrictive - it rules out all slowly propagating phase boundaries. Instead a criterion based on viscosity and capillarity is proposed. The viscosity-capillarity condition is studied and shown to imply that the state on one side of a phase boundary specifies both the speed of the phase boundary and the state of the other side of the phase boundary (a result which is different from classical gas dynamics).

Stabilization of boundary control systems

Abstract Feedback stabilization of a linear hyperbolic boundary value control system is implemented. Consistent with finite dimensional control theory, the property of controllability of the control system yields the asymptotic stability of the feedback system.

Instability of steady shearing flows in a non-linear viscoelastic fluid

This paper proves the non-existence of global smooth solutions to an equation for a viscoelastic fluid shearing flow. The non-existence of smooth solutions is interpreted physically as the formation of a vortex sheet and an instability in the fluid motion.

Trajectories joining critical points

In this paper we shall be concerned with the existence of trajectories joining a pair of critical points, or more generally, a pair of compact invariant sets. Our framework will be abstract dynamical systems. We shall present applications and examples among both finite and infinite dimensional differential equations. The existence of connecting orbits has attracted the attention of many researchers. (In Gelfand [ 19631, a footnote on page 299 explicitly indicates that this is an interesting problem.) A variety of techniques have been employed to cope with the problem. The tools which are employed include consideration of degree and index theories, isolating blocks, dimensionality of stable and unstable manifolds, bifurcation type arguments, fixed point theory, etc. (Consult Conley [1976], Conley and Smoller [1975, 19781, Conlon [1980], Foy [1964], Gordon [ 19741, and Howard and Koppel [ 19751. Some sort of differentiability and smoothness is required in any of the cited works and therefore the applicability to infinite dimensional problems is limited. In some of these works the global connecting orbit was found only in a local neighborhood of some critical point. In this paper we pursue a rather naive approach. It would not be applicable unless some a priori demands on the global behavior of solutions are met. However, when the method works it is capable of handling infinite

An Admissibility Criterion for Fluids Exhibiting Phase Transitions

This paper shows how Korteweg’s theory of capillarity can be used as an admissibility criterion for weak solutions of the equations of compressible fluid flow. Included in the theory are fluids capable of exhibiting liquid-vapor changes of phase.

Positively Invariant Regions for a Problem in Phase Transitions

Positively invariant regions for the system v t + p(w) x =e v xx ,w t − v x =e w xλ are constructed where p′ β, p′(w)= 0, α ≦ w ≦ β, e > 0. Such a choice of p is motivated by the Maxwell construction for a van der Waals fluid. The method of an analysis is a modification of earlier ideas of Chueh, Conley, & Smoller [1]. The results given here provide e independent L ∞, bounds on the solution (w, v).