On Two-Dimensional Sonic-Subsonic Flow

A compensated compactness framework is established for sonic-subsonic approximate solutions to the two-dimensional Euler equations for steady irrotational flows that may contain stagnation points. Only crude estimates are required for establishing compactness. It follows that the set of subsonic irrotational solutions to the Euler equations is compact; thus flows with sonic points over an obstacle, such as an airfoil, may be realized as limits of sequences of strictly subsonic flows. Furthermore, sonic-subsonic flows may be constructed from approximate solutions. The compactness framework is then extended to self-similar solutions of the Euler equations for unsteady irrotational flows.     

A proof of the uniform boundedness of solutions to the wave equation on slowly rotating Kerr backgrounds

We consider Kerr spacetimes with parameters a and M such that |a|≪M, Kerr-Newman spacetimes with parameters |Q|≪M, |a|≪M, and more generally, stationary axisymmetric black hole exterior spacetimes (M,g)(\mathcal{M},g) which are sufficiently close to a Schwarzschild metric with parameter M>0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ _g ψ=0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ _τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ, a suitable energy flux through Σ _τ is bounded by C times the initial energy flux through Σ₀. This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ, in terms of a higher order initial energy on Σ₀. Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.     

Hyperbolic balance laws with dissipative source relaxing to adiabatic conservation laws

The paper discusses the long time behavior of BV solutions to the Cauchy problem for hyperbolic systems of balance laws with partial dissipation, when the relaxed system is adiabatic.

     

Sensitivity analysis for the asymmetric network equilibrium problem

We consider the asymmetric continuous traffic equilibrium network model with fixed demands where the travel cost on each link of the transportation network may depend on the flow on this as well as other links of the network and we perform stability and sensitivity analysis. Assuming that the travel cost functions are monotone we first show that the traffic equlibrium pattern depends continuously upon the assigned travel demands and travel cost functions. We then focus on the delicate question of predicting the direction of the change in the traffic pattern and the incurred travel costs resulting from changes in the travel cost functions and travel demands and attempt to elucidate certain counter intuitive phenomena such as ‘Braess' paradox’. Our analysis depends crucially on the fact that the governing equilibrium conditions can be formulated as a variational inequality. This work was supported by the Program of University Research, U.S. Department of Transportation (Project number DTRS 5680-C-00007).     

The modulation equations of nonlinear geometric optics

The purpose of this paper is to present an introduction to the central aspects of the modulation equations of nonlinear geometric optics. That means regularity properties and existence in the large.     

Black Hole Formation from a Complete Regular Past

An open problem in general relativity has been to construct an asymptotically flat solution to a reasonable Einstein-matter system containing a black hole and yet causally geodesically complete to the past, containing no white holes. We construct such a solution in this paper–in fact a family of such solutions, stable in a suitable sense–where matter is described by a self-gravitating scalar field.     

Partitionable variational inequalities with applications to network and economic equilibria

In this paper, we describe a useful class of finite-dimensional variational inequalities which we call partitionable. These variational inequalities are characterized by state functions which can be thought of as nonlinear separable functions added to antisymmetric linear functions. In the case of partitionable variational inequalities, questions of the monotonicity and coercivity of the state function can be addressed by considering the monotonicity and coercivity of a series of lower-dimensional functions. These functions are generally simpler to investigate than the state function. In the applications, these lower-dimensional functions are usually the natural functions to consider. To demonstrate, we conclude the paper by reviewing several models in the recent literature which give rise to partitionable variational inequalities.