Constraints on unified models for dark matter and dark energy using <Emphasis Type="Italic">H</Emphasis>(<Emphasis Type="Italic">z</Emphasis>)

The differential age data of astrophysical objects that have evolved passively during the history of the universe (e.g. red galaxies) allows us to test theoretical cosmological models through the predicted Hubble function expressed in terms of the redshift z, H(z). We use the observational data for H(z) to test unified scenarios for dark matter and dark energy. Specifically, we focus our analysis on the Generalized Chaplygin Gas (GCG) and the viscous fluid (VF) models. For the GCG model, it is shown that the unified scenario for dark energy and dark matter requires some priors. For the VF model we obtain estimations for the free parameters, which may be compared with further analysis mainly at perturbative level.     

Uniqueness and asymptotic stability of Riemann solutions for the compressible Euler equations

We prove the uniqueness of Riemann solutions in the class of entropy solutions in for the system of compressible Euler equations, under usual assumptions on the equation of state for the pressure which imply strict hyperbolicity of the system and genuine nonlinearity of the first and third characteristic families. In particular, if the Riemann solutions consist of at most rarefaction waves and contact discontinuities, we show the global -stability of the Riemann solutions even in the class of entropy solutions in with arbitrarily large oscillation for the system. We apply our framework established earlier to show that the uniqueness of Riemann solutions implies their inviscid asymptotic stability under perturbation of the Riemann initial data, as long as the corresponding solutions are in and have local bounded total variation satisfying a natural condition on its growth with time. No specific reference to any particular method for constructing the entropy solutions is made. Our uniqueness result for Riemann solutions can easily be extended to entropy solutions , piecewise Lipschitz in , for any 0$">.

     

Uniqueness for Multidimensional Hyperbolic Systems with Commuting Jacobians

We consider nonlinear hyperbolic systems of conservation laws in several space dimensions with Jacobian matrices that commute, and more generally systems that need not be conservative. Generalizing a theorem by Bressan and LeFloch for one-dimensional systems, we establish that the Cauchy problem admits at most one entropy solution depending continuously upon its initial data. The uniqueness result is proven within the class (introduced here) of locally regular BV functions with locally controlled oscillation. These regularity conditions are modeled on well-known properties in the one-dimensional case. Our uniqueness theorem also improves upon the known results for one-dimensional systems.     

Periodic solutions of conservation laws constructed through Glimm scheme

We present a periodic version of the Glimm scheme applicable to special classes of systems for which a simplication first noticed by Nishida (1968) and further extended by Bakhvalov (1970) and DiPerna (1973) is available. For these special classes of systems of conservation laws the simplification of the Glimm scheme gives global existence of solutions of the Cauchy problem with large initial data in , for Bakhvalov's class, and in , in the case of DiPerna's class. It may also happen that the system is in Bakhvalov's class only at a neighboorhood of a constant state, as it was proved for the isentropic gas dynamics by DiPerna (1973), in which case the initial data is taken in with , for some constant which is for the isentropic gas dynamics systems. For periodic initial data, our periodic formulation establishes that the periodic solutions so constructed, , are uniformly bounded in , for all 0$">, where is the period. We then obtain the asymptotic decay of these solutions by applying a theorem of Chen and Frid in (1999) combined with a compactness theorem of DiPerna in (1983). The question about the decay of Nishida's solution was proposed by Glimm and Lax in (1970) and has remained open since then. The classes considered include the -systems with , , , which, for , model isentropic gas dynamics in Lagrangian coordinates.

     

Uniqueness of Solutions to Hyperbolic Balance Laws in Several Space Dimensions

We introduce the concept of modular family of entropy vectors for general r x r systems of balance laws. We then define the notion of entropy solution to the Cauchy problem compatible with the modular family, assuming that the system admits such a family. We show that this concept reduces to the usual one, introduced by S.N. Kruzkov, in the scalar case and when we restrict ourself to Lipschitz continuous solutions. We also show how the compatibility condition appears in the cases of symmetric systems, 2 x 2 psystems and equations of hyperelasticity and electromagnetism, the last two considered earlier by C.M. Dafermos. We demonstrate that generalized Oleinik's condition implies our compatibility condition in the case of symmetric systems. We prove the uniqueness and stability relatively to the initial data of the entropy solutions compatible with the modular family. This theorem has as corollary uniqueness results due to O.A. Oleinik, S.N. Kruzkov, A.E. Hurd, R.J. DiPerna and C. Bardos. We give also two uniqueness theorems to solutions of equations of hyperelasticity and electromagnetism.     

Boundary layers in parabolic perturbations of scalar conservation laws

We consider the problem of estimating the boundary layer thickness for vanishing viscosity solutions of boundary value problems for parabolic perturbations of a scalar conservation law in a space strip in R^d . For the boundary layer thickness () we obtain that one can take ()= ^r, for any r<1/2, arbitrarily close to 1/2.     

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBV _loc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].     

Extended Divergence-Measure Fields and the Euler Equations for Gas Dynamics

 A class of extended vector fields, called extended divergence-measure fields, is analyzed. These fields include vector fields in L ^p and vector-valued Radon measures, whose divergences are Radon measures. Such extended vector fields naturally arise in the study of the behavior of entropy solutions of the Euler equations for gas dynamics and other nonlinear systems of conservation laws. A new notion of normal traces over Lipschitz deformable surfaces is developed under which a generalized Gauss-Green theorem is established even for these extended fields. An explicit formula is obtained to calculate the normal traces over any Lipschitz deformable surface, suitable for applications, by using the neighborhood information of the fields near the surface and the level set function of the Lipschitz deformation surfaces. As an application, we prove the uniqueness and stability of Riemann solutions that may contain vacuum in the class of entropy solutions of the Euler equations for gas dynamics. Received: 7 May 2002 / Accepted: 2 December 2002 Published online: 2 April 2003 Communicated by P. Constantin