Equilibrium and stability of relativistic stars in extended theories of gravity

We study static, spherically symmetric equilibrium configurations in extended theories of gravity (ETG) following the notation introduced by Capozziello et al. We calculate the differential equations for the stellar structure in such theories in a very generic form i.e., the Tolman–Oppenheimer–Volkoff generalization for any ETG is introduced. Stability analysis is also investigated with special focus on the particular example of scalar–tensor gravity.     

On a nonlinear hyperbolic problem arising in two-phase flows in naturally fractured reservoirs

A system of two first-order quasilinear equations consisting of one nonhomogenous hyperbolic conservation law and an ordinary differential equation is investigated in two spatial dimensions. The initial boundary-value problem is solved for the system and existence, uniqueness, and stability theorems are proved. We also obtain a result on the behavior of the solution when time goes to infinity which agrees with practical experience. These results offer mathematical validation to computer models in current usage for the numerical simulation of multiphase flow in naturally fractured reservoirs.     

Asymptotic Stability of Riemann Solutions for a Class of Multidimensional Systems of Conservation Laws with Viscosity

We prove the asymptotic stability of two-state nonplanar Riemann solutions for a class of multidimensional hyperbolic systems of conservation laws when the initial data are perturbed and viscosity is added. The class considered here is those systems whose flux functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. In particular, we obtain the uniqueness of the self-similar L^∞ entropy solution of the two-state nonplanar Riemann problem. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in L_loc¹ of the space of directions ξ = x/t. That is, the solution u(t, x) of the perturbed problem satisfies u(t, tξ)→R(ξ) as t→∞, in L_loc¹(ℝⁿ), where R(ξ) is the self-similar entropy solution of the corresponding two-state nonplanar Riemann problem.     

Solution to the initial boundary-value problem for the regularized Buckley-Leverett system

We solve the initial boundary-value problem for the regularized Buckley-Leverett system, which describes the flow of two immiscible incompressible fluids through a porous medium. This is the case of the flow of water and oil in an oil reservoir. The system is formed by a hyperbolic equation and an elliptic equation coupled by a vector field which represents the total velocity of the mixture. The regularization is done by means of a filter acting on the velocity field. We consider the critical situation in which we inject pure water into the reservoir. At this critical value for the water saturation, the spatial components of the characteristics of the hyperbolic equation vanish and this motivates the use of a new technique to prove the achievement of the boundary condition for the hyperbolic equation. We treat the case of a horizontal plane reservoir. We also prove that the time averages of the saturation component of the solution converge to one, as the time interval increases indefinitely, for almost all points of the reservoir, with a rate of convergence which depends only on the flux function.     

Uniqueness and Stability of Riemann Solutions¶with Large Oscillation in Gas Dynamics

We prove the uniqueness of Riemann solutions in the class of entropy solutions in with arbitrarily large oscillation for the 3 × 3 system of Euler equations in gas dynamics. The proof for solutions with large oscillation is based on a detailed analysis of the global behavior of shock curves in the phase space and the singularity of centered rarefaction waves near the center in the physical plane. The uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily large perturbation of the Riemann initial data, as long as the corresponding solutions are in L ^∞ 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 needed. The uniqueness result for Riemann solutions can easily be extended to entropy solutions U(x,t), piecewise Lipschitz in x, for any t > 0, with arbitrarily large oscillation. Received: 23 April 2001 / Accepted: 20 September 2001     

An abstract version of the Morse index theorem and its application to hypersurfaces of constant mean curvature

In [5] Smale generalized the Morse index theorem (originally proved by Morse in [3]) to elliptic partial differential systems in several independent variables. Smale's result was used by Simons in [4] to obtain the index theorem for minimal submanifolds. The purpose of this paper is to give an abstract version of the Morse index theorem and use it to prove an index theorem for hypersurfaces of constant mean curvature. This was sugested by Barbosa and do Carmo in [1].     

Nonstrictly hyperbolic systems of conservation laws of the conjugate type

We consider an inverse boundary problem for a general second order self-adjoint elliptic differential operator on a compact differential manifold with boundary. The inverse problem is that of the reconstruction of the manifold and operator via all but finite number of eigenvalues and traces on the boundary of the corresponding eigenfunctions of the operator. We prove that the data determine the manifold and the operator to within the group of the generalized gauge transformations. The proof is based upon a procedure of the reconstruction of a canonical object in the orbit of the group. This object, the canonical Schrödinger operator, is uniquely determined via its incomplete boundary spectral data.     

Decay of Entropy Solutions of Nonlinear Conservation Laws

. (Accepted May 14, 1998)     

Divergence‐Measure Fields and Hyperbolic Conservation Laws

. We analyze a class of vector fields, called divergence‐measure fields. We establish the Gauss‐Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of fields. Then we apply this theory to analyze entropy solutions of initial‐boundary‐value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class of systems with affine characteristic hypersurfaces. The analysis in also extends to . (Accepted July 16, 1998)