Leonardo Rendón

Convergence of Approximated Solutions to a Nonstrictly Hyperbolic System

Abstract In this paper, four classes of special entropy-entropy flux pairs of Lax type for the nonstrictly hyperbolic system of type (1.1) (Le Roux system) are constructed based on the solutions of the standard Fuchsion equation. The second derivatives of these entropies are all singular at the point (0,O). A careful computation for these entropies at the singular point shows the compactness of η(u1, v1)t + q(u1, v1)x in H-1loc:(R x R+) with respect to the approximated solutions constructed by using viscosity method or Friedrichs-Lax scheme method. These entropies provide a convergence theorem in the strong topology for the artifical viscosity method or Friedrichs-Lax scheme method when applied to the Cauchy problem (1.1),(1.4) and used together with the theory of compensated compactness.

Global Solutions for a Simplified Shallow Elastic Fluids Model

The Cauchy problem for a simplified shallow elastic fluids model, one system of Temple’s type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth . This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for strictly hyperbolic system and (Heibig, 1994) for strictly hyperbolic system with smooth Riemann invariants.

Delta Shock Wave for the Suliciu Relaxation System

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. This system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issue is that the considered system is such that every characteristic field is linearly degenerate. We show an explicit solution for the Cauchy problem with initial data in . We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established.In this paper we study the one-dimensional Riemann problem for a new hyperbolic system of three conservation laws of Temple class. This systems it is a simplification of a recently propose system of five conservations laws by Bouchut and Boyaval that model viscoelastic fluids. An important issues is that the considered 3 × 3 system is such that every characteristic field is linearly degenerate. Then, in despite of the fact that it is of Temple class, the analysis of the Cauchy problem is more involved since general results for such a systems are not yet available. We show a explicit solution for the Cauchy problem with initial data in L. We also study the Riemann problem for this system. Under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta-shock type solutions are established. On license from Universidad Pedagógica y Tecnológica de Colombia UPTC. e-mail: richard.delacruz@uptc.edu.co or radelacruzg@unal.edu.co † e-mail: jcgalvisa@unal.edu.co ‡ e-mail: jcjuajibioyo@unal.edu.co § e-mail: lrendona@unal.edu.co

Delta shock wave for a 3 × 3 hyperbolic system of conservation laws

We study the one-dimensional Riemann problem for a hyperbolic system of three conservation laws of Temple class. The system is a simplification of a recently proposed system of five conservations laws by Bouchut and Boyaval that models viscoelastic fluids. An important issue is that the considered 3×3 system is such that every characteristic field is linearly degenerate. We study theRiemann problemfor this system and under suitable generalized Rankine-Hugoniot relation and entropy condition, both existence and uniqueness of particular delta shock type solutions are established.

On Global Lipshitz-continuous Solutions of Isentropic Gas Dynamics

Building on work in [Yunguang Lu (1993). The global Hölder continuous solution of isentropic gas dynamics. Proc. Roy. Soc. Edinburgh Sect ., 123A , 231-238.] we consider the Cauchy problem for the isentropic equations of gas dynamics in Eulerian coordinates. For a certain class of initial data we prove existence of Lipshitz-continuous solution for a wide class of the pressure functions.