Global existence of solutions to resonant system of isentropic gas dynamics

Hyperbolic Approximation on System of Elasticity in Lagrangian Coordinates

“…Note 1. The idea to use the flux perturbation coupled with the vanishing viscosity was well applied by the author in [24] to control the super-line, source terms and to obtain the L ∞ estimate for the nonhomogeneous system of isentropic gas dynamics.…”

Existencia global y estabilidad para un sistema de dinámica de gases isotérmico con una fuerza externa

“…are the Riemann invariants of (4) and M > 1 is a constant. Then, for fixed ϵ, δ, the Cauchy problem (6) and 7has a global solution…”

“…To construct the approximate solutions of the Cauchy problem (1)-(2), besides adding the classical viscosity parameter ϵ to the right-hand side of (1), we approximate the flux by adding a term depending on another parameter δ > 0 which vanishes when δ = 0. An obvious advantage of this kind of approximation on the flux functions is to obtain the positive lower bound ρ ≥ 2δ > 0, directly from the first equation in (6), which grantees that the term ρu 2 = m 2 ρ is well defined. Moreover, for any weak entropy-entropy flux pair (η(ρ, m), q(ρ, m)) of system (4), we can easily prove that…”

Global Existence of Solutions for a Nonstrictly Hyperbolic System

“…Without the condition 0 ( ) ≤ 0 or 0 ( ) ≤ 0, the reasonable estimate, depending on the variable : ( , ) ≤ −(2( −1)/( +1)) , > 0 was first obtained in [2] for system (1) when ( ) = , 1 < < 5/3 by using a modified Godunov scheme, and in [14] for general pressure function ( ) and smooth function ( ) by using the compensated compactness.…”