scholarly journals Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

2006 ◽  
Vol 230 (2) ◽  
pp. 465-480
Author(s):  
Hermano Frid ◽  
Leonardo Rendón
Keyword(s):  
Asymptotic Stability ◽  
Conservation Laws ◽  
Multidimensional Systems ◽  
Riemann Solutions ◽  
Systems Of Conservation Laws

ASYMPTOTIC STABILITY OF NON-PLANAR RIEMANN SOLUTIONS FOR MULTI-D SYSTEMS OF CONSERVATION LAWS WITH SYMMETRIC NONLINEARITIES

2004 ◽  
Vol 01 (03) ◽  
pp. 567-579 ◽  
Author(s):  
HERMANO FRID
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Asymptotic Stability ◽  
Conservation Laws ◽  
Entropy Solution ◽  
Smooth Functions ◽  
Riemann Solutions ◽  
The Cauchy Problem ◽  
Systems Of Conservation Laws ◽  
Self Similar

We study the asymptotic behavior of entropy solutions of the Cauchy problem for multi-dimensional systems of conservation laws of the form [Formula: see text], where the gα are real smooth functions defined in [0,+∞), and when the initial data are perturbations of two-state nonplanar Riemann data. Specifically, if R0(x) is such Riemann data and ψ∈L∞(ℝd)n satisfies ψ(Tx)→0 in [Formula: see text], as T→∞, then an entropy solution, u(x,t), of the Cauchy problem with u(x,0)=R0(x)+ψ(x) satisfies u(ξt,t)→R(ξ) in [Formula: see text], as t→∞, where R(x/t) turns out to be the unique self-similar entropy solution of the corresponding Riemann problem.

LARGE VISCOUS SOLUTIONS FOR SMALL DATA IN SYSTEMS OF CONSERVATION LAWS THAT CHANGE TYPE

2008 ◽  
Vol 05 (02) ◽  
pp. 257-278 ◽  
Author(s):  
VÍTOR MATOS ◽  
DAN MARCHESIN
Keyword(s):  
Conservation Laws ◽  
Small Amplitude ◽  
Elliptic Boundary ◽  
High Amplitude ◽  
Small Data ◽  
Riemann Solutions ◽  
Type Iv ◽  
Systems Of Conservation Laws ◽  
Viscous Solutions ◽  
Special Singularity

We study a quadratic system of conservation laws with an elliptic region. The second order terms in the fluxes correspond to type IV in Schaeffer and Shearer classification. There exists a special singularity for the EDOs associated to traveling waves for shocks. In our case, this singularity lies on the elliptic boundary. We prove that high amplitude Riemann solutions arise from Riemann data with arbitrarily small amplitude in the hyperbolic region near the special singularity. For such Riemann data there is no small amplitude solution. This behavior is related to the bifurcation of one of the codimension-3 nilpotent singularities of planar ODEs studied by Dumortier, Roussarie and Sotomaior.

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