The Riemann problem for a resonant nonlinear system of conservation laws with Dirac-measure solutions

1998 ◽  
Vol 128 (1) ◽  
pp. 81-94 ◽  
Author(s):  
Jiaxin Hu
Keyword(s):  
Nonlinear System ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Additional Condition ◽  
Dirac Measure ◽  
Viscosity Approximation ◽  
Riemann Solution ◽  
System Of Conservation Laws ◽  
Genuine Nonlinearity ◽  
Contact Wave

The Riemann problem for a resonant nonlinear system of conservation laws is considered here. The Riemann solution was constructed by employing the viscosity approximation approach. One kind of new discontinuity, which is called the Dirac-contact wave, appeared in the Riemann solution. Because the strict hyperbolicity as well as the genuine nonlinearity of the system considered failed, the solution we obtained in this paper is not unique for some initial data. An additional condition was explored to guarantee the uniqueness of the Riemann problem.

The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity

1995 ◽  
Vol 125 (4) ◽  
pp. 675-699 ◽  
Author(s):  
Michael Shearer ◽  
Yadong Yang
Keyword(s):  
Shock Waves ◽  
Conservation Laws ◽  
Riemann Problem ◽  
Mixed Type ◽  
Vector Fields ◽  
Heteroclinic Orbits ◽  
Cubic Nonlinearity ◽  
System Of Conservation Laws ◽  
Admissible Solutions ◽  
Image Position

Using the viscosity-capillarity admissibility criterion for shock waves, we solve the Riemann problem for the system of conservation lawswhere σ(u) = u3 − u. This system is hyperbolic at (u, v) unless . We find that the Riemann problem has a unique solution for all data in the hyperbolic regions, except for a range of data in the same phase (i.e. on the same side of the nonhyperbolic strip). In the nonunique cases, there are exactly two admissible solutions. The analysis is based upon a formula describing all saddle-to-saddle heteroclinic orbits for a family of cubic vector fields in the plane.

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