Riemann Problem for Conservation Laws with an Umbilic Point

2003 ◽  
pp. 315-323
Author(s):  
Fumioki Asakura ◽  
Mitsuru Yamazaki
Keyword(s):  
Conservation Laws ◽  
Riemann Problem ◽  
Umbilic Point

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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