Shock Waves and Conservation Laws

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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Pointwise convergence to shock waves for viscous conservation laws

Communications on Pure and Applied Mathematics ◽

10.1002/(sici)1097-0312(199711)50:11<1113::aid-cpa3>3.0.co;2-d ◽

1997 ◽

Vol 50 (11) ◽

pp. 1113-1182 ◽

Cited By ~ 73

Author(s):

Tai-Ping Liu

Keyword(s):

Shock Waves ◽

Conservation Laws ◽

Pointwise Convergence ◽

Viscous Conservation Laws

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Delta shock waves with Dirac delta function in both components for systems of conservation laws

Journal of Differential Equations ◽

10.1016/j.jde.2014.08.009 ◽

2014 ◽

Vol 257 (12) ◽

pp. 4369-4402 ◽

Cited By ~ 35

Author(s):

Hanchun Yang ◽

Yanyan Zhang

Keyword(s):

Shock Waves ◽

Conservation Laws ◽

Delta Function ◽

Dirac Delta Function ◽

Dirac Delta ◽

Delta Shock ◽

Systems Of Conservation Laws ◽

Delta Shock Waves

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Symmetries of conservation laws

Publications de l Institut Mathematique ◽

10.2298/pim0591029k ◽

2005 ◽

Vol 77 (91) ◽

pp. 29-51

Author(s):

Sanja Konjik

Keyword(s):

Shock Waves ◽

Conservation Laws ◽

Gas Dynamics ◽

Hyperbolic Conservation Laws ◽

Strong Association ◽

Group Analysis ◽

Symmetry Groups ◽

Dynamics Model ◽

Hyperbolic Conservation ◽

Systems Of Conservation Laws

We apply techniques of symmetry group analysis in solving two systems of conservation laws: a model of two strictly hyperbolic conservation laws and a zero pressure gas dynamics model, which both have no global solution, but whose solution consists of singular shock waves. We show that these shock waves are solutions in the sense of 1-strong association. Also, we compute all project able symmetry groups and show that they are 1-strongly associated, hence transform existing solutions in the sense of 1-strong association into other solutions.

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