The Lax Entropy Condirion

1. The Lax Entropy Condirion • We have discussed the entropy condition for the scalar equations: • This means that f’(u) is a monotonic function of u.

2. Genuinely Nonlinear • The characteristics field is said to be genuinely nonlinear if • where

3. Notice that in the scalar case m=1, • This condition reduces to the convexity requirement:

4. This implies The characteristic speed f’(u) is a montonically decreasing or increasing function as u vries. • For a system of equations, the condition means that te characteristics speed • is monotonically increasing or decreasing function of u along an integral curve of the vector field .

5. A straightforward generalization of the scalar entropy condition is the lax entropy condition: • A jump in the p’th field is admissible only if

6. Example • Isothermal gas

7. Example • Suppose are connected by a one shock; then • Since

8. Example • The entropy condition becomes • and this is satisfied if and only if

9. Example • For 2-Shocks the entropy condition is satisfied if and only if

10. Example • Any Hugoniot locus is symmetric to the states • while entropy condition is not. • It is important in entropy conditions to know which state is the left state. • It is preferable to draw the Hugoniot locus to satisfy the entropy condition.

11. Example • States that can be connected to • by an entropy satisfying shock • by an entropy satisfying shock

12. Example • States that can be connected to • by an entropy satisfying shock.

13. Linearly degenerate • The p’th field is called this if • The discontnuity in a linearly degenerate field are called Contact discontinuity (from gas dynamics) • It can be shown that for contact discontinuity

14. So, the entropy condition for both genuinely nonlinear and linearly degenerate fields should be

15. Rarefaction waves and integral curves