I have to prove that for a flux function $f$ convex the two entropy conditions are equivalent:
(Lax) Discontinuities at form
$f'(u_l)>s>f'(u_r)$(Oleinik) Discontinuities at form
$\dfrac{f(u)-f(u_l)}{u-u_l}\geq s \geq \dfrac{f(u)-f(u_l)}{u-u_l}\qquad \forall u ~;~ u_l<u<u_r $
I've got two doubts:
1) It's easy to prove that L $\implies$ O if I consider $f$ continuously differentiable, and so $f(y)-f(x)\leq f'(x)(y-x)$. Can I prove this even if $f$ is not continuously differentiable?
2) Taking the limits, O $\implies f'(u_l)\geq s \geq f'(u_r)$… How could I turn the $\geq's$ into $>'s$ to get L?
Here, $s=\dfrac{[f]}{[u]}$ (Rankine-Hugoniot condition).
Many thanks for any help.
Best Answer
I talked with the professor and get:
1) The $f$ is considered continuously differentiable.
2) Is not possible change $\leq$ by $\lt$ (this diference has relation with genuine nonlinearity and linear degeneracy).
Many thanks.