# Riemann problem - Are two shocks possible?

Given the hyperbolic conservation law \begin{align*} u_t + f(u)_x = 0, \; x \in \mathbb{R}, t > 0 \\ u(x,0) = u_0(x), \; x \in \mathbb{R} \end{align*} with $$f : \mathbb{R} \to \mathbb{R}$$, $$f''(u) > 0$$ for all $$u \in \mathbb{R}$$ and the initial condition $$$$u_0(x) = \begin{cases} u_l &, x < 0 \\ u_r &, x \geq 0 \end{cases}$$$$ for $$u_l, u_r \in \mathbb{R}$$. Is it possible that there exist's a weak solution $$u$$ which is piecewise constant and has exactly two shocks?

So far what I got is:
Assume $$u \in L_{loc}(\mathbb{R} \times \mathbb{R}^+)$$ is a weak solution of the riemann problem with two shocks $$\psi_1$$ and $$\psi_2$$. Since $$u$$ has two shocks, it has 2 formations of discontinuity and therefore under the assumption that $$u$$ is piecewise constant
$$$$u(x,t) = \begin{cases} u_l &, x < at \\ u^* &, at \leq x \leq bt \\ u_r &, x > bt \end{cases}$$$$ for some $$a,b \in \mathbb{R}$$ with $$b > a$$. Now $$u$$ is a weak solution of the hyperbolic conservation law if $$$$\int^{\infty}_{0}{\int_{\mathbb{R}}{u \Phi_t + f(u) \Phi_x dx dt}} = -\int_{\mathbb{R}}{u_0(x) \Phi(x,0) dx} \mbox{ for all } \Phi \in C^1_0(\mathbb{R}^2).$$$$ By partial integration I got \begin{align} \int^{\infty}_{0}{\int_{\mathbb{R}}{u \Phi_t dx dt}} = &-\int_{\mathbb{R}}{u(x,0) \Phi(x,0) dx} \\ &- a(u_l - u^*)\int^{\infty}_{0}{\Phi(at, t) dt} \\ &- b(u^* - u_r)\int^{\infty}_{0}{\Phi(bt, t) dt} \end{align} and $$$$\int^{\infty}_{0}{\int_{\mathbb{R}}{f(u) \Phi_x dx dt}} = \Big(f(u_l) - f(u^*)\Big) \int^{\infty}_{0}{\Phi(at, t) dt} + \Big(f(u^*) - f(u_r) \Big)\int^{\infty}_{0}{\Phi(bt, t) dt}.$$$$ Now if $$\psi_1'(t) = \frac{f(u_l) - f(u^*)}{u_l - u^*}$$ and $$\psi_2'(t) = \frac{f(u^*) - f(u_r)}{u^* - u_r}$$ are the shock velocities there can be exactly two shocks and $$u$$ is a piecewise constant weak solution of the riemann problem. I am not sure if this is right.I read that there cannot be exactly two shocks for a piecewise constant function, but I can't see why.

In your case for a strictly convex flux $$f''(u) > 0$$, however, you cannot have two shocks. For a more general $$f$$, it is possible. I argue below why that is.
By the method of characteristics, we look for a curve $$x(t)$$ along which the solution of the PDE stays constant: \begin{align} \frac{\mathrm d}{\mathrm dt} u\big(x(t), t \big) &\overset{!}{=}0 \\ \overset{\text{Chain rule}}{\Rightarrow}\partial_t u + \big(\partial_x u\big) x'(t) &\overset{!}{=}0 \end{align} Now compare this to your PDE $$\Big($$under the assumption that $$f = f(u) \neq f(x) \Big)$$: $$\partial_u + f(u)_x = \partial_u + f'(u)u_x \overset{!}{=}0$$ Thus, the ODE for the characteristic reads: $$x'(t) = f'(u) = f'\Big(u\big(x(t), t\big) \Big)$$ The assumption that $$u$$ stays constant along characteristics $$x(t)$$ implies now that $$u\big(x(t), t \big)$$ is exactly the same as at the initial time $$t = t_0 = 0$$: $$u\big(x(t), t \big) = u\big(x(0), 0\big) = u_0\big(x(0)\big).$$ As a consequence, the ODE for the characteristic is trivial: \begin{align} x'(t) &= f'\Big(u_0\big(x(0)\big) \Big) = \text{const} \\ \Rightarrow x(t) &= f'\Big(u_0\big(x(0) \big) \Big) \cdot t + x(0)\end{align}
Consquently, you have exactly two types of characterists: One with $$x(0) < 0$$ have slope $$f'(u_L)$$ and the other starting at $$x(0) > 0$$ with slope $$f'(u_R)$$.
Now, if $$u_L > u_R$$, you have a single shock with speed given by the Rankine-Hugoniot condition. In the opposite case, $$u_R > u_L$$ the entropy conditions dictate a rarefaction solution: $$u(x, t) = \begin{cases} u_L & x \leq f'(u_L) t \\ [f']^{-1}(x / t) & f'(u_L) t < x < f'(u_R) \\ u_R & x \geq f'(u_R) \end{cases}$$ Note how the strictly convex flux $$f''(u) > 0 \: \forall \: u \: \in \mathbb{R}$$ ensures that $$f'(u)$$ is monotonically increasing, thus invertible.
In the case of a non strictly convex / concave flux, you have to get your hands dirty and find a way to "join" the states $$u_L, u_R$$ in an entropy-admissible way. Then you can also have two shocks with a rarefaction in between, see for instance this notes, page 27.