# Show that a solution of $u_t+(|u|^\alpha)_x=0$ violates entropy condition

Consider $$u_t+(|u|^\alpha)_x=0, \quad\alpha>1$$ Given the initial condition $$u(x,0)=\begin{cases} 0, x<0\\1,x>0\end{cases}$$

a) Find a solution for $$u(x, t)$$ that is continuous for all $$t > 0$$ and satisﬁes the initial condition.

b) Find a solution for $$u(x, t)$$ that includes a shock obeying the appropriate jump condition.

(c) Show that one of these solutions does not satisfy the entropy condition.

My attempt:

We can rewrite the equation $$u_t+(|u|^\alpha)_x=u_t+\alpha u |u|^{\alpha-2}u_x$$ then we parametrize \begin{align} x(0,r) &= r \\ t(0,r) &= 1 \\ u(0,r) &= u_0 \end{align}

The characteristics satisfying IVP are

\begin{align} x_s &= \alpha u |u|^{\alpha-2} \\ \implies x &= \alpha u |u|^{\alpha-2}t+r \\ t_s &= 1 \\ \implies t &= s \\ u_s &=0 \\ \implies u &= u_0 \end{align}

The projection on $$(x,t)$$-plane is given by $$t = \frac{x-r}{\alpha u |u|^{\alpha-2}}$$ So if $$x<0$$, the denominator is $$0$$. How do I find the solution?

The characteristics are the curves $$x=x_0+\alpha u|u|^{\alpha-2}t$$ along which $$u=u(x_0,0)=F(x_0)$$ is constant. With the given initial data, we have u(x,t) = \left\lbrace \begin{aligned} &0 &&\text{if}\quad x <0\\ &U (x/t) &&\text{if}\quad 0 \leq x \leq \alpha t\\ &1 &&\text{if}\quad \alpha t where $$U (x/t)$$ is a rarefaction wave solution. The latter is obtained from assuming a smooth solution of the form $$u (x,t) = U (\xi)$$ with $$\xi=x/t$$. Using this self-similarity Ansatz and the PDE, one obtains $$U (x/t) = \left.\left (\xi\to\alpha \xi|\xi|^{\alpha-2}\right)^{-1}\right|_{\xi=x/t} = \left(\frac{x}{\alpha t}\right)^\frac {1}{\alpha-1} .$$ Concerning the shock wave solution u(x,t) = \left\lbrace \begin{aligned} &0 &&\text{if}\quad x with $$s = \frac{|1|^\alpha - |0|^\alpha}{1 - 0} = 1$$ following from the Rankine-Hugoniot condition, one shows that this solution does not satisfy the Lax entropy condition since $$\alpha\, 0\, |0|^{\alpha-2} < s < \alpha\, 1\, |1|^{\alpha-2} .$$ Indeed, the Lax entropy condition requires opposite inequalities. Finally, the entropy solution is the rarefaction wave.
• How can we find s, the jump condition gives us $s=\frac{|u^-|^\alpha-|u^+|^\alpha}{u^- - u^+}$, and how can we show that the shockwave solution is a weak solution? I know the formula I do not see how that condition is easy to check – dxdydz Dec 9 '18 at 16:26