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Exercise 2. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function and assume that $u(t, x)$ solves the conservation law $\partial_t u + \partial_x (f(u)) = 0$, or equivalently $\partial_t u + f'(u)\partial_x u = 0$, on the strip \{$(t, x) : 0 \le t < t_{max}$\} in the space-time. Show that $u$ is constant along suitable segments in the space-time; namely, for every $\lambda \in \mathbb{R}$, show that there exists $\mu \in \mathbb{R}$ (depending on $\lambda$) such that $u(s, \lambda + \mu s)$ is constant for $0 \le s < t_{max}$. Hint: use the method of characteristics and use the equation to show that $u$ is constant along each trajectory $\gamma(s)$; deduce that the speed $\gamma'$ is constant.

          Exercise 2. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a smooth function and assume that $u(t, x)$ solves the conservation law
$\partial_t u + \partial_x (f(u)) = 0$,
or equivalently
$\partial_t u + f'(u)\partial_x u = 0$,
on the strip \{$(t, x) : 0 \le t < t_{max}$\} in the space-time. Show that $u$ is constant along suitable
segments in the space-time; namely, for every $\lambda \in \mathbb{R}$, show that there exists $\mu \in \mathbb{R}$ (depending
on $\lambda$) such that $u(s, \lambda + \mu s)$ is constant for $0 \le s < t_{max}$.
Hint: use the method of characteristics and use the equation to show that $u$ is constant along
each trajectory $\gamma(s)$; deduce that the speed $\gamma'$ is constant.

Exercise 2. Let f: ℝ→ℝ be a smooth function and assume that u(t, x) solves the conservation law
u + (f(u)) = 0,
or equivalently
u + f'(u) u = 0,
on the strip {(t, x) : 0 ≤ t < tmax} in the space-time. Show that u is constant along suitable
segments in the space-time; namely, for every λ∈ℝ, show that there exists μ∈ℝ (depending
on λ) such that u(s, λ + μ s) is constant for 0 ≤ s < tmax.
Hint: use the method of characteristics and use the equation to show that u is constant along
each trajectory γ(s); deduce that the speed γ' is constant.

Added by Adriana G.

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