📂Partial Differential Equations

Rankine-Hugoniot Condition and Entropy Condition

Definition

$$ \begin{cases} u_{t} + u u_{x} = 0 & , t>0 \\ u(t,x) = f(x) & , t=0 \end{cases} $$

Let’s say the solution of the given inviscid Burgers’ equation is $u$, and its rupture time is $t_{\ast}$.

When the solution of the inviscid Burgers’ equation ruptures, it is connected by a line segment so that the areas to the left and right become equal, as shown above. This adjustment of the solution to be physically interpretable is called the equal area rule.

Explanation

If the position of the discontinuity is $\sigma (t)$, then

$$ \begin{cases} \displaystyle u^{+} (t) := \lim_{x \to \sigma (t)^{+}} u(t,x) \\ \displaystyle u^{-} (t) := \lim_{x \to \sigma (t)^{-}} u(t,x)\end{cases} $$

and it satisfies the following conditions:

1. Rankine-Hugoniot condition: $\displaystyle \sigma ' (t) = {{d} \over {dt}} \sigma (t) = {{u^{+}(t) + u^{-}(t) } \over {2}}$
2. Entropy condition: $u^{+} (t) \le \sigma '(t) \le u^{-} (t)$

To elaborate, 1. means that the travel time of the rupture position is represented by the average of the left and right limits.

2. seems obvious because, if it were not for $u^{+} (t) \le u^{-} (t)$, the rupture would not have occurred in the first place. 1. is a necessary and sufficient condition for $u$ to be the solution; if the solution does not satisfy this condition, it can be confirmed that it was incorrectly derived.