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Riemann data $u_{L} = 3$ , $u_{R} = 1$ : shock speed and entropy coefficients, and a rarefaction check

Example

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the cited Lebesgue weak/entropy formulation.

Take $u_{L} = 3 > u_{R} = 1$ in the shock of Riemann shock and rarefaction waves for $u_{t} + (u^{2} / 2)_{x} = 0$ are weak solutions, and the shock is an entropy solution. The shock speed is $s = \frac{u_{L} + u_{R}}{2} = 2$ . For the entropy pair $(η_{k}, q_{k})$ :

for $k = 2$ , which lies strictly between $u_{R} = 1$ and $u_{L} = 3$ , the interface coefficient is $s [η_{2}] - [q_{2}] = 1$ ;
for $k = 0$ , which lies outside the interval $[1, 3]$ , the coefficient is $s [η_{0}] - [q_{0}] = 0$ ;

both nonnegative, in accordance with the theorem's closed formula $s [η_{k}] - [q_{k}] = - (k - u_{L}) (k - u_{R})$ in the interior case and $= 0$ in the exterior case.

For the rarefaction with $u_{L} = 1 < u_{R} = 3$ , the fan $u = x / t$ satisfies $u_{t} + u u_{x} = 0$ identically, and the entropy residuals vanish because the fan is continuous.

Facts & Assumptions

Given: Countable choice. The Riemann data above and the theorem of Riemann shock and rarefaction waves for $u_{t} + (u^{2} / 2)_{x} = 0$ are weak solutions, and the shock is an entropy solution.

[L1]

For the shock $u_{L} > u_{R}$ : $u$ is a weak solution with $s = (u_{L} + u_{R}) / 2$ , and $s [η_{k}] - [q_{k}] = - (k - u_{L}) (k - u_{R})$ when $u_{R} < k < u_{L}$ , while $s [η_{k}] - [q_{k}] = 0$ when $k \geq u_{L}$ or $k \leq u_{R}$ (Riemann shock and rarefaction waves for $u_{t} + (u^{2} / 2)_{x} = 0$ are weak solutions, and the shock is an entropy solution, Weak and entropy solutions of the scalar conservation law $u_{t} + (u^{2} / 2)_{x} = 0$ ).

[L2]

For the rarefaction $u_{L} < u_{R}$ , the fan branch $u (t, x) = x / t$ has $u_{t} = - x / t^{2}$ and $u_{x} = 1 / t$ (Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, $C^{k}$ Euclidean maps and diffeomorphisms, Weak and entropy solutions of the scalar conservation law $u_{t} + (u^{2} / 2)_{x} = 0$ ).

Verification

technique · direct; substitute the explicit data into the closed formulas

1.1givenL1algebra

For $k = 2$ : $[η_{2}] = ∣ u_{R} - 2 ∣ - ∣ u_{L} - 2 ∣ = 1 - 1 = 0$ , and $[q_{2}] = sgn (u_{R} - 2) \frac{u_{R}^{2} - 4}{2} - sgn (u_{L} - 2) \frac{u_{L}^{2} - 4}{2} = (- 1) \frac{1 - 4}{2} - (1) \frac{9 - 4}{2} = \frac{3}{2} - \frac{5}{2} = - 1$ . Hence $s [η_{2}] - [q_{2}] = 0 - (- 1) = 1 = - (2 - 3) (2 - 1)$ , the interior formula.

1.2givenL1algebra

For $k = 0$ : $[η_{0}] = 1 - 3 = - 2$ and $[q_{0}] = \frac{1}{2} - \frac{9}{2} = - 4$ , so $s [η_{0}] - [q_{0}] = 2 (- 2) - (- 4) = 0$ , the exterior value; both computed coefficients are $\geq 0$ as the theorem requires.

1.3givenL2algebra

For the rarefaction at $t = 1$ and $1 < x < 3$ : $u = x$ , and by [L2] with $t = 1$ , $u_{t} = - x$ , $u_{x} = 1$ , so $u_{t} + u u_{x} = - x + x = 0$ .

2.1step 1.1step 1.2step 1.3L1L2algebra∎

The computations instantiate the theorem's weak-solution and entropy identities at explicit data: the interior entropy coefficient is positive, the exterior one vanishes, and the rarefaction branch satisfies the equation pointwise.

Remarks

This is a finite check of the closed formulas; it does not test the general weak formulation beyond the stated Riemann profiles, and the numerical values $3, 1, 2, 0$ are chosen only to exhibit both the interior and the exterior case of the entropy coefficient.

Riemann data $u_{L} = 3$ , $u_{R} = 1$ : shock speed and entropy coefficients, and a rarefaction check

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Riemann data uL=3uL​=3, uR=1uR​=1: shock speed and entropy coefficients, and a rarefaction check

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources

Riemann data $u_{L} = 3$ , $u_{R} = 1$ : shock speed and entropy coefficients, and a rarefaction check