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A local genuinely nonlinear curve generates a centered rarefaction fan

Statement

Let $r_{i}$ be a $C^{1}$ genuinely nonlinear field on a neighborhood, with $D λ_{i} (u) r_{i} (u)$ of constant nonzero sign. If $σ \neq 0$ satisfies $σ D λ_{i} (u_{0}) r_{i} (u_{0}) > 0$ and is sufficiently small, then the left/right data $u_{0}, R_{i} (σ; u_{0})$ are connected by a centered rarefaction fan. It is a self-similar weak solution with those Riemann data.

Facts & Assumptions

Given: A local $C^{1}$ characteristic field, its eigenvalue, and its rarefaction curve.

[F1]

Strictly hyperbolic conservation systems and characteristic families: $D f (u) r_{i} (u) = λ_{i} (u) r_{i} (u)$ , and genuine nonlinearity means $D λ_{i} r_{i}$ has one nonzero sign locally.

[F2]

Local characteristic rarefaction curve: $R_{i}^{'} = r_{i} (R_{i})$ with $R_{i} (0; u_{0}) = u_{0}$ .

Proof

1.1givenF1F2algebra

Along the curve, $\frac{d}{d s} λ_{i} (R_{i} (s; u_{0})) = D λ_{i} (R_{i} (s; u_{0})) r_{i} (R_{i} (s; u_{0}))$ keeps its nonzero sign. Therefore the map $s \mapsto λ_{i} (R_{i} (s; u_{0}))$ is strictly monotone, and for each speed between its endpoint values there is a unique curve parameter $s$ . Since $σ D λ_{i} (u_{0}) r_{i} (u_{0}) > 0$ and the sign remains constant, $λ_{i} (R_{i} (σ; u_{0})) > λ_{i} (u_{0})$ .

2.1step 1.1F1F2constructalgebra

Define $q (ξ) = u_{0}$ for $ξ \leq λ_{i} (u_{0})$ , $q (ξ) = R_{i} (σ; u_{0})$ for $ξ \geq λ_{i} (R_{i} (σ; u_{0}))$ , and inside this interval let $q (ξ) = R_{i} (s; u_{0})$ for the unique $s$ with $ξ = λ_{i} (R_{i} (s; u_{0}))$ . In the fan, differentiating this identity shows $q^{'} (ξ)$ is parallel to $r_{i} (q)$ ; hence $D f (q) q^{'} = λ_{i} (q) q^{'} = ξ q^{'}$ . For $u (t, x) = q (x / t)$ the chain rule gives $u_{t} + D f (u) u_{x} = t^{- 1} (D f (q) - (x / t) I) q^{'} = 0$ .

3.1step 2.1algebra∎

The profile is continuous at the two fan edges, so there is no jump measure there. The fan occupies a bounded $x$ -interval of length proportional to $t$ ; outside it the solution equals the initial left or right state. Hence its local $L^{1}$ trace as $t ↓ 0$ is the Riemann step data, and the classical equation in step 2.1 is also the distributional equation.

Depends on

Used by

Local small-data Riemann solver near a strictly hyperbolic state Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Alberto Bressan, Hyperbolic Conservation Laws: An Illustrated Tutorial (standard reference, not scraped)
Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)