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Convex entropy admissibility for a scalar conservation law

Definition

For the scalar case of Strictly hyperbolic conservation systems and characteristic families, use its definition of a weak solution on $Q = (0, \infty) \times R$ . An entropy pair on $G$ is a function $η \in C^{2} (G)$ that is convex on each interval component of $G$ , together with $q \in C^{1} (G)$ satisfying $q^{'} (u) = η^{'} (u) f^{'} (u)$ . A weak solution $u$ is entropy-admissible for this pair when $η \circ u, q \circ u \in L_{l o c}^{1} (Q)$ and $\partial_{t} η (u) + \partial_{x} q (u) \leq 0$ in distributions; equivalently, $\int_{Q} (η (u) φ_{t} + q (u) φ_{x}) d t d x \geq 0$ for every nonnegative $φ \in C_{c}^{\infty} (Q)$ . For a single jump of speed $s$ , the entropy inequality requires $q (u_{R}) - q (u_{L}) - s (η (u_{R}) - η (u_{L})) \leq 0$ .

Depends on

Strictly hyperbolic conservation systems and characteristic families

Used by

A Roe jump property does not imply entropy admissibility Counterexample

Dependency tree · two levels

3 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

Randall J. LeVeque, Numerical Methods for Conservation Laws (standard reference, not scraped)