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  "body": "\n## Summary\n\nA hyperbolic conservation law is a system of [[conservation laws|8RA7AMWSAF]] whose flux Jacobian, in every direction, has real eigenvalues and a full set of eigenvectors.\nThe eigenvalues are the speeds at which small disturbances travel, so information moves at finite speed along several families of waves.\nThe equations of gas dynamics, shallow water and traffic flow are of this type.\nTheir solutions typically form [[shock waves|JSDMGGTG7G]], and the main results concern weak entropy solutions.\nSee the [[definition|WBYJ7A0HY7]].\n\n## Explanations\n\n### L1 human\n\nIn a gas, a small push creates sound waves that travel left and right at finite speeds.\nA hyperbolic conservation law is a conservation law with this behaviour: every small disturbance splits into waves with real speeds.\nThe condition is that a certain matrix built from the flux has only real eigenvalues and enough eigenvectors.\nFor a single equation $u_t + F(u)_x = 0$ the matrix is the number $F'(u)$, which is always real, so every scalar conservation law is hyperbolic.\n",
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