= Entropy solution
{wiki}
An <entropy solution> of a <scalar conservation law> is a <weak solution> satisfying additional inequalities that select the admissible continuation through <shock waves>. For a bounded solution of $u_t+f(u)_x=0$ on the real line with $f\in C^1$, require every <Kruzhkov entropy inequality>
$$
\partial_t|u-k|+\partial_x\!\left(\operatorname{sgn}(u-k)[f(u)-f(k)]\right)\leq0,\qquad k\in\mathbb R,
$$
in the sense of <distributions>, together with the <initial condition> as a local $L^1$ trace. Equivalently, the interior integral of $|u-k|\varphi_t+q_k(u)\varphi_x$ is nonnegative for every nonnegative compactly supported <test function> away from $t=0$.
For a smooth <convex function> $\eta$, the corresponding <entropy flux for a scalar conservation law> satisfies $q'=\eta'f'$. A <viscous scalar conservation law> yields $\partial_t\eta(u)+\partial_xq(u)=\varepsilon\partial_{xx}\eta(u)-\varepsilon\eta''(u)u_x^2$; the final term is nonpositive. Suitable bounded strongly convergent <vanishing viscosity approximations> therefore motivate the inequalities. <Convexity> of the flux itself is unnecessary for the <local L1 contraction for scalar conservation laws> and the resulting uniqueness; for strictly convex fluxes, the inequalities exclude expansion jumps and select compressive <entropy shocks>.
Back to article page
