Entropy conditions and Kružkov solutions

For a scalar flux f, a smooth convex function $\eta$ has a compatible entropy flux q determined up to a constant by $q'=\eta'f'$. Smooth inviscid solutions satisfy an entropy equality by the chain rule. Adding positive viscosity produces an extra divergence term and a nonpositive term involving $\eta''u_x^2$. This calculation explains the sign of the entropy inequality in a sufficiently controlled vanishing-viscosity limit; establishing that limit still requires bounds and compactness rather than a formal deletion of terms.

The mathematical entropy is taken convex, so the selected distributional inequality has nonpositive production. A physical thermodynamic entropy may use the opposite concavity or sign convention. At a moving jump, the distributional coefficient is the entropy-flux jump minus shock speed times entropy jump, using the same right-minus-left convention as conservation. This coefficient must be nonpositive. Checking it is different from checking the state-flux Rankine–Hugoniot equality.

For every real constant k, use absolute distance from k as entropy and the signed difference of fluxes as its entropy flux. The family detects how a candidate solution compares with every constant state, not only with zero. A bounded weak solution is a Kružkov entropy solution when all these inequalities hold and it attains its initial datum strongly in local L¹ as time decreases to zero. The sign at equality can be set to zero; its value there does not alter the flux difference.

For the autonomous scalar Cauchy problem on the whole line, a twice continuously differentiable flux and essentially bounded initial data admit a unique bounded entropy solution, represented continuously in time with values in local L¹. Boundedness on each finite time strip and the initial trace are part of the class in which uniqueness is asserted. Neither strict convexity of f nor bounded variation of the initial datum is required for this scalar theorem. Boundary data, spatially discontinuous fluxes, unbounded solution classes, and systems require additional or different theory.

If two bounded scalar entropy solutions have an integrable initial difference, their L¹ distance cannot increase. Individual states need not both have finite L¹ norms: their difference is what must be integrable for the global inequality to be finite. More generally a local contraction estimate enlarges the initial observation interval by the maximal characteristic speed over the relevant bounded state range. This combines admissibility with finite propagation in a way that yields uniqueness for identical initial data.

The proof uses a doubling-of-variables comparison and the whole entropy family. A calculation for one shock or one quadratic entropy illustrates the mechanism but does not replace that proof for arbitrary bounded data. A numerical entropy diagnostic is useful for the same reason: a violated necessary inequality exposes an error, while passing one scalar test does not establish every hypothesis of a convergence or uniqueness theorem. Keep the solution class, entropy family, initial trace, and comparison norm visible when using the result.