By means of the comparisons with the formulas in statistical mechanics and thermodynamics, in this paper it is demonstrated that for the single conservation law $\partial_{t} u + \partial_{x} f(u) = 0$, if the flux function $f (u)$ is convex (or concave), then, the physical entropy is $S = -f (u)$; Furthermore, if we assume this result can be generalized to any $f (u)$ with two order continuous derivative, from the thermodynamical principle that the local entropy production must be non-negative, one entropy inequality is derived, by which the O.A. Olejnik's famous E-condition can be explained successfully in physics.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9173.html} }