I am self-studying Numerical Methods for Conservation Laws by LeVeque. I have a question about the derivation of the entropy inequality for the convex scalar conservation law $$ u_t+f(u)_x=0\tag{1} $$ which allows us to select unique weak solutions. I am sure that my question has been answered somewhere on stack exchange, but I haven't been able to express the mathematical verbiage that would allow me to find it.


Let $\eta(u)$ be our "entropy" function which satisfies a conservation law of the form

$$ \eta(u)_t+\psi(u)_x=0 $$

where $\psi(u)$ is some "entropy flux" function. Assume that both $\eta(u)$ and $\psi(u)$ are convex. Multiplying (1) by $\eta'(u)$, we have $$ \eta'(u)u_t+\eta'(u)f'(u)u_x=0 $$ from which we infer $\psi'(u)=\eta'(u)f'(u)$. Since we care about weak solutions admitting initial data with discontinuities in $u$, we consider the viscous equation for (1) as $\epsilon \to 0$. $$ u_t+f(u)_x=\epsilon u_{xx} $$ From which we can write $$ \eta(u)_t+\psi(u)_x=\epsilon\eta'(u)u_{xx} $$ Manipulating the right-hand side, we have $$ \eta(u)_t+\psi(u)_x=\epsilon(\eta'(u)u_x)_x-\epsilon \eta''(u)u_x^2 $$ Now, consider a rectangle $[x_1,x_2] \times [t_1,t_2]$ where $u(x,t)$ has some discontinuity in the limiting solution as $\epsilon \to 0$. $$ \int_{t_1}^{t_2}\int_{x_1}^{x_2} \eta(u)_t+\psi(u)_x dxdt = \epsilon \int_{t_1}^{t_2}\int_{x_1}^{x_2} (\eta'(u)u_x)_x dxdt -\epsilon \int_{t_1}^{t_2}\int_{x_1}^{x_2} \eta''(u)u_x^2 dxdt $$ LeVeque writes

As $\epsilon \to 0$, the first term on the right hand side vanishes. (This is clearly true if $u$ is smooth at $x_1$ and $x_2$, and can be shown more generally.) The other term, however, involves integrating $u_x^2$ over $[x_1,x_2] \times [t_1,t_2]$. If the limiting weak solution is discontinuous along a curve in this rectangle, then this term will not vanish in the limit.

The entropy condition for weak solutions $$ \eta(u)_t+\psi(u)_x\leq 0 $$ follows.


Can you help me understand why $$ \lim_{\epsilon\to 0}\left[\epsilon \int_{t_1}^{t_2}\int_{x_1}^{x_2} (\eta'(u)u_x)_x dxdt\right]=0 $$ but $$ \lim_{\epsilon\to 0}\left[\epsilon \int_{t_1}^{t_2}\int_{x_1}^{x_2} \eta''(u)u_x^2 dxdt\right]\neq 0 $$




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