# Derivation of entropy inequality for scalar conservation laws from viscous equation - discontinuous or infinite integrand

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.

Background

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.

Question

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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