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$ϵ\to 0$ 并不是$ϵ=0$，否则不会有...if $u$ is smooth at $x_1$ and $x_2$...。在粘度趋向于很小的时候，不连续变成具备一定厚度的光滑解，同样承认有厚度的激波。所以
$$ϵ{\int }_{\mathrm{\Omega }}\frac{\mathrm{\partial }}{\mathrm{\partial }u}\left(\frac{\mathrm{\partial }\eta }{\mathrm{\partial }u}\right){\left(\frac{\mathrm{\partial }u}{\mathrm{\partial }x}\right)}^2\mathrm{d}x\mathrm{d}t\ge 0$$
另外，
$$\begin{array}{}\text{(1)}& {\int }_{x_2}^{x_1}(ϵ({\eta }_q{q}_x{)}_x\mathrm{d}x-ϵ\eta ({\eta }_q{)}_q{q}_x^2)\mathrm{d}x=ϵ({\eta }_q{q}_x{|}_{x=x_1}-{\eta }_q{q}_x{|}_{x=x_2})-ϵ\eta ({\eta }_q{)}_q{q}_x^2\mathrm{\Delta }x\end{array}$$
考虑一个非常小的$ϵ=1e-10$，在控制体内$ϵ({\eta }_q{q}_x{|}_{x=x_1}-{\eta }_q{q}_x{|}_{x=x_2})\to 0$，$ϵ\eta ({\eta }_q{)}_q{q}_x^2\mathrm{\Delta }x$还是大于0.