Consider the claim: $$\frac{u(x+hv,t+h)-u(x,t)}{h}\leq L(v)\implies v\cdot Du(x,t)+u_t(x,t)\leq L(v)\text{ when }h\to0^{+}.$$
As far as I can recall, $$v\cdot Du(x,t) = \lim_{h\to 0}\frac{u(x+hv,t+hv)-u(x,t)}{h}$$ where $Du(x,t)$ is the gradient of $u(x,t).$ Using this and the standard definition for $u_t$ I am unable to deduce this inequality. Perhaps someone can explain?