let $u$ and $v$ be two solutions to some PDE with initial data $u_0$ and $v_0$. Is $$|u(t)-v(t)|_{L^1} \leq |u_0-v_0|_{L^1}$$ a contraction principle?
I read that "contraction principle gives comparison principle" but I don't see how. A comparison principle would be: "if $u_0 \geq v_0$ then $u \geq v$".
What would this be called: $$|(u(t)-v(t))^+| \leq |(u_0-v_0)^+|$$?
Basically my question is how to get a comparison principle from the contraction principle at the top?
On
If the PDE conserves the integral over time, i.e. $\int u(t) \,dx = \int u_0 \,dx$ for all $t > 0$, and a technical condition on the domain of valid initial states is satisfied, then this indeed follows from Proposition 1 of the paper "Some relations between nonexpansive and order preserving mappings" of Crandall and Tartar.
I don't know how the inequality $|(u(t)-v(t))^+| \leq |(u_0-v_0)^+|$ could be called, but it certainly implies the comparison principle. Indeed, if $u_0\le v_0$, then the right hand side is zero, hence the left hand side is zero, hence $u(t)\le v(t)$.
Maybe it could be called the comparison-contraction principle.