Suppose $\theta=\theta(x,t)\in \mathbb R$ for $x\in\mathbb R^N,t\in[0,T]$ is a classical solution to $$\theta_t + u\cdot \nabla \theta = 0$$ with initial data $\theta_0\in C^\infty_c$, and some incompressible, smooth velocity $u$ (possibly determined from $\theta$), i.e.
$$ \nabla \cdot u = 0.$$
I believed that this equation would preserve all $L^p$ norms, essentially by noticing that the solution $\theta(x,t)$ can be written $\theta_0(X(t,x_0))$ (where $X$ is the particle trajectory map) which is a measure preserving(since incompressible) rearrangement of $\theta_0$, and $L^p$ norms are invariant under rearrangement. I believe you can also prove this by testing the equation against $\theta |\theta|^{p-2}$.
But I've just read two pieces of work in a row only claiming that an inequality holds, i.e. $\|\theta\|_{L^p} \le \|\theta_0\|_{L^p}$ and I'm feeling a little paranoid. Naturally, when computing estimates only one direction is needed, but am I mistaken?