The classical heat equation read $\partial_tu-\Delta u=0$ in $\Bbb R^3$. If $u$ is sufficintly smooth, then we can multiply both sides by $|u|^{p-2}u$ to get the $L^p$ estimate, where $1<p<\infty$. That is, $$\frac{1}{p}\frac{d}{dt}||u||_{L^p}^p+c(p)||\nabla |u|^\frac{p}{2}||_{L^2}^2=0.$$

What about the $L^1$ case? I have not seen this in a book or paper.

Maybe we can use the heat kernel to do such one. But can we just do $L^1$ estimate on the solution. In fact, when considering axiysmmetric Navier-Stokes equations, the swirl component of the voriticity divides by $r$: $\Omega=\omega/r$ satisfies $$\partial_t\Omega+(u^r\partial_r+u^z\partial_z)\Omega-(\Delta+\frac{2}{r}\partial_r)\Omega=f$$ for some $f$. In this case, how can we do $L^1$ estimate?