I have two questions related to the continuity equation.
(1) In fluid mechanics, we have the continuity equation
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0$$
I am interested in deriving an expression for the squared norm of $\rho v$:
$$\left\lVert \rho(t, x) v(t, x) \right\lVert^{2} = \cdots$$
Can any one help me derive the expression on the right-hand side?
(2) The Fisher information of the density is given by:
$$\int \left\lVert \nabla \log \rho(t, x) \right\lVert^{2} \rho(t, x) \mathrm{d}x$$
What is the gradient with respect to? Is it w.r.t. x?
Thank you!
For (1), you need a equation of $v(t,x)$ as: $$ \frac{\partial v}{\partial t}=F(v,p,\rho) $$
Then you can combine the equations of $v$ and $\rho$ as: $$ v\frac{\partial v}{\partial t}+\rho\frac{\partial v}{\partial t}=\frac{\partial \rho v}{\partial t}=... $$
Finally, you get: $$ \rho v \frac{\partial \rho v}{\partial t}=\frac{\partial}{\partial t}\frac{(\rho v)^2}{2}=... $$
which is the equation of the norm you are seeking for.
For (2), I guess it is w.r.t. x (space).