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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!

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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).

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  • $\begingroup$ In the continuity equation, there is a divergence operator. Can't we simply express the term $(\rho v)$ in terms of (1) the partial derivative $\partial \rho / \partial t$ and (2) the divergence operator? $\endgroup$ Commented Apr 26, 2022 at 5:38
  • $\begingroup$ If you want the tendency of the norm of $\rho v$, you definite need the tendency of $v$, as the divergence operator is only about the spatial variations. If you just want an expression of the norm, I am a little confused about what you are expecting on the rhs... $\endgroup$
    – MiniUFO
    Commented Apr 27, 2022 at 9:30

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