Find conserved quantities with respect to parameters in PDE

Question

I have a standard, steady-state convection-diffusion PDE given as: $$ -p_1 \nabla^2 c(x,y) + \vec{v} \cdot \vec{\nabla}c(x,y) = f(x,y) $$ With $\vec{v} := \frac{p_2}{\sqrt{2}}\langle 1,1\rangle$. Here $p_1$ and $p_2$ are parameters each on their own domain (i.e. $p_1 \in [a,b]$ and $p_2 \in [c,d]$, for $a,b,c,d \in \mathbb{R}$). Given the solution is known, is there a way to define a functional on the solution, $F[c(x,y;p_1,p_2),x,y]$ such that $$ \frac{\partial F}{\partial p_1} = 0 \:\:\: \text{but}\:\:\: \frac{\partial F}{\partial p_2} \neq 0 \:\:\: \text{and vice versa} $$ This boils down to finding conserved quantities on $c(x,y)$ with respect to my parameters. I figured if this is possible this might use some form of a variational principle, but I can't quite figure out how that would be implemented. On the other hand, if this is not possible, could someone explain why?

Answer 1

I don't know of such a functional but perhaps this can be of help on the way. To get a conservation law, first consider: $$-p_1 \nabla^2 c + \vec{v} \cdot \nabla c =0$$ Now, assuming $p_1$, $p_2$, and $\vec{v}$ are constants we can rewrite the equation as :

$$- \nabla \cdot \nabla (p_1 c) + \nabla \cdot (\vec{v} c) = 0$$

This is because: $$ \nabla \cdot (\vec{v} c) = c \nabla \cdot \vec{v}+\vec{v} \cdot \nabla c $$

and since $\vec{v}$ is constant we have: $$\nabla \cdot \vec{v}=0$$

Collecting terms we get: $$ \nabla \cdot (-p_1 \nabla c + \vec{v} c) = 0$$

We can now integrate over a control volume and get:

$$\int_\Omega \nabla \cdot (-p_1 \nabla c + \vec{v} c) dV= 0$$ Using Gauss theorem we get: $$\int_{\partial \Omega} \mathbf{n} \cdot (-p_1 \nabla c + \vec{v} c) dS= 0$$ where $\mathbf{n}$ is the normal vector on the control surface $\partial \Omega$ enclosing $\Omega$. This means that the quantity: $$-p_1 \nabla c + \vec{v} c$$ is a conserved quantity in terms of all of $p_1$, $p_2$, $c$.

However, you also have $f$ injecting "mass" into the control volume which makes the equation as a whole non-conservative. You can write this on conservative form even in this case provided you can find a $\vec{q}$ such that $f=\nabla \cdot \vec{q}$ since then the equation can be written: $$ \nabla \cdot (-p_1 \nabla c + \vec{v} c-\vec{q}) = 0$$ So that now: $$-p_1 \nabla c + \vec{v} c-\vec{q}$$ is the conserved quantity.