How do I determine if the equation is a conservation law?

Question

We have the PDE $\frac{\partial u}{\partial t}+a(x,y)\frac{\partial u}{\partial x}+b(x,y)\frac{\partial u}{\partial y}=0$. What would be conditions on $a$ and $b$ for the equation to constitute a conservation law? I think that $a=b$ would be sufficient, since then the flux of $u$ into a small control volume in a Cartesian coordinate system would affect the growth rate $\partial u / \partial t$ in the same way regardless of the direction which the flux of $u$ comes from. What do you think?

Answer 1

A conservation law has the following structure: $$u_t=-\mathrm{div} (\mathbf{F}(x,y,u))$$ where $\mathbf{F}=(F^1,F^2)^T$ is a vector field that, in principle, can be nonlinear.

The last PDE can be rewritten in the following form: $$u_t=-{F^1}_x-{F^2}_y-\mathbf{F}_u\cdot \nabla u$$

In your case we have ${F^1}_x={F^2}_y=0$ and this implies $F^1=f(y,u)$ and $F^2=g(x,u)$.

Moreover $\mathbf{F}_u=(a(x,y),b(x,y))^T$ and this means that $\mathbf{F}$ must be linear with respect to $u$. Hence we have $\mathbf{F}_u=(f(y)u+c_1,g(x)u+c_2)^T$, where $c_1$ and $c_2$ are two constants.

EDIT (to answer @user1952009)

What you are trying to do is something different. The question which can you pose to my answer is:

Are your calculations independent with respect a change of coordinate? Will the function keep the same structure changing the set of coordinates?

The answer is clearly o because at some point I started to use the definition of divergence in Cartesian coordinate. In order to take into account the general setting (differential calculus on a manifold) the answer would be more complex.

Apparently you have deleted the comment down from my answer.