Suppose I have a simple linear first order PDE of the form: $$au_x+bu_y=0$$ I want to create a PDE in 3 variables, $x,y,t$ (so the function is now $u(x,y,t)$) such that given an initial condition $u(x,y,t)=u_0(x,y)$ (depending on just the first two variables):
If $u_0$ satisfies the original PDE, then the solution to the new PDE is equal to $u_0$ at all times.
Generally, as $t\rightarrow \infty$, the solution to the new PDE converges to a function of just two variables $u(x,y)$ which satisfies the original PDE. (converges in what topology? Naturally I'd prefer the topology to be as strong as possible, but I'm open to anything interesting)
So in a sense, I want my PDE to stabilize over time to a PDE of 'less' variables. For instance, I can create a PDE which satisfies the first condition by writing: $$u_t=au_x+bu_y$$ But it does not generally satisfy the 2nd condition (I checked). An example which comes to mind is the heat equation, which usually stabilizes over time to a harmonic function. But I want something which fits my specific first order PDE. Does anyone have any ideas on how to do this?
This is the first part of my question. The harder part is - if we can find such a method for the PDE above, can we generalize this method for a system of 2 coupled PDE's of the form: $$au_x+bv_y=0$$ $$cu_y+dv_x=0$$
Thanks in advance!