Advection-diffusion with periodic boundary conditions

Any $rho$ which solves the equation on the whole torus must also be a solution locally on every subset. In particular, it must be solution on the (non-toroidal) open $L times L $ square. Since solutions on the torus are a subset of the solutions on the square, the question becomes: Do there exist solutions on the square which happen to match at the boundaries?

On this square, we can define $V = U - mathbb{x} cdot mathbb{q}$, and we have an ordinary advection-diffusion equation. We know there exist solutions of the form $alpha e^{-V(mathbb{x})}$. We also know that $U$ is periodic, so $V$ can only be periodic if $mathbb{q} = mathbb{0}$. However $e^{-V}$ could still be periodic if $mathbb{q}$ is imaginary. Specifically, we have periodic solutions for $mathbb{q} = frac{2pi i}{L}mathbb{n}, mathbb{n} in mathbb{Z}^2$.

For other $mathbb{q}$, solutions proportional to $ e^{-V(mathbb{x})} $ cannot extend to solutions on the whole torus. The remaining question: Are such solutions the whole solution space?

Now, Matthew Kvalheim points to Zeeman, 1988. Theorem 3 reads

Let $U$ be a vector field on a compact manifold $X$ without boundary, and let $epsilon$ > 0. Then the Fokker-Planck equation for $U$ with $epsilon$-diffusion has a unique steady state, and all solutions tend to that steady state.

The torus is a compact manifold without boundary, Zeeman's $U$ is our $-nabla V$, and we have $epsilon = 1$, so the theorem tells us a solution $rho$ must exist and is unique (up to an overall scalar). Unfortunately, this proof is not constructive.

In one dimension, variation of parameters gives the solution $$rho = C_1 e^{-V}left(C_2 + int_0^x e^Vright)$$ and the requirement $rho(0) = rho(L)$ fixes $C_2$. We can try to extend this to two dimensions as follows: Assume $rho$ is of the form $alpha(x)e^{-V}$. Then the equation becomes $$ nabla cdot [-nabla V alpha(x)e^{-V} - nabla (alpha(x) e^{-V})] = 0 $$ which simplifies to $$ nabla cdot (nablaalpha(x) e^{-V}) = 0 $$ Solutions are $$ nablaalpha(x) e^{-V} = nabla times mathbf{psi} $$ for $mathbf{psi} = mathbf{e}_z $ and $g$ some scalar function. Then $$ nablaalpha(x) = e^V(nabla times mathbf{psi}) $$ If $$ nabla times (e^V(nabla times mathbf{psi})) = 0 $$ then this has solution $$ alpha(x,y) = C + left(int_0^x -e^V g_y dxright) + left(int_0^y e^V g_x dyright) $$ The requirement of periodic boundary conditions picks out some unique $g$, $C$ up to an overall constant. We need $$ alpha(x,0) = alpha(x,L)e^{-Lq_y} $$ or $$ C + left(int_0^x -e^V g_y dxright) = Ce^{-Lq_y} + left(int_0^x -e^V g_y dxright)e^{-Lq_y} + left(int_0^L e^V g_x dyright)e^{-Lq_y} $$ At $x = 0$ this simplifies to $$ C = frac{1}{e^{Lq_y} - 1}int_0^L e^V g_x(0,y) dy $$ It remains to find $g$.

I'm not sure that there is a nice expression for the solution in general. Some miscellaneous thoughts:

• When $U = 0$, $rho = C$ is a solution, which corresponds to $alpha = e^V, g = xq_y-yq_x$. This shows that $g$ may be defined only on the square, not on the torus.
• When $nabla U gg q$ or $q gg nabla U $, we can start with the nearby known solution and series expand.