I am developing a pseudo-spectral code for the vorticity transport equation. In order to validate the correct computation of the nonlinear (convective) term in the physical space, I have first solved the 1D Burgers equation and compared the numerical result with an analytical solution for a periodic domain. This worked fine.
Now I aim to do the same thing with the 2D Burgers equation or any other PDE containing a nonlinear term such as $u_j\frac{\partial u_i}{\partial x_j}$,
$\frac{\partial u_i}{\partial t}+u_j\frac{\partial u_i}{\partial x_j}=\nu\frac{\partial^2 u_i}{\partial x_j^2}$.
However, I am struggling to find an exact solution for the 2D Burgers equation in a periodic domain (maybe it doesn't exist, I don't know..). If it doesn't, which other PDE of this family could I use for this purpose? Or in which other way could I validate my numerical solution?