I'm trying to numerically solve the following equation
$$\frac{\partial{u}}{\partial{t}}=D\nabla^2u-\vec{v}.\nabla(u)$$
for which -1 $\le$ x $\le$ 1, -1 $\le$ y $\le$ 1, 0 $\le$ t $\le$ 1 with initial condition u(0,x,y)=sin(πxy) and u(t,x,y)=sin(πxy) and D=0.1 and $\vec{v}$ is {y,-x}. I've tried doing the following
eqn2 = Inactive[0.1*Laplacian[u[t, x, y], {t, x, y}] - {y, -x}*Gradient[u[t,x,y],{t, x, y}]]
pdesol = DSolve[{eqn2, u[0, x, y] == Sin[\[Pi] x y], u[t, x, y] == Sin[Pi x y]}, u[t, x, y], {x, -1, 1}, {y, -1, 1}, {t, 0, 1}]
But an error occurs. I have also tried to use NDSolve, but I get the same error, which is, "equation or list of equations expected instead of...". How could I fix this?