I have a question concerning solving PDE in my research. Supposing there is a Langevin Equation. $$ dX/dt = F(X)$$ I want to decompose it into two components which are orthogonal to each other. $$dX/dt = F(X) = -grad(U) + Fr$$ with $(grad(U), Fr) = 0$.
However, I know that this nonlinear PDE does not always admit a solution. $$(grad(U), F + grad(U))=0$$
For example, $dx/dt = xy$, $dy/dy = 0$
Or $dx/dt = -x + xy$, $dy/dt = -y$
But how to rigorously prove that it does not have solutions for some cases?
Many thanks,
Joseph Zhou
