I wonder if one knows that there exists a unique classical solution of a pde (for instance: Fokker-Planck equation), is one able to conclude that there isn't any weak solution of the pde, which differs from the classical one? To be more specific:
I'm facing the following Fokker-Planck Equation:
$-\partial_t g(x,t) - \Delta g(x,t) -div(g(x,t) b(x,t)) = 0$ with initial condition $g(x,0)=g_0(x)$
and where $g_0$ is a density function and $b:\mathbb{R}^n\times(0,T)$ is a continuous, bounded and lipschitz in $x$ vector field. I know from the literature, that there is a classical solution $g \in C^{2,1}(\mathbb{R}^n\times(0,T))$ of this equation.
What i know aswell is that there is an absolute continuous probability measure (the law of the stochastic process, which satisfies the corresponding SDE), which satisfies this fokker-planck equation in a distributional sense and as a consequence, the density function of this measure satisfies the equation in a weak sense. I now wonder if this density function is a classical solution, because of the statement given above. Does anyone know if this is true? Thanks a lot!