Does existence and uniqueness of a classical solution impose uniqueness of weak solutions to a pde? 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:
$−∂tg(x,t)−Δg(x,t)−div(g(x,t)b(x,t))=0$ with initial condition $g(x,0)=g0(x)$
and where $g_0$ is a density function and $b:ℝn×(0,T)$ is a continuous, bounded and lipschitz in x vector field. I know from the literature, that there is a classical solution $g∈C2,1(ℝn×(0,T))$ 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 in advance!
 A: I apologize if the following is a bit too general or hand-wavy, but hopefully it will be helpful.
The characteristic property of a differential operator that tells you that a weak solution is equivalent to a strong solution is generally called "regularity".
Luckily, there are a whole host of theorems for differential operators in which this property is identified, particularly for differential operators classified as hypoelliptic. Examples of hypoelliptic operators include the Laplace operator (responsible for driftless diffusion) and the heat equation operator. Assuming your $b$ is not sufficiently pathological (which you appear to indicate is the case), the Fokker-Planck operator you describe above should fall into this category.
In short, you can be sure that—assuming you don't have initial conditions/$b$ functions that are too "funky", I think—a weak solution you find to your problem will always be a strong solution. Theorems to prove this property tend to be a bit extensive: a good description of this property is given in this book by Norio Shimakura. (Unfortunately the classic PDE book by Lawrence Evans doesn't touch on this property very much, if at all.)
