Consider the following parabolic equation on $\mathbb{R}^d$:
\begin{equation} \partial_t\mu=\mathrm{div}(b\mu) + \mathrm{div}(D\nabla\mu), \end{equation} where the drift $b:\mathbb{R}^d\rightarrow\mathbb{R}$ is sufficiently smooth say $C^2$ and the diffusion matrix $D:\mathbb{R}^d\rightarrow \mathbb{R}^{d\times d}$ is symmetric, strictly positive definite with entries sufficiently smooth, say $C^2$ as well. Here notation $C^2$ means continously differentiable upto second order.
Such an equation arises as the forward-kolmogorov equation for an stochastic differential equation (SDE).
Question: Following the connection from SDEs assume that at time $t=0$, $\mu|_{t=0}=\nu$ where $\nu\in L^1(\mathbb{R}^d)$. How does one prove regularity for the solution of such a PDE?
I have heard that the solution of this equation should be smooth (upto C^2?), but I have no idea how to prove this. Any suggestions and references are very welcome.
Thanks in advance.