since I am not very familiar with this area, I have some queries(basic) about the regularity of solutions of a parabolic type PDE. Suppose we have the following differential operator:

$Lu(x)=b(x)^\top\nabla_x u(x)+\nabla\cdot(A(x)\nabla_x u(x))$,

where $A(.)$ is a matrix of order $n\times n$ that is uniformly elliptic and the entries are infinitely differentiable. Also, $b(.)$ is a vector valued function which is infinitely differentiable. Now, for a function $f\in L^p(\mathbb{R}^n)$, we consider the following PDE:

$$u_t= Lu \ \ \text{in} \ \mathbb{R}^+\times\mathbb{R}^n $$ $$u(0,x)=f(x) \ \ \text{in} \ \mathbb{R}^n$$

You may assume that $A,b$ are Lipschitz continuous. My question is that what type of assumptions do we need further(specially on $f$) to ensure that the solution is infinitely differentiable?


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