Consider the parabolic equation in $p: \mathbb R^2\to\mathbb R$
$$\partial_t p + b(t)\partial_x p + D(t,x)\partial^2_{xx}p=0,$$
where $b$, $D$ are nice enough functions. I look for the continuity of the derivatives $\partial_t p$, $\partial_x p$ of the solution. It is known by Nash's paper (Continuity of Solutions of Parabolic and Elliptic Equations) that, under very reasonable conditions on $b$, $D$, we have the Hölder-type continuity of $p$. Is there any work concerning such continuity analysis of $\partial_t p$, $\partial_x p$?
PS: My idea is to consider $q\mathrel{:=}\partial_x p$. Then
$$\partial_t q + b(t)\partial_xq + D(t,x)\partial^2_{xx}q=-\partial_x D(t,x)\partial_{xx}p$$
is a similar parabolic equation for $q$ with an additional source. But the term $\partial_x D(t,x)\partial_{xx}p$ contains $\partial_{xx}p$, which makes the estimation even harder….
Best Answer
The equation for $q = p_x$ can be written in divergence form as $$q_t + b(t)q_x + (D(t,\,x)q_x)_x = 0,$$ so Nash's theorem (which applies to divergence-form equations) implies that $p_x$ is Holder continuous under mild hypotheses on the coefficients (boundedness and measurability).
If the coefficients are more regular (e.g. in Holder classes) then parabolic Schauder estimates (e.g. in the book of Lieberman) give higher regularity of $p$, the general principle being that $p$ has twice as many spatial derivatives as temporal ones (by the scaling of the equation).