We study interior $C^{ 1, \al}$ regularity of viscosity solutions of the parabolic Monge-Amp\'ere equation $$u_t = b(x,t) \ddua,$$ with exponent $p >0$ and with coefficients $b$ which are bounded and measurable. We show that when $p$ is less than the critical power $\frac{ 1}{ n-2}$ then solutions become instantly $C^{ 1, \al}$ in the interior. Also, we prove the same result for any power $p>0$ at those points where either the solution separates from the initial data, or where the initial data is $C^{ 1, \beta}$.
Categories: math.AP