Second order Sobolev regularity results for the generalized $p$-parabolic equation

We study a general class of parabolic equations $$u_t-\mathrm{\mid }Du{\mathrm{\mid }}^{\gamma }(\mathrm{\Delta }u+(p-2){\mathrm{\Delta }}_{\mathrm{\infty }}^{N}u)=0,$$ which can be highly degenerate or singular. This class contains as special cases the standard parabolic $p$-Laplace equation and the normalized version that arises from stochastic game theory. Utilizing the systematic approach developed in our previous work we establish second order Sobolev regularity together with a priori estimates and improved range of parameters. In addition we derive second order Sobolev estimate for a nonlinear quantity. This quantity contains many useful special cases. As a corollary we also obtain that a viscosity solution has locally $L^2$-integrable Sobolev time derivative.