Second order Sobolev regularity for normalized parabolic $p(x)$-Laplace equations via the algebraic structure

Denote by $\mathrm{\Delta }$ the Laplacian and by ${\mathrm{\Delta }}_{\mathrm{\infty }}$ the $\mathrm{\infty }$-Laplacian. A fundamental inequality is proved for the algebraic structure of $\mathrm{\Delta }v{\mathrm{\Delta }}_{\mathrm{\infty }}v$: for every $v\in C^{\mathrm{\infty }}$, $$\mid \mathrm{\mid }{D}^{2}vDv{\mathrm{\mid }}^2-\mathrm{\Delta }v{\mathrm{\Delta }}_{\mathrm{\infty }}v-\frac{1}{2}[\mathrm{\mid }{D}^{2}v{\mathrm{\mid }}^2-(\mathrm{\Delta }v{)}^2]\mathrm{\mid }Dv{\mathrm{\mid }}^2\mid \le \frac{n-2}{2}[\mathrm{\mid }{D}^{2}v{\mathrm{\mid }}^2\mathrm{\mid }Dv{\mathrm{\mid }}^2-\mathrm{\mid }{D}^{2}vDv{\mathrm{\mid }}^2]$$ Based on this, we prove the result: When $n\ge 2$ and $p(x)\in (1,2)\cup (2,3+\frac{2}{n-2})$, the viscosity solutions to parabolic normalized $p(x)$-Laplace equation have the $W_{loc}^{2,2}$-regularity in the spatial variable and the $W_{loc}^{1,2}$-regularity in the time variable.