Approximations for singularly perturbed parabolic equations of arbitrary order (Q2763759)

The approximations for singularly perturbed parabolic equations NEWLINE\[NEWLINE\partial_\tau \nu^\varepsilon (\tau,x)= \varepsilon\sum_{|k|\leq 2p} A_k (\tau,x) \partial_x^k \nu^\varepsilon (\tau,x),\quad \nu^\varepsilon (0,x)= \varphi(x),\quad (\tau,x)\in (0,T/ \varepsilon] \times R^dNEWLINE\]NEWLINE are studied. Time scaling reduces this equation to NEWLINE\[NEWLINE\partial_t u^\varepsilon (t,x)=\sum_{|k|\leq 2p}A_k(t/ \varepsilon,x) \partial^k_x u^\varepsilon (t,x),\quad u^\varepsilon (0,x)= \varphi(x),\quad (t,x)\in(0,T] \times R^d.\tag{*}NEWLINE\]NEWLINE Under mild conditions, (*) has a \(\mathbb{C}^N\)-valued solution, whose arbitrary order \(x\)-derivatives are compared (as \(\varepsilon\to +0\) on \((0,T]\times R^d)\) with derivatives of the unique solution of an ``averaged'' equation \(\partial_t u(t, x)= \sum_{|k|\leq 2p} A_k^0(x) \partial^k_xu(t,x)\), \(u(0,x)= \varphi (x)\). The authors show that \(\partial^n_x (u^\varepsilon-u)\) tends to zero (as \(\varepsilon\to +0)\) uniformly over \((0,T]\times R^d\) in a weighted Hölder norm or, in the unscaled case, that \(\partial_x^n |\nu^\varepsilon (\tau,x)-u (\varepsilon\tau,x)|\) converges to 0 uniformly on \((0,T/ \varepsilon] \times R^d\).