Lemma 12 ([Interior estimate with an $L^2$ input]). Let U_1 U_2 be fixed bounded spatial domains and b_0<b_1<b , with b-b_1 bounded above. Suppose that the coefficient assumptions of Lemma 11 ([Local Schauder estimates]) hold on …

Lemma 12 ([Interior estimate with an $L^2$ input]). Let U_1 U_2 be fixed bounded spatial domains and b_0<b_1<b , with b-b_1 bounded above. Suppose that the coefficient assumptions of Lemma 11 ([Local Schauder estimates]) hold on U_2×(b_0,b) and that Lu=f there. Then N_α(u;U_1×[b_1,b)) ≤C(|f|_ α;U_2×(b_0,b) +\|u\|_ L^2(U_2×(b_0,b)) ). The constant depends on the spatial gap, b_1-b_0 , the upper bound for b-b_1 , and the coefficient constants, but requires no terminal trace. The estimate is unchanged if the upper face is included and the solution is defined there. In particular it holds on fixed nested annuli through an open upper time endpoint.