Remarks on the pressure error estimatesfor

The purpose of this note is to correct some errors in the proofs of pressure error estimates in 1] and 2]. The origin of the errors is the following incorrect inequality 1 (1) c 2 kuk 2 ?1 (A ?1 u; u) ; (see (2.1) in 1] and (2.7) in 2]), which the author incorrectly derived by identifying kuk V 0 with kuk ?1 for u 2 H. 2 The correct inequality is: (2) c 2 kuk 2 V 0 (A ?1 u; u) : However, the incorrect proofs induced by the error (1) can all be xed as indicated below. More precisely, the proofs for the velocity error estimates in 1, 2] are all valid provided some minor changes of notations; the pressure error estimates still hold, but their proofs necessitate some additional estimates. In summary, all the results presented in 1, 2] remain valid provided some additional regularity assumptions on the exact solution are made. In the following we provide the details for the aforementioned modiications and corrections. 1 The incorrectness of (1) was pointed out to me by R. Temam to whom I am grateful. Later on J.L. Guermond has also expressed to me his doubts about (1) 2 The confusion can be partly explained by a technical sublety. We know that V H 1 0 () d and hence H 1 0 () d = V V ?. By the Riesz theorem and elementary properties of Hilbert spaces, V 0 is isomorphic to and could be identiied with a subspace of H ?1 () d. However this identiication must not be made because, as it is usual with evolution equations, we already identiied H with a subspace of V 0 (V H V 0). Hence this double identiication is not allowed Numerische Mathematik Electronic Edition { page numbers may diier from the printed version page 513 of Numer.