Arch. Rational Mech. Anal. 128 (1994) 313-327. 9 Springer-Verlag 1994 Singular Limits in Compressible Fluid Dynamics H. BEIR,~O DA VEIGA Communicated by P.-L. LIONS Main Notation We start by introducing the main notation. As usual, IR § is the set of positive reals and IR~- = IR + • {0}. We denote by [. [p and I[" IIm the canonical norms in the space L p = LP(f~), 1 <__ p __< oe, and in the L2-Sobolev space H m =//m(~)), respec- tively. We set II" II = IF" [Io. The point x belongs to the n-dimensional toms, n > 2, identified here with the set f~ = [0, 1]". We denote by II. Ilm, r and [']m,r the canonical norms in L~ T; H ~) and L2(0, T; H"), respectively. The function x ~ f(t, x), for a fixed t, is sometimes denoted byf(t). We denote by ko the smallest integer larger than n/2 and by k a fixed integer satisfying k __> ko + 1. The parameters n, k, Po, Vo, and/~o are fixed. Positive constants that depend at most on these quantities are denoted by c. Below we also introduce the positive constants ci, that play here an important role (see (1.6)-(1.8)). Positive constants that depend at most on cl are denoted by C~; positive constants that depend at most on c~ and c2 by C2; and positive constants that depend at most on ca, c2, c3 by C3. Distinct constants C~ are denoted by the same symbol provided that they depend on the same basic constants cl. Introduction We study the dependence of solutions to the equations of motion of compressible fluids on the Mach number 2-1 and on the viscosity coefficients v and #. We assume that 2 > 1 (we shall be interested in letting 2 tend to ~ ) and that v ~ [0, %], /~ s [0, #o] for arbitrary, but fixed, constants vo and #o- Viscous and inviscid fluids are studied together, since 0 is an admissible value for the viscosity coeff• # and v. We denote by v the velocity field, by p the density of the fluid, and by p(2, p) the pressure p as a function of the density p and the Mach number. It is worth noting that our results and proofs apply if p(2, p) enjoys the properties assumed in the papers [BV1,2]. In order to avoid technicalities, we assume here that (1.1) PO,, P) = ,~2p(p)