This paper studies the incompressible limit of global strong solutions to the threedimensional compressible Navier-Stokes equations associated with Navier's slip boundary condition, provided that the time derivatives, up to first order, of solutions are bounded initially. The main idea is to derive a differential inequality with decay, so that the estimates are bounded uniformly both in the Mach number ǫ ∈ (0, ǫ 0 ] (for some ǫ 0 > 0) and the time t ∈ [0, +∞). * This work was initiated when the author Ou visited The Institute of Mathematical Science, The Chinese University of Hong Kong in 2012. He would like to thank Prof. Zhouping Xin for the invitation and helpful discussions.pressure, respectively. And the matrix S ≡ S(u) = 2µD(u) + λdivuI 3 is the viscous stress tensor for newtonian fluids, where D(u) = (∇u + ∇u t )/2. The constant ǫ ∈ (0, 1] is the Mach number of the highly subsonic fluids. The constants µ, λ are viscosity coefficients with µ > 0, µ + 3λ/2 ≥ 0. In this paper, we suppose that the pressure p = p(ρ) is a C 3 function satisfying that p ′ (ρ) > 0 for ρ > 0 .In the physical viewpoint, the motions of highly subsonic compressible fluids would behave similarly to the incompressible ones (see [17]). Formally, as the Mach number ǫ tends to zero, the solutions to (1.1)-(1.2) will converge to the solution (u, π) of the incompressible Navier-Stokes equations, namely u t + u · ∇u − µ∆u + ∇π = 0, divu = 0.
In this paper, we establish the local existence of strong solutions to an Oldroyd-B model for the incompressible viscoelastic fluids in a bounded domain R d , d D 2 or 3, via the incompressible limit. The main idea is to derive the uniform estimates with respect to the Mach number for the linearized system of compressible Oldroyd equations.
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