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# Regular flows of weakly compressible viscoelastic fluids and the incompressible limit

• We consider compressible viscoelastic fluids satisfying the Oldroyd constitutive law. We prove the local existence (and uniqueness) of flows by a classical fixed point argument. We also prove some global properties of the solutions. In particular, we obtain some a priori estimates which are uniform in the Mach number and prove global existence of weakly compressible fluids flows. We show that weakly compressible flows with well-prepared initial data converge to incompressible ones when the Mach number converges to zero.
Mathematics Subject Classification: Primary: 76A05, 76A10, 76N10; Secondary: 35B30.

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