In this paper, we are concerned with the global existence and optimal rates of strong solutions for three-dimensional compressible viscoelastic flows. We prove the global existence of the strong solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H 2 -framework. If additionally the initial data belong to L 1 , the optimal convergence rates of the solutions in L p -norm with 2 ≤ p ≤ 6 and optimal convergence rates of their spatial derivatives in L 2 -norm are obtained.

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We investigate stability of an equilibrium state to a nonhomogeneous incompressible viscoelastic fluid driven by gravity in a bounded domain Ω ⊂ R 3 of class C 3. First, we establish a critical number κ C , which depends on the equilibrium density and the gravitational constant, and is a threshold of the elasticity coefficient κ for instability and stability of the linearized perturbation problem around the equilibrium state. Then we prove that the equilibrium sate is exponential stability provided that κ > κ C and the initial disturbance quantities around the equilibrium state satisfy some relations. In particular, if the equilibrium densityρ is a Rayleigh-Taylor (RT) type andρ ′ is a constant, our result strictly shows that the sufficiently large elasticity coefficient can prevent the RT instability from occurrence.

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