Global existence and decay estimate of solution to rate type viscoelastic fluids

Abstract

We are concerned with the global well-posedness of rate type viscoelastic fluids system (1.1)–(1.2). First, we prove the global existence and decay estimate of solution to (1.1)–(1.2) with stress diffusion near a constant state in HN(R3) (N≥4). We only assume that the H3-norm of initial data is small, but the L2-norm of the higher order derivatives can be arbitrary large. If the initial data belongs to homogeneous Sobolev or Besov spaces, we obtain the optimal decay rates of the solution and its higher order derivatives. Next, we establish the local existence of smooth solution to (1.1)–(1.2) without stress diffusion and present a blow-up criterion in R2. From which and the Bony decomposition, we prove the global existence of smooth solution.