Global well-posedness and large time behavior for the inviscid Oldroyd-B model

Abstract

In this paper, we consider the 3-dimensional incompressible inviscid Oldroyd-B model. Firstly, we establish the global existence of the solutions for the inviscid Oldroyd-B model with different coupling coefficient k>0. Then, we show the connection between the solution with the parameter k that is the reciprocal of Weissenberg number. On one hand, we prove that the solutions (u,τ) depend continuously on the parameter k>0, though (u,τ) corresponds to different decays rate for different k. On the other hand, when k→0 in large time, we prove that there is a gap between the L2 norm of solution uk(t,x) to the model of parameter k>0 with the L2 norm of solution u0(t,x) to the model of parameter k=0. In a word, we prove that the larger positive k induces the faster decay rate of the solution, but k cannot go to zero, or the dissipation will vanish instantly.