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

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, tau) depend continuously on the parameter k > 0, though (u, tau) 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 L-2 norm of solution u(k)(t, x) to the model of parameter k > 0 with the L-2 norm of solution u(0)(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.