Regularity of the solution map for the Oldroyd–B model with fractional stress dissipation

The data-to-solution map of the incompressible Euler equations is continuous but not uniformly continuous on Sobolev spaces, whereas viscosity makes the Navier–Stokes solution map Lipschitz continuous on small data, uniformly in time. This paper identifies the behavior of the solution map for a hybrid system sitting between these two extremes: the incompressible Oldroyd–B model in $\mathbb{R}^d$, $d=2,3$, in the high-Reynolds-number, high-Weissenberg-number regime. The velocity equation is a forced Euler equation and the only dissipation is a fractional diffusion $η(-Δ)^β$, $\frac12\leβ\le1$, acting on stress. We prove that, on the ball of small $H^s$ data ($s>1+\frac d2$) on which global solutions exist, the solution map is Lipschitz continuous with respect to the $H^{r}$-norm, $r=s-(2β-1)$, uniformly for all time, and Lipschitz into the natural dissipation spaces; the constant is explicit and tends to $\sqrt3$ as the size of the data tends to zero. Consequently the map is Hölder continuous into $C_b([0,\infty);H^{s'})$ for every $s'<s$, and is Lipschitz in $H^s$ itself when $β=\frac12$. Thus, although the velocity field is only transported, the solution map behaves like that of the Navier–Stokes equations rather than that of the Euler equations. The proof exploits the damped wave structure generated by the coupling between $u$ and $\nabla\cdotτ$ through a Lyapunov functional with a cross term $(δu,\nabla\cdotδτ)_{H^{r-β}}$, which transfers the stress dissipation to the velocity difference, together with a low-frequency interpolation argument that keeps all Gronwall coefficients integrable on $(0,\infty)$. The loss of $2β-1$ derivatives is exactly the gap between the hidden velocity dissipation of order $2(1-β)$ and the dissipation required by the Lipschitz estimate, and we show that it cannot be reduced by the energy method.