Random Data Cauchy Theory for the Generalized Incompressible Navier–Stokes Equations

Abstract

In this paper, we consider the generalized Navier–Stokes equations where the space domain is ${\mathbb{T}^N}$ or ${\mathbb{R}^N, N\geq3}$ . The generalized Navier–Stokes equations here refer to the equations obtained by replacing the Laplacian in the classical Navier–Stokes equations by the more general operator (−Δ)^α with ${\alpha\in (\frac{1}{2},\frac{N+2}{4})}$ . After a suitable randomization, we obtain the existence and uniqueness of the local mild solution for a large set of the initial data in ${H^s, s\in[-\alpha,0]}$ , if ${1 < \alpha < \frac{N+2}{4}, s\in(1-2\alpha,0]}$ , if ${\frac{1}{2} < \alpha\leq 1}$ . Furthermore, we obtain the probability for the global existence and uniqueness of the solution. Specially, our result shows that, in some sense, the Cauchy problem of the classical Navier–Stokes equation is local well-posed for a large set of the initial data in H ⁻¹⁺, exhibiting a gain of ${\frac{N}{2}-}$ derivatives with respect to the critical Hilbert space ${H^{\frac{N}{2}-1}}$ .