Incompressible generalised Navier–Stokes equations with supercritical initial data: almost sure existence and optimal decay rates of global weak solutions

Abstract

In this paper, we study the initial value problem of the incompressible generalised Navier–Stokes equations with fractional dissipation (−Δ)αu in Rd, where d⩾2, and the initial data lies in the supercritical regime of negative order Sobolev spaces H˙s(Rd) with s<0. By employing a probabilistic approach of data randomisation, the almost sure existence of global weak solutions is obtained for the problem with dissipation exponents α∈(12,d+24] and s∈(−α+(1−α)+,0), in which it overcomes a technical difficulty as α⩽23 that limited authors’ previous study in (2026 J. Math. Anal. Appl. 555 130042) to the range α>23. The proof mainly relies on a refined analysis of the corresponding integral equation near t=0, incorporating novel functional frameworks and bilinear estimates. Furthermore, an optimal decay rate in time of the Lx2 norm of the solution is derived by using the Fourier splitting method, it also provides a framework potentially applicable for studying the large time behaviour of other fluid models with low-regularity initial data. Additionally, we obtain the uniqueness of weak solutions to this problem when the critical dissipation exponent α=d+24 with d⩾2, which extends the well-known result for the two-dimensional Navier–Stokes equations.