On Analysis of Fractional Navier-Stokes Equations via Nonsingular Solutions and Approximation

Abstract EN

Until now, all the investigations on fractional or generalized Navier-Stokes equations have been done under some restrictions on the different values that can take the fractional order derivative parameter β .

In this paper, we analyze the existence and stability of nonsingular solutions to fractional Navier-Stokes equations of type ( u t + u · ∇ u + ∇ p - R e - 1 ( - ∇ ) β u = f in Ω × ( 0 , T ] ) defined below.

In the case where β = 2 , we show that the stability of the (quadratic) convergence, when exploiting Newton’s method, can only be ensured when the first guess U 0 is sufficiently near the solution U .

We provide interesting well-posedness and existence results for the fractional model in two other cases, namely, when 1 / 2 < β < 1 and β ≥ 1 / 2 + ( 3 / 4 ) .