On analysis of fractional Navier-Stokes equations via nonsingular solutions and approximation (Q1665007)

Summary: 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 \(\beta\). In this paper, we analyze the existence and stability of nonsingular solutions to fractional Navier-Stokes equations of type (\(\mathbf{u}_t + \mathbf{u} \cdot \nabla \mathbf{u} + \nabla p - \mathrm{Re}^{- 1}(- \nabla)^\beta \mathbf{u} = \mathbf{f} \text{ in } \Omega \times(0, T]\)) defined below. In the case where \(\beta = 2\), we show that the stability of the (quadratic) convergence, when exploiting Newton's method, can only be ensured when the first guess \(\mathcal{U}^0\) is sufficiently near the solution \(\mathcal{U}\). We provide interesting well-posedness and existence results for the fractional model in two other cases, namely, when \(1 / 2 < \beta < 1\) and \(\beta \geq \left(1 / 2\right) +(3 / 4) \).