On the Cauchy problem for hyperdissipative Navier-Stokes equations in super-critical Besov and Triebel-Lizorkin spaces (Q2094281)

The authors study well-posedness of the hyperdissipative Navier-Stokes equations in the \(n\)-dimensional whole space. This system is obtained from the classical equations for incompressible Newtonian fluids by replacing the negative of the Laplace operator with a higher-order integer power \((-\Delta)^\alpha\), \(1<\alpha\in\mathbb{N}\). For initial data in a supercritical Besov or Triebel-Lizorkin space, existence of local-in-time mild solution is shown by a fixed-point argument using time-decay properties of the Gauss-Weierstrass semigroup generated by \((-\Delta)^\alpha\). It is further shown that the established mild solutions are strong solutions and locally stable.