Navier–Stokes Cauchy Problem with |v0(x)|2 Lying in the Kato Class K3

Abstract

We investigate the 3D Navier–Stokes Cauchy problem. We assume the initial datum v0 is weakly divergence free, supR3∫R3|v0(y)|2|x−y|dy<∞ and |v0(y)|2∈K3, where K3 denotes the Kato class. The existence is local for arbitrary data and global if supR3∫R3|v0(y)|2|x−y|dy is small. Regularity and uniqueness also hold.

On the Stability of Steady-State Solutions to the Navier–Stokes Equations in the Whole Space

“…Actually, L p α , with α = 1 − 3 p , is another example of scaling invariant norm (in the sense proposed in [1]) for the global existence with small data. In this regards we notice that for p = 2 the above space was considered in [4] in the different context of sufficient conditions for the existence of regular solutions. For p = 3 , L 3 α ≡ L 3 (that is α = 0) and the result goes back to the well known theory.…”

“…Remark 4. In [4], the weighted L 2 -space is restricted to the so called Kato class. In this way the authors are able to give an estimate of the interval of existence (a priori finite) without employing auxiliary functions (density property).…”

A new proof of existence in the L3-setting of solutions to the Navier-Stokes Cauchy problem

“…In this connection, the recent paper [2] seems to be an exception. It is employed the dimensionless weighted functional ||U 0 || 2 wt := sup x R 3…”