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.
“…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.…”
Section: Formulation Of the Problem And Statement Of The Main Resultsmentioning
confidence: 99%
“…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).…”
Section: Formulation Of the Problem And Statement Of The Main Resultsmentioning
We prove asymptotic stability of steady-state solutions to the Navier–Stokes equations in the whole space. One of the novelties of this work consists in considering perturbations that show a distinct pointwise behavior (in space and time) and correspond to initial data only belonging to suitable weighted Lebesgue spaces.
“…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.…”
Section: Formulation Of the Problem And Statement Of The Main Resultsmentioning
confidence: 99%
“…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).…”
Section: Formulation Of the Problem And Statement Of The Main Resultsmentioning
We prove asymptotic stability of steady-state solutions to the Navier–Stokes equations in the whole space. One of the novelties of this work consists in considering perturbations that show a distinct pointwise behavior (in space and time) and correspond to initial data only belonging to suitable weighted Lebesgue spaces.
We investigate on the existence of solutions with initial datum U 0 in L 3 . Our chief goal is to establish the existence interval (0, T ) uniquely considering the size and the absolute continuity of |U 0 (x)| 3 .
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