Strong solutions of the Navier-Stokes equation in Morrey spaces.

*(English)*Zbl 0781.35052The author establishes the well-posedness of the nonstationary free Navier-Stokes equation in \(\mathbb{R}^ +\times \mathbb{R}^ m\) in Morrey spaces. It is shown that if a certain seminorm of the initial velocity is sufficiently small, then a time local solution exists and is unique within certain restrictions; and if the norm of the initial velocity is sufficiently small, then a global solution exists. A similar result with initial velocity replaced by initial vorticity is proved. The initial velocity is an element of \(M_{p,m-p}\), \(1\leq p<\infty\), and hence in case \(p=1\) it may be a certain measure. The large time decay and regularity of the solutions are also established.

The materials presented as preliminaries would be very instructive for the study of Morrey spaces as well as various operators such as the heat operator, translation operators, singular integral operators, etc. acting in these spaces.

The materials presented as preliminaries would be very instructive for the study of Morrey spaces as well as various operators such as the heat operator, translation operators, singular integral operators, etc. acting in these spaces.

Reviewer: H.Tanabe (Toyonaka)

##### MSC:

35Q30 | Navier-Stokes equations |

47B38 | Linear operators on function spaces (general) |

35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |

##### Keywords:

\(C_ 0\)-semigroup; well-posedness; nonstationary free Navier-Stokes equation; Morrey spaces; initial velocity; local solution; global solution; initial vorticity; large time decay; regularity
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\textit{T. Kato}, Bol. Soc. Bras. Mat., Nova Sér. 22, No. 2, 127--155 (1992; Zbl 0781.35052)

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