Hokkaido Mathematical Journal

Well-chosen weak solutions of the instationary Navier-Stokes system and their uniqueness

Reinhard FARWIG and Yoshikazu GIGA

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Abstract

We clarify the notion of well-chosen weak solutions of the instationary Navier-Stokes system recently introduced by the authors and P.-Y. Hsu in the article {\em Initial values for the Navier-Stokes equations in spaces with weights in time, Funkcialaj Ekvacioj} (2016). Well-chosen weak solutions have initial values in $L^{2}_{\sigma}(\Omega)$ contained also in a quasi-optimal scaling-invariant space of Besov type such that nevertheless Serrin's Uniqueness Theorem cannot be applied. However, we find universal conditions such that a weak solution given by a concrete approximation method coincides with the strong solution in a weighted function class of Serrin type.

Article information

Source
Hokkaido Math. J., Volume 47, Number 2 (2018), 373-385.

Dates
First available in Project Euclid: 18 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1529308824

Digital Object Identifier
doi:10.14492/hokmj/1529308824

Mathematical Reviews number (MathSciNet)
MR3815298

Zentralblatt MATH identifier
06901711

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 35B65: Smoothness and regularity of solutions 76D05: Navier-Stokes equations [See also 35Q30] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

Keywords
Navier-Stokes equations initial values strong $L^s_\alpha(L^q)$-solutions well-chosen weak solutions Serrin's uniquenes theorem

Citation

FARWIG, Reinhard; GIGA, Yoshikazu. Well-chosen weak solutions of the instationary Navier-Stokes system and their uniqueness. Hokkaido Math. J. 47 (2018), no. 2, 373--385. doi:10.14492/hokmj/1529308824. https://projecteuclid.org/euclid.hokmj/1529308824


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