## Hokkaido Mathematical Journal

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

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

Reinhard FARWIG and Yoshikazu GIGA

#### 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