### Abstract

In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u_{0}, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L^{2 / α} over(H, ̇)^{α} × L^{2 / β} over(H, ̇)^{β} × P → R, where α, β belong to [ 0, 1 ]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].

Original language | English (US) |
---|---|

Pages (from-to) | 373-428 |

Number of pages | 56 |

Journal | Journal of Differential Equations |

Volume | 226 |

Issue number | 2 |

DOIs | |

State | Published - Jul 15 2006 |

### Fingerprint

### Keywords

- Leray solutions
- Multipliers
- Navier-Stokes equations
- Paraproduct
- Sobolev spaces
- Weak-strong uniqueness

### ASJC Scopus subject areas

- Analysis

### Cite this

**Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations.** / Germain, Pierre.

Research output: Contribution to journal › Article

*Journal of Differential Equations*, vol. 226, no. 2, pp. 373-428. https://doi.org/10.1016/j.jde.2005.10.007

}

TY - JOUR

T1 - Multipliers, paramultipliers, and weak-strong uniqueness for the Navier-Stokes equations

AU - Germain, Pierre

PY - 2006/7/15

Y1 - 2006/7/15

N2 - In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u0, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L2 / α over(H, ̇)α × L2 / β over(H, ̇)β × P → R, where α, β belong to [ 0, 1 ]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].

AB - In this article, we describe spaces P such that: if u is a weak (in the sense of Leray [J. Leray, Sur le mouvement d'un fluide visqueux remplissant l'espace, Acta Math. 63 (1934) 193-248]) solution of the Navier-Stokes system for some initial data u0, and if u belongs to P, then u is unique in the class of weak solutions. We say then that weak-strong uniqueness holds. It turns out that the proof of such results relies on the boundedness of a trilinear functional F : L2 / α over(H, ̇)α × L2 / β over(H, ̇)β × P → R, where α, β belong to [ 0, 1 ]. In order to find optimal conditions for the boundedness of F, we are led to describing spaces of multipliers and of paramultipliers (that is, functions which map, by classical pointwise product or by paraproduct, a given Sobolev spaces in another given Sobolev space). The study of these spaces enables us to give conditions for weak-strong uniqueness which generalise all previously known results, from the famous Serrin criterion [J. Serrin, The initial value problem for the Navier-Stokes equations, in: R.E. Langer (Ed.), Nonlinear Problems, Univ. of Wisconsin Press, 1963, pp. 69-98], to the recent conditions formulated by Lemarié-Rieusset [P.-G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Chapman and Hall, 2003].

KW - Leray solutions

KW - Multipliers

KW - Navier-Stokes equations

KW - Paraproduct

KW - Sobolev spaces

KW - Weak-strong uniqueness

UR - http://www.scopus.com/inward/record.url?scp=33646507669&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33646507669&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2005.10.007

DO - 10.1016/j.jde.2005.10.007

M3 - Article

VL - 226

SP - 373

EP - 428

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 2

ER -