Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Time-dependent singularities in the Navier-Stokes system

Pages: 3039 - 3057, Volume 35, Issue 7, July 2015      doi:10.3934/dcds.2015.35.3039

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Grzegorz Karch - Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email)
Xiaoxin Zheng - Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland (email)

Abstract: We show that, for a given Hölder continuous curve in $\{(\gamma(t),t)\,:\, t>0\} \subset \mathbb{R}^3 \times \mathbb{R}^+$, there exists a solution to the Navier-Stokes system for an incompressible fluid in $\mathbb{R}^3$ which is regular outside this curve and singular on it. This is a solution of the homogeneous system outside the curve, however, as a distributional solution on $\mathbb{R}^3 \times \mathbb{R}^+$, it solves an analogous Navier-Stokes system with a singular force concentrated on the curve.

Keywords:  Navier-Stokes system, incompressible fluid, time-dependent singularity, Slezkin-Landau solutions.
Mathematics Subject Classification:  Primary: 35Q40; Secondary: 76D05.

Received: April 2014;      Revised: October 2014;      Available Online: January 2015.