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July 2015, 35(7): 3039-3057. doi: 10.3934/dcds.2015.35.3039
Time-dependent singularities in the Navier-Stokes system
Grzegorz Karch 1, and Xiaoxin Zheng 2,
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław
Instytut Matematyczny, Uniwersytet Wroc lawski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Received April 2014 Revised October 2014 Published January 2015
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: time-dependent singularity, Navier-Stokes system, incompressible fluid, Slezkin-Landau solutions..
Mathematics Subject Classification: Primary: 35Q40; Secondary: 76D0.
Citation: Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039
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Grzegorz Karch Xiaoxin Zheng