In this paper, we constructed solutions to the Cauchy problem of the $ n $-dimensional incompressible Navier-Stokes equations, which were classical strain flows and extended ones presented in Majda and Bertozzi's classical book Vorticity and Incompressible Flow. Then we studied the asymptotic behaviors on these solutions. More precisely, under specific constraints of the initial data, we proved that the constructed solutions converged to the corresponding ones of the Euler equations uniformly on the time $ t\in [0, +\infty) $ as the viscosity vanished. This was very different from the previous result which held on $ t\in [0, T] $ for any $ T > 0 $. Furthermore, we showed that for some specific initial data, the constructed solutions converged to the corresponding ones of the Euler equations when $ t\to \infty $ for any fixed viscosity and the convergence rate was exponential one.
Citation: Quansen Jiu, Zhihong Yan. On strain flow solutions to the Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2026, 18(4): 877-888. doi: 10.3934/cam.2026034
In this paper, we constructed solutions to the Cauchy problem of the $ n $-dimensional incompressible Navier-Stokes equations, which were classical strain flows and extended ones presented in Majda and Bertozzi's classical book Vorticity and Incompressible Flow. Then we studied the asymptotic behaviors on these solutions. More precisely, under specific constraints of the initial data, we proved that the constructed solutions converged to the corresponding ones of the Euler equations uniformly on the time $ t\in [0, +\infty) $ as the viscosity vanished. This was very different from the previous result which held on $ t\in [0, T] $ for any $ T > 0 $. Furthermore, we showed that for some specific initial data, the constructed solutions converged to the corresponding ones of the Euler equations when $ t\to \infty $ for any fixed viscosity and the convergence rate was exponential one.
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Quansen Jiu, Zhihong Yan. On strain flow solutions to the Navier-Stokes equations[J]. Communications in Analysis and Mechanics, 2026, 18(4): 877-888. doi: 10.3934/cam.2026034
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