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arXiv 2406.09587math.AP
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无界区域上 Navier-Stokes-Fourier 系统解的长时间行为

On the long time behaviour of solutions to the Navier-Stokes-Fourier system on unbounded domains

Elisabetta Chiodaroli, Eduard Feireisl

AI总结:

本文研究 $R^3$ 无界区域上满足远场条件的 NSF 系统,证明其任意全局弱解按遍历平均收敛至零密度平衡态,并据此推出总动量不可能全局守恒。

摘要:

我们考虑欧几里得空间 $R^3$ 中一个无界区域上的 Navier-Stokes-Fourier 系统,并为相变量补充远场条件,具体为:当 $|x|\to\infty$ 时,$\rho\to 0$、$\vartheta\to\vartheta_\infty$、$u\to 0$。我们研究解的长时间行为,并证明 NSF 系统的任意时间全局弱解在时间趋于无穷时,都按遍历平均意义趋于平衡态 $\rho_s=0$、$\vartheta_s=\vartheta_\infty$、$u_s=0$。作为该收敛结果与总质量守恒相结合的推论,我们可以表明:时间全局弱解的总动量永远不会全局守恒。

英文摘要:

We consider the Navier-Stokes-Fourier system on an unbounded domain in the Euclidean space $R^3$, supplemented by the far field conditions for the phase variables, specifically: $ρ\to 0,\ \vartheta \to \vartheta_\infty, \ u \to 0$ as $\ |x| \to \infty$. We study the long time behaviour of solutions and we prove that any global-in-time weak solution to the NSF system approaches the equilibrium $ρ_s = 0,\ \vartheta_s = \vartheta_\infty,\ u_s = 0$ in the sense of ergodic averages for time tending to infinity. As a consequence of the convergence result combined with the total mass conservation, we can show that the total momentum of global-in-time weak solutions is never globally conserved.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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