傅里叶热传导可压缩流体的Cattaneo-Christov近似
The Cattaneo-Christov approximation of Fourier heat-conductive compressible fluids
AI总结：
本文研究三维及以上空间中热传导可压缩流动的NSC近似模型，在临界正则框架下证明其小初值全局适定性，得到大时间渐近与NSF系统的误差估计，首次给出三维不适定初值下该松弛过程的强收敛结果。
摘要：
我们研究了$\u211d^d$（$d\geq3$）中的Navier-Stokes-Cattaneo-Christov（NSC，纳维-斯托克斯-卡塔内奥-克里斯托夫）系统，这是一种热传导可压缩流动模型，可作为Navier-Stokes-Fourier（NSF，纳维-斯托克斯-傅里叶）系统的有限传播速度近似。由于Oldroyd（奥尔德罗伊德）上随流导数的存在，NSC系统存在“双曲性缺失”的问题，这使得其适定性的建立极具挑战性，尤其是在多维情形下。本文在临界正则性函数框架内，证明了NSC系统对于常平衡态的小扰动初值具有全局时间适定性，且该结果关于近似参数$\varepsilon>0$一致成立。基于这一结果，我们进一步得到了NSC系统的精确大时间渐近行为，并针对所有时间$t>0$，推导了NSC系统与NSF系统解之间的定量误差估计。据我们所知，本研究首次给出了三维情形下、针对不适定初值的该松弛过程的强收敛结果。\nNSC系统是部分耗散的，同时包含部分扩散和部分阻尼机制。为处理这些特性并保证解的大时间稳定性，我们基于欠强制性理论构造了频率局部化的扰动能量泛函。更具体地说，我们的分析依赖于将频率空间划分为低、中、高三个不同的频段。在每个频段内，我们引入了有效未知量和Lyapunov（李雅普诺夫）泛函，揭示了谱意义上预期的耗散结构。
英文摘要：
We investigate the Navier-Stokes-Cattaneo-Christov (NSC) system in $\mathbb{R}^d$ ($d\geq3$), a model of heat-conductive compressible flows serving as a finite speed of propagation approximation of the Navier-Stokes-Fourier (NSF) system. Due to the presence of Oldroyd's upper-convected derivatives, the system (NSC) exhibits a \textit{lack of hyperbolicity} which makes it challenging to establish its well-posedness, especially in multi-dimensional contexts. In this paper, within a critical regularity functional framework, we prove the global-in-time well-posedness of (NSC) for initial data that are small perturbations of constant equilibria, uniformly with respect to the approximation parameter $\varepsilon>0$. Then, building upon this result, we obtain the sharp large-time asymptotic behaviour of (NSC) and, for all time $t>0$, we derive quantitative error estimates between the solutions of (NSC) and (NSF). To the best of our knowledge, our work provides the first strong convergence result for this relaxation procedure in the three-dimensional setting and for ill-prepared data. The (NSC) system is partially dissipative and incorporates both partial diffusion and partial damping mechanisms. To address these aspects and ensure the large-time stability of the solutions, we construct localized-in-frequency perturbed energy functionals based on the hypocoercivity theory. More precisely, our analysis relies on partitioning the frequency space into \textit{three} distinct regimes: low, medium and high frequencies. Within each frequency regime, we introduce effective unknowns and Lyapunov functionals, revealing the spectrally expected dissipative structures.
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