AI 中文总结
本文研究非自治松弛系数的部分耗散双曲系统，在临界正则性下证明全局强解存在并给出最优代数衰减率，通过引入混合 Besov 空间处理时变频率阈值，揭示时间依赖系数对耗散和长时间动力学的影响。
AI 中文摘要
我们研究了在 $\mathbb{R}^d$ ($d\geq1$) 中具有非自治松弛系数的拟线性可对称化部分耗散双曲系统。对于满足所谓的 Shizuta-Kawashima (SK) 和熵条件的系统，我们在临界正则性框架下建立了全局强解的存在性。当初始数据额外在较低正则性范数下有界时，我们证明相应的解以最优代数衰减速率收敛到平衡态。此外，我们表明解的保守部分渐近地表现为非自治抛物方程的解。我们的结果适用于速度方程中具有时间依赖阻尼系数 $\frac{K}{(1+t)^{\alpha}}$ ($\alpha<1$, $K>0$ 或 $\alpha=1$, $K\gg 1$) 的可压缩欧拉系统。自治理论中自然的低/高频分裂在非自治情形下仍然存在，但频率阈值随时间演化。为了处理这种移动的频率结构，我们引入了一类新的适应于时间依赖阈值的混合 Besov 空间，并在每个频率区域推导了 hypocoercive 估计。我们的结果揭示了一般时间依赖松弛系数对耗散和长时间动力学的定性和定量影响。
英文摘要
We study quasilinear symmetrizable partially dissipative hyperbolic systems with non-autonomous relaxation coefficients in $\mathbb{R}^d$ ($d\geq1$). The existence of global strong solutions is established in a critical regularity setting for systems satisfying the so-called Shizuta-Kawashima (SK) and entropy conditions. When the initial data are additionally bounded in a lower-regularity norm, we prove that the corresponding solutions converge to equilibrium at optimal algebraic decay rates. Furthermore, we show that the conservative part of the solution behaves asymptotically as the solution of a non-autonomous parabolic equation. Our results apply to the compressible Euler system with the time-dependent damping coefficient $\frac{K}{(1+t)^α}$ ($α<1$, $K>0$ or $α=1$, $K\gg 1$) in the velocity equation. The natural low/high-frequency splitting of the autonomous theory persists in the non-autonomous setting, but with a frequency-threshold that evolves in time. To handle this moving frequency structure, we introduce a new class of hybrid Besov spaces adapted to time-dependent thresholds and derive hypocoercive estimates in each frequency regime. Our results reveal the qualitative and quantitative effects of general time-dependent relaxation coefficients on dissipation and large-time dynamics.
