具有双尺度遗传粘性的纳维-斯托克斯方程：超临界范数膨胀与临界空间中的全局适定性
The Navier-Stokes equations with dual-scale hereditary viscosity: supercritical norm inflation and global well-posedness in critical spaces
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中文总结 AI 辅助
研究具有双尺度遗传粘性的纳维-斯托克斯方程柯西问题，通过建立衰减估计确定临界阈值。超临界时利用映射和不等式证不适定性，拓扑极限时借助插值证明时间平滑作用，最终为特定小初始数据建立全局适定性。
中文摘要 AI 辅助
本文研究了由双尺度遗传记忆控制的不可压缩流体流动的柯西问题，它是经典纳维-斯托克斯方程的粘弹性变体，可捕捉异常动量传输。非局部耗散打破了精确的全局尺度不变性，需要在Hörmander类$S^{-2}_{1,0}$内进行伪微分分析，其中空间梯度会导致分数时间惩罚。我们严格建立了$L^q - L^p$衰减估计，并确定了临界勒贝格阈值$p_c = N (\frac{1+\alpha_\infty}{1-\alpha_\infty})$。在超临界区域$1 < p < p_c$，通过将调频双线性流直接映射到傅里叶空间来绕过空间局部化的损失；利用伯恩斯坦不等式，证明了原点处的瞬时范数膨胀并确认了内在不适定性。相反，在拓扑极限$p \to \infty$时，证明双尺度记忆在最大临界贝索夫空间$\dot{B}^{-1}_{\infty, \infty}$中从结构上防止了传统流体中观察到的崩溃。通过在Bony的准微分演算中利用不对称插值，证明了时间平滑超过了高高对流共振级联。这种微妙的分析平衡将不适定性限制在贝索夫拓扑不可分离的高频尾部，从而为在$\dot{B}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$中具有高频附着的小初始数据建立了全局时间的哈达玛适定性，该适定性区域比可分离的小贝索夫闭包$\dot{b}^{-\kappa}_{\infty, \infty}(\mathbb{R}^N)$严格更宽，其中$\kappa = \frac{1-\alpha_\infty}{1+\alpha_\infty}$。
英文摘要
This manuscript investigates the Cauchy problem for an incompressible fluid flow governed by a dual-scale hereditary memory, representing a viscoelastic variant of the classical Navier-Stokes equations that captures anomalous momentum transport. The non-local dissipation breaks exact global scale invariance, dictating a pseudo-differential analysis within the Hörmander class $S^{-2}_{1,0}$ where the spatial gradient induces a fractional temporal penalty. We rigorously establish $L^q-L^p$ decay estimates and identify the critical Lebesgue threshold $p_c = N (\frac{1+α_\infty}{1-α_\infty})$. In the supercritical regime $1 < p < p_c$, we bypass the loss of spatial localization by mapping the frequency-modulated bilinear flow directly into Fourier space; by utilizing Bernstein's inequalities, we prove instantaneous norm inflation at the origin and confirm intrinsic ill-posedness. Conversely, in the topological limit $p \to \infty$, we demonstrate that the dual-scale memory structurally prevents the collapse traditionally observed for classical fluids within the maximal critical Besov space $\dot{B}^{-1}_{\infty, \infty}$. By exploiting an asymmetric interpolation within Bony's para-differential calculus, we prove that the temporal smoothing overpowers the high-high convective resonant cascade. This delicate analytical balance confines ill-posedness to the non-separable high-frequency tail of the Besov topology, thereby establishing global-in-time Hadamard well-posedness for small initial data possessing high-frequency adherence within $\dot{B}^{-κ}_{\infty, \infty}(\mathbb{R}^N)$, a well-posedness regime strictly broader than the separable little Besov closure $\dot{b}^{-κ}_{\infty, \infty}(\mathbb{R}^N)$, where $κ= \frac{1-α_\infty}{1+α_\infty}$.
