发表机构
Università degli Studi dell’Aquila; Technische Universität Berlin; Universität Basel（阿奎拉大学; 柏林工业大学; 巴塞尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究Kraichnan型粗糙输运噪声驱动的二维活性标量SPDE，证明反常正则化与可积性，并在Euler情形建立涡度反常耗散。
AI 中文摘要
我们研究由Kraichnan型不可压缩输运噪声驱动的二维活性标量所关联的SPDE，其中正则性指数为$\alpha\in (0,1)$；我们的例子包括Euler、SQG和IPM系统。我们研究在线性Kraichnan模型中已被充分理解的若干“湍流”现象是否在非线性设定中持续存在，并一致地适用于消失粘性近似。首先，对于适当的$\alpha$值和$L^p_x$中的初始数据，我们建立了反常正则化估计，其以正则性$\beta=\beta(\alpha,p)>0$的适当端点Besov型空间度量。值得注意的是，这些结果允许参数$\alpha,p$的某些尺度超临界区域；另一方面，对于（次）临界参数，我们恢复了与线性情形相同的正则性指数$\beta=1-\alpha$。其次，在（次）临界情形中，我们进一步证明了反常可积性，即解在正时间瞬时成为$L^\infty_x$值，且一致地适用于粘性；此外，在这种情况下，我们建立了无粘SPDE解的强存在性和路径唯一性，这些解被恢复为唯一的消失粘性极限。最后，在二维Euler情形中，对于$\alpha\in (0,1/2)$，我们建立了涡度的反常耗散以及反常正则化的锐利性。
英文摘要
We study SPDEs associated with $2$D active scalars driven by incompressible transport noise of Kraichnan type, with regularity exponent $α\in (0,1)$; our examples include the Euler, SQG and IPM systems. We investigate whether a number of ``turbulent'' phenomenologies, which are well understood in the linear Kraichnan model, persist in this nonlinear setting, uniformly in vanishing viscosity approximations. First, for suitable values of $α$ and initial data in $L^p_x$, we establish anomalous regularization estimates, measured in appropriate endpoint Besov-type spaces of regularity $β=β(α,p)>0$. Remarkably, these results allow for some scaling supercritical regimes of the parameters $α,p$; on the other hand, for (sub)critical parameters, we recover the same regularity exponent $β=1-α$ as in the linear case. Second, in the (sub)critical case, we further prove anomalous integrability, namely solutions becoming instantaneously $L^\infty_x$-valued at positive times, uniformly in the viscosity; moreover, in this case we establish strong existence and pathwise uniqueness of solutions to the inviscid SPDE, which are recovered as the unique vanishing viscosity limit. Finally, in the $2$D Euler case, for $α\in (0,1/2)$, we establish anomalous dissipation of enstrophy and sharpness of anomalous regularization.
Comments50 pages, 2 figures. All comments are welcome!
