二维完美流体中的对数螺线
Logarithmic spirals in 2d perfect fluids
AI总结：
研究二维欧拉方程对数螺线解，证明局部与全局适定性，刻画长时间行为：解要么爆破要么完全均匀化，并实现涡片动力学为光滑解的极限。
摘要：
我们研究二维不可压缩欧拉方程的对数螺线解，这些解在$\mathbb{S}$上求解一个非线性输运系统。我们证明该系统在$L^p, p\geq 1$以及原子测度（即对数螺线涡片）情形下是局部适定的。特别地，我们将对数涡片的动力学实现为对数解（可能在角度上光滑）的良定义极限。此外，我们的公式不仅为不对称多分支对数螺线涡片的存在性和分岔提供了简单证明，还为研究自相似动力学的渐近稳定性提供了框架。\n我们给出了对数螺线长时间行为的完整刻画。我们证明了有界对数螺线以及允许至多对数奇点的数据的全局适定性。这源于观察到原点周围涡量的局部环量是时间的严格单调量。然后我们能够证明长时间行为中的二分法：解要么爆破（有限或无限时间），要么完全均匀化。特别地，有界对数螺线应收敛到常数稳态。对于对数螺线涡片，二分法被证明更为剧烈，其中仅可能且确实发生有限时间爆破或流体的完全均匀化。
英文摘要：
We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on $\mathbb{S}$. We show that this system is locally well-posed in $L^p, p\geq 1$ as well as for atomic measures, that is logarithmic spiral vortex sheets. In particular, we realize the dynamics of logarithmic vortex sheets as the well-defined limit of logarithmic solutions which could be smooth in the angle. Furthermore, our formulation not only allows for a simple proof of existence and bifurcation for non-symmetric multi branched logarithmic spiral vortex sheets but also provides a framework for studying asymptotic stability of self-similar dynamics. We give a complete characterization of the long time behavior of logarithmic spirals. We prove global well-posedness for bounded logarithmic spirals as well as data that admit at most logarithmic singularities. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (either in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals should converge to constant steady states. For logarithmic spiral sheets, the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.
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