时间中非平凡绝对连续部分的异常耗散度量
Nontrivial absolutely continuous part of anomalous dissipation measures in time
AI总结：
研究通过构建新的纳维-斯托克斯方程解，揭示了时间中非平凡绝对连续部分的异常耗散度量及湍流零定律关联。
摘要：
我们积极回答了[Bruè, De Lellis, 2023]中的问题2.2和问题2.3，在维度4中通过构建新的解于强制的4d不可压缩纳维-斯托克斯方程的解，这些解表现出异常耗散，与湍流的零定律[K41]相关。我们还证明了4d纳维-斯托克斯方程的唯一光滑解v_ν在粘性参数ν→0时，以L^∞-弱*收敛到强制欧拉方程的解v_0。此外，序列ν|∇v_ν|^2在测度意义下弱*收敛（至多子序列）到μ∈M((0,1)×T^4)，并且μ_T=π_#μ具有非平凡的绝对连续部分，其中π是时间变量的投影。此外，我们还证明μ在H^{-1}_{t,x}意义上接近强制欧拉方程解v_0的Duchon-Robert分布mathcal{D}[v_0]。最后，v_0的动能分布时间平滑。我们的结果依赖于一个新的异常耗散结果，即对于具有无散3d自主速度场的对流-扩散方程，以及对3+1/2维不可压缩纳维-斯托克斯方程的研究。这项研究激发了一些开放性问题。
英文摘要：
We positively answer Question 2.2 and Question 2.3 in [Bruè, De Lellis, 2023] in dimension $4$ by building new examples of solutions to the forced $4d$ incompressible Navier-Stokes equations, which exhibit anomalous dissipation, related to the zeroth law of turbulence [K41]. We also prove that the unique smooth solution $v_ν$ of the $4d$ Navier--Stokes equations with time-independent body forces is $L^\infty$-weakly* converging to a solution of the forced Euler equations $v_0$ as the viscosity parameter $ν\to 0$. Furthermore, the sequence $ν|\nabla v_ν|^2$ is weakly* converging (up to subsequences), in the sense of measure, to $μ\in \mathcal{M} ((0,1) \times \mathbb{T}^4)$ and $μ_T = π_{\#} μ$ has a non-trivial absolutely continuous part where $π$ is the projection onto the time variable. Moreover, we also show that $μ$ is close, up to an error measured in $H^{-1}_{t,x}$, to the Duchon--Robert distribution $\mathcal{D}[v_0]$ of the solution to the $4d$ forced Euler equations. Finally, the kinetic energy profile of $v_0$ is smooth in time. Our result relies on a new anomalous dissipation result for the advection--diffusion equation with a divergence free $3d$ autonomous velocity field and the study of the $3+\frac{1}{2} $ dimensional incompressible Navier--Stokes equations. This study motivates some open problems.
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