论文标题
通过涡旋的不可压缩欧拉方程的弱解决方案节能
Energy conservation of weak solutions for the incompressible Euler equations via vorticity
论文作者
论文摘要
由Cheskidov，Lopes Filho，Nussenzveig Lopes和Shvydkoy的作品的动机[8，Commun。数学。物理。 348：129-143，2016]和陈和Yu [5，J。Math。 Pures Appl。 131：1-16，2019]，我们解决了$ l^p $涡度控制如何影响本文中不可压缩的均匀和非均匀欧拉方程的能源保护。对于周期性域或整个空间中的均匀流，我们为标准提供了一个独立的证据，$ω= \ text {curl} u \ in l^{3}（0，t; t; l^{\ frac {3nn} {3n} {n+2}}}}}}}}}（ω）\，（ω）\，（ω） Onsager临界时空空间。 Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and $\nabla\sqrtρ$ belongs to $L^{\infty}(0,T;L^{n}(\mathbb{T}^{n}))\,(n=2,3)$, which gives an affirmative answer to a problem proposed陈和Yu在[5]中。
Motivated by the works of Cheskidov, Lopes Filho, Nussenzveig Lopes and Shvydkoy in [8, Commun. Math. Phys. 348: 129-143, 2016] and Chen and Yu in [5, J. Math. Pures Appl. 131: 1-16, 2019], we address how the $L^p$ control of vorticity could influence the energy conservation for the incompressible homogeneous and nonhomogeneous Euler equations in this paper. For the homogeneous flow in the periodic domain or whole space, we provide a self-contained proof for the criterion $ω=\text{curl}u\in L^{3}(0,T;L^{\frac{3n}{n+2}}(Ω))\,(n=2,3)$, which generalizes the corresponding result in [8] and can be viewed as in Onsager critical spatio-temporal spaces. Regarding the nonhomogeneous flow, it is shown that the energy is conserved as long as the vorticity lies in the same space as before and $\nabla\sqrtρ$ belongs to $L^{\infty}(0,T;L^{n}(\mathbb{T}^{n}))\,(n=2,3)$, which gives an affirmative answer to a problem proposed by Chen and Yu in [5].
