Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD
Remarks on singularities, dimension and energy dissipation for ideal hydrodynamics and MHD
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DOI：
10.1007/s002200050067
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发表时间：
1997-03-01
影响因子：
2.4
通讯作者：
Steele, G
中科院分区：
文献类型：
--
作者：
Caflisch, RE;Klapper, I;Steele, G
对于不可压缩欧拉方程的弱解，如果速度位于 Besov 空间 B-s(3) 且 s 大于 1/3，则存在能量守恒。 B-s(P) 由以 L-P 范数测量的 Lip(s)（即指数 s 连续的 Holder）函数组成。此处，该结果应用于速度场 Lip(alpha(0))，除了在其上为 Lip(alpha(1)) 的一组余维 kappa(1) 上，其均匀性将在下面精确说明。我们证明，Frisch-Parisi 多重分形形式对于这一函数是有效的（至少在一个方向上），并且如果 min(alpha)(3 alpha + kappa(alpha)) > 1，则存在能量守恒。针对能量和螺旋度的不可压缩理想 MHD（即零粘度和电阻率）方程得出了类似的守恒结果。此外，推广了理想流体动力学的 Beale-Kato-Majda 条件，推导了理想 MHD 奇点发展的必要条件。
For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B-s(3) with s greater than 1/3. B-s(P) consists of functions that are Lip(s) (i.e., Holder continuous with exponent s) measured in the L-P norm. Here this result is applied to a velocity field that is Lip(alpha(0)) except on a set of co-dimension kappa(1) on which it is Lip(alpha(1)), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if min(alpha)(3 alpha + kappa(alpha)) > 1. Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity. In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.