Research on a refinement of the energy inequality for weak solutions to the Navier-Stokes equations

描述不可压缩流体运动的Navier-Stokes方程的数学研究由来已久，但其理论研究尚未完成。特别是关于弱解的正则性和唯一性问题，我们只有部分答案。一个弱解只在某种弱意义下满足方程，因此它可能有奇点。如果一个解是光滑的，那么它相对于时间变量保持能量。对于某些弱解，我们可以证明其能量不增，但不确定它们是否保持能量。这就是所谓的“能源不平等”。“对于弱解，时间导数的可积性仅从定义中不清楚。这就是为什么我们不能显示能量的保存。这一事实表明，时间导数是关于“时间”的奇异度量。我们知道，对于离散莫尔斯流方法构造的弱解，利用分数阶时间导数可以从下面估计能量的减少。这是一个新的估计。本研究的目的是研究对任何弱解进行这种加细的可能性。准确地说，我们澄清了以下事实。我们考虑弱解作为映射到平方可积函数空间的函数。能量的降低与解的Nikol'skii意义下的1/2时间差的极限有关。如果我们假设极限为零，有或没有速度，那么我们可以用一个额外的项来补偿能量的减少。而且，在不假设极限存在的情况下，我们可以用另一个不同表达式的附加项来证明能量恒等式。这种表达上的差异来自于时间差极限收敛的拓扑上的差异。

Though the mathematical research on the Navier-Stokes equations that describe the motion of incompressible fluid has a long history, the theory has not completed yet. In particular concerning the problems of the regularity and uniqueness of weak solutions, we have only partial answers. A weak solution satisfies equations only in some weak sense, and therefore it may have singular points. If a solution is smooth, then it preserves energy with respect to time variable. For some weak solutions we can show the non-increasing property of energy, but it is uncertain whether they preserve energy or not. It is called the "energy inequality." For weak solutions the integrability of time-derivative is unclear only from the definition. This is why we cannot show the preservation of energy. This fact suggests that time-derivative is a singular measure with respect to "time". Consequently we must consider the integral of this "singular" part to show the preservation of energy.We know that it is possible to estimate the decrease of energy from below by use of fractional time-derivative for the weak solution constructed by the method of discrete Morse flow. This is a new estimate. The purpose of this research is to study the possibility of such a refinement for any weak solution. For precise, we clarified the following facts. We consider weak solutions as functions which map to the space of square-integrable functions. Then the decrease of energy is related to the limit of the 1/2-time-difference of solutions in the sense of Nikol'skii. If we assume that the limit is zero with or without some speed, then we can show the energy identity with an additional term which compensates the decrease of energy. Furthermore without the assumption of the existence of limit, we can show the energy identity with another additional term of different expression. The difference of expression comes from that of topology of convergence of limit of time-difference.