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

尽管描述不可压缩流体运动的Navier-Stokes方程的数学研究已有很长的历史，但其理论尚未完成。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。弱解仅在某种弱意义上满足方程，因此它可能有奇点。如果解是光滑的，那么它相对于时间变量是守恒的。对于一些弱解，我们可以证明能量的非增性，但不能确定它们是否守恒。它被称为“能源不平等”。对于弱解，时间导数的可积性仅从定义上是不清楚的。这就是为什么我们不能表现出能量的保存。这一事实表明，时间导数是关于“时间”的单一度量。因此，我们必须考虑“奇异”部分的积分，以表明能量的守恒。我们知道，对于离散Morse流方法构造的弱解，可以利用分数时间导数来估计从下到下的能量减少。这是一个新的估计。这项研究的目的是研究对任何弱解进行这种求精的可能性。准确地说，我们澄清了以下事实。我们把弱解看作映射到平方可积函数空间的函数。能量的降低与解的1/2时间差在Nikol‘skii意义下的极限有关。如果我们假设有或没有某一速度的极限为零，那么我们可以用一个附加的项来证明能量恒等式，它补偿了能量的减少。此外，在不假定极限存在的情况下，我们可以用另一个不同表达式的附加项来表示能量恒等式。表达的差异来自于时差极限收敛的拓扑的差异。

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.