Hard Problems in Hard Analysis

我们调查了四个主要的未解决的问题，或接近谐波分析。这些问题包括Kakeya问题、Lipschitz微分问题、约束问题和Navier-Stokes方程的全局可解性。对于Kakeya问题，我们继续了Tao在和-差方法中改进指数的工作;对于Lipschitz向量场微分问题，我们试图应用我们在任意方向上的极大函数上的工作来理解反例的限制是什么，并且很显然不可能存在。对于约束问题，我们尝试将新的结果应用于Kakeya，并更好地理解Bourgain将Kakeya结果转化为约束结果的机制。对于Navier Stokes，我们首先将所有的结果离散为一种广义小波系数的形式。在与Pavlovic的合作中，这已经将Caffarelli-Kohn-Nirenberg定理推广到了超耗散的情况，我们希望从这个角度发现非线性项级联效应的一种局部色散性质。然后，我们希望将粘土问题与一个并矢模型联系起来，在这个模型中，色散是给定的。可以想象，我们不太可能解决所有这些问题。分析涉及通过检查所有部分的贡献来证明有趣系统的估计。一个这样的系统是纳维尔斯托克斯方程，它控制着不可压缩粘性流体的行为。一个重要的开放性问题是，这个方程是否从光滑的初始数据开始，可以在没有强迫项的情况下发展奇点。这就像一个人的浴缸里突然刮起的旋风。这似乎不太可能，但分析工具还没有强大到足以排除它。我们的方法是将问题离散化，也就是说，试图用一个有限数量的对象来近似问题，并通过组合学来研究这些对象之间可能的相互作用。大多数物理产生的数学都可以这样看待，因为物质不是连续的，而是由粒子组成的。我们将研究上述问题和其他一些可以用同样观点来处理的重要问题。

We investigate four major unsolved problems in or bordering on harmonicanalysis. These are the Kakeya problem, the Lipschitz differentiationproblem, the restriction problem, and global solvability for theNavier Stokes equation. For the Kakeya problem, we continue ourwork with Tao on improving exponents in the sums-differences approach.For the problem of differentiation by Lipschitz vector fields, we tryto apply our work on maximal functions in arbitrary directions tounderstand what are the limitations on a counterexample and hopefullythat none can exist. For the restriction problem, we try to applythe new results on Kakeya and to better understand Bourgain'smachine for converting Kakeya results to ones about restriction.For Navier Stokes, we first discretize everything in theform of a kind of generalized wavelet coefficients. In work withPavlovic, this has already produced a generalization ofthe Caffarelli-Kohn-Nirenberg theorem to the case of hyperdissipation.We hope from this point of view to discover a sort of local dispersionproperty for the cascading effect from the nonlinear term. Then wehope to tie in the Clay problem with a dyadic model in which thisdispersion is a given. As might be imagined, we are unlikely to solveall these problems.Analysis concerns the proof of estimates on interesting systems byexaming the contributions of all their parts. One such system isthe Navier Stokes equation which governs the behaviour of incompressibleviscous fluids. An important open problem is whether this equationstarting with smooth initial data can develop singularities withouta forcing term. This would akin to a cyclone beginning spontaneouslyin one's bathtub. It seems rather unlikely but the tools of analysisare not yet strong enough to rule it out. Our approach is todiscretize the problem, that is to try to approximate the problemby one about a finite number of objects and investigate possibleinteractions of those objects by means of combinatorics. Mostphysically arising mathematics can be looked at this way because matteris not continuous but rather composed of particles. We will workon the above problem and some other important problems which may beapproached with the same point of view.