Active Scalar Equations and a Geodynamo Model

主动标量方程描述了流体动力学中出现的一些物理现象。由于它们在物理上的重要性和具有挑战性的数学性质，关于这些方程的文献非常广泛。然而，与方程的非线性有关的许多问题仍然没有解决，特别是当漂移速度U是非局部的时候。这个项目解决了其中的一些问题，特别强调了一些例子，其中算子M编码了与活动标量有关的U的物理问题是奇异的和无界的。在一定分数阶拉普拉斯级数范围内，研究了奇异活动标量方程中扩散效应与非线性的相互作用。对于一定的幂次，将证明系统是非线性不稳定的；对于较低的幂次，系统是利普希茨不适定的。这些结果将通过构造演化方程的不稳定特征值来获得，演化方程线性化到一个适当的平衡。连分式技术将被用来产生不稳定特征值的增长率作为物理参数的函数的显式下界。该项目还将研究消失扩散率的高度奇异极限。这个极限通常在物理上是适当的，并且可能与由活动标量方程的弱解模拟的湍流现象密切相关。该项目涉及一般类型的活动标量方程：特殊的物理例子包括不可压缩多孔介质方程和磁地转方程。在这两个例子中，相关算子M的傅里叶乘数符号相对于波数矢量是偶的。这防止了非线性引起的能量估计的某些消去。因此，这个问题比一些经常研究的方程更微妙，比如曲面拟等转方程，其中类似的傅立叶乘子符号是奇数，并且可以使用换向子估计。对模拟流体运动的偏微分方程的数学研究是许多应用的重要基础。这些方程是非常复杂和具有挑战性的。这些方程的一个子集是所谓的“活动标量方程”，其中标量量（如流体的密度）的时间演化由流体的运动决定，其中速度本身随标量场而变化。这种反馈产生了复杂的非线性。弗里德兰德使用“硬”分析技术来研究这类方程的一般类别。一个特别的例子是地球发电机模型。这是地球磁场通过流体核心的运动产生和维持的过程，流体核心由快速旋转的、密度分层的、导电的流体组成。弗里德兰德将证明该模型确实可以产生发电机作用，通过证明存在一种强烈的不稳定性，其增长率可以由一个明确的表达式从下面限定，该表达式取决于地球流体核心的物理参数。

Active scalar equations describe a number of physical phenomena that arise in fluid dynamics. Because of their physical importance and challenging mathematical nature there is a very extensive literature on such equations. However, many problems connected with the nonlinearity of the equation remain open, particularly when the drift velocity U is nonlocal. This project addresses some of these questions with particular emphasis on examples where the operator M that encodes the physics of the problem relating U to the active scalar is singular and unbounded. It is proposed to study the interplay of diffusive effects and the nonlinearity in singular active scalar equations for a range of fractional powers of the Laplacian. For certain powers it will be shown that the system is nonlinearly unstable: for lower powers the system is Lipschitz ill-posed. These results will be obtained via the construction of unstable eigenvalues for the evolution equation linearised about an appropriate equilibrium. Techniques of continued fractions will be employed to produce an explicit lower bound on the growth rate of the unstable eigenvalues as a function of the physical parameters. The project will also study the highly singular limit of vanishing diffusivity. This limit is often physically appropriate and may be closely connected with turbulent phenomena modeled by weak solutions of the active scalar equations. The project addresses general classes of active scalar equations: particular physical examples include the incompressible porous media equation and the magnetogeostrophic equation. In both examples the Fourier multiplier symbol for the relevant operator M is even with respect to the wave number vector. This prevents certain cancellations in energy estimates arising in the nonlinearity. Hence the problem is more subtle than some frequently studied equations, such as the surface quasigeostrophic equation, where the analogous Fourier multiplier symbol is odd and commutator estimates can be employed.The mathematical study of the partial differential equations that model fluid motion forms an essential foundation for many applications. These equations are highly complex and challenging. A subset of these equations are so-called "active scalar equations" where the evolution in time of a scalar quantity such as the density of the fluid is governed by the motion of the fluid where the velocity itself varies with this scalar field. This feedback produces an intricate nonlinearity. Friedlander uses techniques of "hard" analysis to study general classes of such equations. A particular example is a model for the geodynamo. This is the process by which the Earth's magnetic field is created and sustained through the motion of the fluid core which is composed of a rapidly rotating, density stratified, electrically conducting fluid. Friedlander will prove that the model can indeed produce dynamo action by demonstrating the existence of a strong instability whose growth rate can be bounded from below by an explicit expression that depends on the physical parameters of the Earth's fluid core.