Active Scalar Equations and a Geodynamo Model

活动标量方程描述了流体动力学中出现的许多物理现象。由于它们的物理重要性和具有挑战性的数学性质，有一个非常广泛的文献，这样的方程。然而，许多问题与非线性方程仍然开放，特别是当漂移速度U是非本地的。这个项目解决了其中的一些问题，特别强调的例子中，运营商M编码的物理问题有关U的活动标量是奇异的和无界的。研究了在拉普拉斯算子的分数阶幂次范围内奇异活动标量方程中扩散效应与非线性的相互作用。对于一定的权力，它将被证明是非线性不稳定的系统：较低的权力，系统是Lipschitz不适定。这些结果将通过构造关于适当平衡点线性化的发展方程的不稳定特征值得到。将采用连分数的技术，以产生一个明确的下限的增长率的不稳定本征值的物理参数的函数。该项目还将研究消失扩散率的高度奇异极限。这个极限在物理上通常是适当的，并且可能与用活动标量方程的弱解模拟的湍流现象密切相关。该项目解决一般类的活动标量方程：具体的物理例子包括不可压缩的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数。这防止了在非线性中产生的能量估计中的某些抵消。因此，这个问题比一些经常被研究的方程（如表面准地转方程，其中类似的傅里叶乘子符号是奇数，可以采用换位子估计）更微妙。对模拟流体运动的偏微分方程的数学研究形成了许多应用的重要基础。这些方程非常复杂，具有挑战性。 这些方程的子集是所谓的“主动标量方程”，其中标量（诸如流体的密度）在时间上的演变由流体的运动支配，其中速度本身随该标量场而变化。这种反馈产生复杂的非线性。弗里德兰德使用“硬”分析技术来研究这类方程的一般类。一个特别的例子是地球发电机的模型。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，密度分层，导电流体组成。弗里德兰德将证明，该模型确实可以产生发电机的行动，证明存在一个强大的不稳定性，其增长率可以从下面的一个明确的表达式，取决于地球的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.