Active Scalar Equations and a Geodynamo Model

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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.