Critical thresholds in Eulerian dynamics

In this thesis, we study the critical regularity phenomena in Eulerian dynamics, $u_t+u\cdot \nabla u=F(u,D u,\cdots),$ here $F$ represents a general force acting on the flow and by regularity we seek to obtain a large set of sub-critical initial data. We analyze three prototype models, ranging from the one-dimensional Euler-Poisson equations to two-dimensional system of Burgers equations to three-dimensional, four-dimensional and even higher-dimensional restricted Euler systems. We begin with the one-dimensional Euler-Poisson equations, where $F$ is the Poisson forcing term together with the usual $\gamma$-law pressure. We prove that global regularity of the Euler-Poisson equations with $\gamma\geq 1$ depends on whether or not the initial configuration crosses an intrinsic critical threshold. Next, we discuss multi-dimensional examples. The first multi-dimensional example that we focus our attention on is the two dimensional pressureless flow, where $F=\epsilon \Delta u$. Our analysis shows that there is a uniform \textsl{BV} bound of the solutions $u^{\epsilon}$. Moreover, if the initial velocity gradient $\nabla u_0$ does not have negative eigenvalues, then its vanishing viscosity limit is the smooth solution of the corresponding equations of the inviscid fluid flow. The second multi-dimensional example we discuss here is the restricted Euler dynamics, where $\nabla F= \displaystyle\frac{1}{n} \mathrm {tr} (\nabla u)^2I_{n\times n}$\,. Our analysis shows that for the three-dimensional case, the finite-time breakdown of the restricted Euler system is generic, and for the four-dimensional case, there is a surprising global existence for sub-critical initial data. Further analysis extends the above result to the general $n$-dimensional ($n>4$) restricted Euler system

The Euler-Poincaré Equations in Geophysical Fluid Dynamics

Recent theoretical work has developed the Hamilton's-principle analog of Lie-Poisson Hamiltonian systems defined on semidirect products. The main theoretical results are twofold: 1. Euler-Poincaré equations (the Lagrangian analog of Lie-Poisson Hamiltonian equations) are derived for a parameter dependent Lagrangian from a general variational principle of Lagrange d'Alembert type in which variations are constrained; 2. an abstract Kelvin-Noether theorem is derived for such systems. By imposing suitable constraints on the variations and by using invariance properties of the Lagrangian, as one does for the Euler equations for the rigid body and ideal fluids, we cast several standard Eulerian models of geophysical fluid dynamics (GFD) at various levels of approximation into Euler-Poincaré form and discuss their corresponding Kelvin-Noether theorems and potential vorticity conservation laws. The various levels of GFD approximation are related by substituting a sequence of velocity decompositions and asymptotic expansions into Hamilton's principle for the Euler equations of a rotating stratified ideal incompressible fluid. We emphasize that the shared properties of this sequence of approximate ideal GFD models follow directly from their Euler-Poincaré formulations. New modifications of the Euler-Boussinesq equations and primitive equations are also proposed in which nonlinear dispersion adaptively filters high wavenumbers and thereby enhances stability and regularity without compromising either low wavenumber behavior or geophysical balances

The Euler–Poincaré Equations and Semidirect Products with Applications to Continuum Theories

We study Euler–Poincaré systems (i.e., the Lagrangian analogue of Lie–Poisson Hamiltonian systems) defined on semidirect product Lie algebras. We first give a derivation of the Euler–Poincaré equations for a parameter dependent Lagrangian by using a variational principle of Lagrange d'Alembert type. Then we derive an abstract Kelvin–Noether theorem for these equations. We also explore their relation with the theory of Lie–Poisson Hamiltonian systems defined on the dual of a semidirect product Lie algebra. The Legendre transformation in such cases is often not invertible; thus, it does not produce a corresponding Euler–Poincaré system on that Lie algebra. We avoid this potential difficulty by developing the theory of Euler–Poincaré systems entirely within the Lagrangian framework. We apply the general theory to a number of known examples, including the heavy top, ideal compressible fluids and MHD. We also use this framework to derive higher dimensional Camassa–Holm equations, which have many potentially interesting analytical properties. These equations are Euler–Poincaré equations for geodesics on diffeomorphism groups (in the sense of the Arnold program) but where the metric is H^1 rather thanL^2

The Geometric Structure of Complex Fluids

This paper develops the theory of affine Euler-Poincar\'e and affine Lie-Poisson reductions and applies these processes to various examples of complex fluids, including Yang-Mills and Hall magnetohydrodynamics for fluids and superfluids, spin glasses, microfluids, and liquid crystals. As a consequence of the Lagrangian approach, the variational formulation of the equations is determined. On the Hamiltonian side, the associated Poisson brackets are obtained by reduction of a canonical cotangent bundle. A Kelvin-Noether circulation theorem is presented and is applied to these examples