关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

A group theoretical approach to hydrodynamics considers hydrodynamics to be the differential geometry of diffeomorphism groups. The principle of least action implies that the motion of a fluid is described by the geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy. Investigation of the geometry and structure of such groups turns out to be useful for describing the global behavior of fluids for large time intervals.

Topological methods in hydrodynamics

A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

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Distinguished material surfaces and coherent structures in three-dimensional fluid flows

The most widely used definitions of a vortex are not objective: they identify different structures as vortices in frames that rotate relative to each other. Yet a frame-independent vortex definition is essential for rotating flows and for flows with interacting vortices. Here we define a vortex as a set of fluid trajectories along which the strain acceleration tensor is indefinite over directions of zero strain. Physically, this objective criterion identifies vortices as material tubes in which material elements do not align with directions suggested by the strain eigenvectors. We show using examples how this vortex criterion outperforms earlier frame-dependent criteria. As a side result, we also obtain an objective criterion for hyperbolic Lagrangian structures.

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An objective definition of a vortex

Onsager conjectured that weak solutions of the Euler equations for incompressible fluids in conserve energy only if they have a certain minimal smoothness (of the order of 1/3 fractional derivatives) and that they dissipate energy if they are rougher. In this paper we prove that energy is conserved for velocities in the function space . We show that this space is sharp in a natural sense. We phrase the energy spectrum in terms of the Littlewood?Paley decomposition and show that the energy flux is controlled by local interactions. This locality is shown to hold also for the helicity flux; moreover, every weak solution of the Euler equations that belongs to conserves helicity. In contrast, in two dimensions, the strong locality of the enstrophy holds only in the ultraviolet range.

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Energy conservation and Onsager's conjecture for the Euler equations

Preface On the Contact Topology and Geometry of Ideal Fluids (Robert Christ) Shock Reflection in Gas Dynamics (Denis Serre) The Mathematical Theory of the Incompressible Limit in Fluid Dynamics (Steven Schochet) Local Regularity Theory of Navier-Stokes Equations (Gregory Seregin) On the Influence of the Earth's Rotation on Geophysical Flows (Isabelle Gallagher and Laure Saint-Raymond) The Foundations of Oceanic Dynamics and Climate Modelling (George R. Sell) Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities (Josef Malek and K.R. Rajagopal) Navier-Stokes System in Domians with Cylindrical Outlets to Infinity (Konstantin Pileckas) Periodic Homogenization Problems in Incompressible Fluid Equations (Carlos Conca and M. Vanninathan) Author Index Subject Index

Handbook of mathematical fluid dynamics

Linearized instability implies nonlinear instability under certain rather general conditions. This abstract theorem is applied to the Euler equations governing the motion of an inviscid fluid. In particular this theorem applies to all 2D space periodic flows without stagnation points as well as 2D space-periodic shear flows.

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Nonlinear instability in an ideal fluid

We present a geometric estimate from below on the growth rate of a small perturbation of a three-dimensional steady flow of an ideal fluid and thus we obtain effective criteria for local instability for Euler's equations. We use these criteria to demonstrate the instability of several simple flows and to show that any flow with a hyperbolic stagnation point is unstable.

Susan Friedlander

Papers

Energy conservation and Onsager's conjecture for the Euler equations

Handbook of mathematical fluid dynamics

Nonlinear instability in an ideal fluid

Instability criteria for the flow of an inviscid incompressible fluid

Global well-posedness for an advection–diffusion equation arising in magneto-geostrophic dynamics