Book contents
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
3 - Energy Methods for the Euler and the Navier–Stokes Equations
Published online by Cambridge University Press: 03 February 2010
- Frontmatter
- Contents
- Preface
- 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows
- 2 The Vorticity-Stream Formulation of the Euler and the Navier-Stokes Equations
- 3 Energy Methods for the Euler and the Navier–Stokes Equations
- 4 The Particle-Trajectory Method for Existence and Uniqueness of Solutions to the Euler Equation
- 5 The Search for Singular Solutions to the 3D Euler Equations
- 6 Computational Vortex Methods
- 7 Simplified Asymptotic Equations for Slender Vortex Filaments
- 8 Weak Solutions to the 2D Euler Equations with Initial Vorticity in L∞
- 9 Introduction to Vortex Sheets, Weak Solutions, and Approximate-Solution Sequences for the Euler Equation
- 10 Weak Solutions and Solution Sequences in Two Dimensions
- 11 The 2D Euler Equation: Concentrations and Weak Solutions with Vortex-Sheet Initial Data
- 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions
- 13 The Vlasov–Poisson Equations as an Analogy to the Euler Equations for the Study of Weak Solutions
- Index
Summary
In the first two chapters of this book we described many properties of the Euler and the Navier–Stokes equations, including some exact solutions. A natural question to ask is the following: Given a general smooth initial velocity field v(x, 0), does there exist a solution to either the Euler or the Navier–Stokes equation on some time interval [0, T)? Can the solution be continued for all time? Is it unique? If the solution has a finite-time singularity, so that it cannot be continued smoothly past some critical time, in what way does the solution becomes singular? This chapter and Chap. 4 introduce two different methods for proving existence and uniqueness theory for smooth solutions to the Euler and the Navier–Stokes equations. In this chapter we introduce classical energy methods to study both the Euler and the Navier–Stokes equations. The starting point for these methods is the physical fact that the kinetic energy of a solution of the homogeneous Navier–Stokes equations decreases in time in the absence of external forcing. The next chapter introduces a particle method for proving existence and uniqueness of solutions to the inviscid Euler equation. As is true for all partial differential evolution equations, the challenge in proving that the evolution is well posed lies in understanding the effect of the unbounded spatial differential operators. The particle method exploits the fact that, without viscosity, the vorticity is transported (and stretched in three dimensions) along particle paths.
- Type
- Chapter
- Information
- Vorticity and Incompressible Flow , pp. 86 - 135Publisher: Cambridge University PressPrint publication year: 2001