Description

Book Synopsis
This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D NavierâStokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) â proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.

Table of Contents
1. Preliminaries; 2. Two-dimensional Navier–Stokes equations; 3. Uniqueness of stationary measure and mixing; 4. Ergodicity and limiting theorems; 5. Inviscid limit; 6. Miscellanies; 7. Appendix; 8. Solutions to some exercises.

Mathematics of TwoDimensional Turbulence 194 Cambridge Tracts in Mathematics Series Number 194

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    A Hardback by Sergei Kuksin, Armen Shirikyan

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      View other formats and editions of Mathematics of TwoDimensional Turbulence 194 Cambridge Tracts in Mathematics Series Number 194 by Sergei Kuksin

      Publisher: Cambridge University Press
      Publication Date: 9/20/2012 12:00:00 AM
      ISBN13: 9781107022829, 978-1107022829
      ISBN10: 1107022827
      Also in:
      Stochastics

      Description

      Book Synopsis
      This book is dedicated to the mathematical study of two-dimensional statistical hydrodynamics and turbulence, described by the 2D NavierâStokes system with a random force. The authors' main goal is to justify the statistical properties of a fluid's velocity field u(t,x) that physicists assume in their work. They rigorously prove that u(t,x) converges, as time grows, to a statistical equilibrium, independent of initial data. They use this to study ergodic properties of u(t,x) â proving, in particular, that observables f(u(t,.)) satisfy the strong law of large numbers and central limit theorem. They also discuss the inviscid limit when viscosity goes to zero, normalising the force so that the energy of solutions stays constant, while their Reynolds numbers grow to infinity. They show that then the statistical equilibria converge to invariant measures of the 2D Euler equation and study these measures. The methods apply to other nonlinear PDEs perturbed by random forces.

      Table of Contents
      1. Preliminaries; 2. Two-dimensional Navier–Stokes equations; 3. Uniqueness of stationary measure and mixing; 4. Ergodicity and limiting theorems; 5. Inviscid limit; 6. Miscellanies; 7. Appendix; 8. Solutions to some exercises.

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