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One-Dimensional Turbulence and the Stochastic Burgers Equation [Mīkstie vāki]

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  • Formāts: Paperback / softback, 192 pages, height x width: 254x178 mm, weight: 369 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-Oct-2021
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470464365
  • ISBN-13: 9781470464363
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One-Dimensional Turbulence and the Stochastic Burgers EquationPalielināt
  • Formāts: Paperback / softback, 192 pages, height x width: 254x178 mm, weight: 369 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 30-Oct-2021
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470464365
  • ISBN-13: 9781470464363
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470464363.html
Keywords:
This book is dedicated to the qualitative theory of the stochastic one-dimensional Burgers equation with small viscosity under periodic boundary conditions and to interpreting the obtained results in terms of one-dimensional turbulence in a fictitious one-dimensional fluid described by the Burgers equation. The properties of one-dimensional turbulence which we rigorously derive are then compared with the heuristic Kolmogorov theory of hydrodynamical turbulence, known as the K41 theory. It is shown, in particular, that these properties imply natural one-dimensional analogues of three principal laws of the K41 theory: the size of the Kolmogorov inner scale, the $2/3$-law, and the Kolmogorov-Obukhov law.

The first part of the book deals with the stochastic Burgers equation, including the inviscid limit for the equation, its asymptotic in time behavior, and a theory of generalised $L_1$-solutions. This section makes a self-consistent introduction to stochastic PDEs. The relative simplicity of the model allows us to present in a light form many of the main ideas from the general theory of this field. The second part, dedicated to the relation of one-dimensional turbulence with the K41 theory, could serve for a mathematical reader as a rigorous introduction to the literature on hydrodynamical turbulence, all of which is written on a physical level of rigor.
Introduction 1(10)
Notation and agreement
8(3)
Part 1 Stochastic Burgers Equation
11(76)
Chapter 1 Basic results
13(36)
1.1 Introduction
13(2)
1.2 Properties of the random force
15(5)
1.3 Deterministic Cauchy problem
20(10)
Preliminaries
20(3)
Well-posedness of equation (B)
23(7)
1.4 Strong solutions of the stochastic Cauchy problem
30(5)
Notion of a strong solution and its existence
30(1)
Basic estimates for strong solutions
31(3)
The balance of energy relation
34(1)
The vorticity stretching
35(1)
1.5 Weak solutions and space-homogeneous solutions
35(2)
Weak solutions of (B)
35(2)
Space-homogeneous random fields and solutions
37(1)
1.6 The two Markov semigroups and the Markov property
37(5)
The semigroup for measures
38(1)
The semigroup for functions
39(1)
Kolmogorov-Chapman relation and Markov property
40(2)
1.7 Weak convergence of measures
42(2)
1.8 Appendix: More on the deterministic Burgers equation
44(5)
Analyticity of the flow of the deterministic equation (B)
44(2)
Burgers equation with a regular force
46(3)
Chapter 2 Asymptotically sharp estimates for Sobolev norms of solutions
49(16)
2.1 Oleinik's estimate
49(4)
A version of Oleinik's estimate
52(1)
2.2 Upper bounds for moments of Sobolev norms of solutions
53(3)
2.3 Energy balance and lower bounds for moments of norms of solutions
56(9)
Bounds for moments of Hm norms, m ≥ 1
57(3)
Bounds for moments of Wm∞ norms, m ≥ 0
60(1)
Bounds for moments of Wmp norms
60(2)
Bounds for moments of Lp norms
62(3)
Chapter 3 Mixing in the stochastic Burgers equation
65(12)
3.1 Stationary measure and the Bogolyubov-Krylov method
65(1)
3.2 Recurrence lemmas and L1-nonexpansion
66(4)
3.3 Uniqueness of the stationary measure and the mixing property
70(3)
3.4 Moments of the stationary measure
73(4)
Other scalings for the force η in equation (B)
75(2)
Chapter 4 Stochastic Burgers equation in the space L1
77(8)
4.1 Setting and definitions
77(3)
4.2 Mixing for L1 solutions
80(5)
Chapter 5 Notes and comments, I
85(2)
Part 2 One-Dimensional Turbulence
87(48)
Chapter 6 Turbulence and Burgulence
89(12)
Chapter 7 Rigorous burgulence
101(18)
7.1 Dissipation scale
102(2)
Energy and inertial ranges
103(1)
7.2 Structure function and intermittency
104(6)
Structure function
104(4)
Dissipation range in the x-presentation
108(1)
Intermittency
109(1)
7.3 Energy spectrum
110(2)
7.4 The limiting in time behaviour
112(2)
The energy spectrum
112(1)
The structure function
113(1)
7.5 Appendix: High-probability versions of the results above
114(5)
Chapter 8 The Inviscid limit and Inviscid Burgulence
119(14)
8.1 Cauchy problem for an auxiliary deterministic equation
119(2)
8.2 Inviscid limit for the deterministic equation
121(3)
8.3 Inviscid limit for the stochastic equation
124(3)
The limiting process in L1
125(2)
8.4 Inviscid Burgulence
127(1)
8.5 Mixing for the entropy solutions
128(5)
Inviscid limit for the stationary measure for eq. (1.1.12)
128(1)
Mixing for the entropy solutions
129(1)
Time-asymptotic for the inviscid energy spectrum and structure function
130(3)
Chapter 9 Notes and comments, II
133(2)
Part 3 Additional Material
135(46)
Chapter 10 Miscellanea
137(20)
10.1 Other types of forces: Estimates, Burgulence and mixing
137(6)
10.2 High-frequency kick forces
143(8)
10.3 Exponential mixing and its consequences
151(2)
Exponential mixing in the Burgers equation with kick-forces and red forces
151(1)
Exponential mixing and ergodicity
152(1)
10.4 Other equations
153(4)
Chapter 11 Appendices
157(18)
11.1 Appendix A: Preliminaries from analysis
157(3)
11.2 Appendix B: Spaces of functions
160(2)
11.3 Appendix C: Measurable spaces, probability spaces and transition probabilities
162(4)
11.4 Appendix D: Kantorovich distance
166(1)
11.5 Appendix E: Random processes and the Wiener process
167(2)
11.6 Appendix F: Filtered probability spaces and Markov processes
169(1)
11.7 Appendix G: Stopping and hitting times and the strong Markov property
170(1)
11.8 Appendix H: Stochastic differential equations and Ito's formula
171(4)
Chapter 12 Solutions for selected exercises
175(6)
Acknowledgements 181(2)
Bibliography 183(8)
Index 191
Alexandre Boritchev, Universite Claude Bernard Lyon 1, Villeurbanne, France.

Sergei Kuksin, Universite Paris-Diderot, France, and Shandong University, Jinan, People's Republic of China.