Amplitude Equations for Stochastic Partial Differential Equations

Dirk Blomker9789812706379, 981-270-637-2

Rigorous error estimates for amplitude equations are well known for deterministic PDEs, and there is a large body of literature over the past two decades. However, there seems to be a lack of literature for stochastic equations, although the theory is being successfully used in the applied community, such as for convective instabilities, without reliable error estimates at hand. This book is the first step in closing this gap.The author provides details about the reduction of dynamics to more simpler equations via amplitude or modulation equations, which relies on the natural separation of time-scales present near a change of stability.For students, the book provides a lucid introduction to the subject highlighting the new tools necessary for stochastic equations, while serving as an excellent guide to recent research.

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Table of contents :
Contents……Page 10
Preface……Page 8
1. Introduction……Page 12
1.1 Formal Derivation of Amplitude Equations……Page 18
1.1.1 Cubic Nonlinearities……Page 19
1.1.2 Other Types of Nonlinearities……Page 21
1.1.3 Quadratic Nonlinearities……Page 22
1.1.4 Large or Unbounded Domains……Page 24
1.2 General Structure of the Approach……Page 28
1.2.1 Meta Theorems……Page 30
1.3 Examples of Equations……Page 32
2. Amplitude Equations on Bounded Domains……Page 36
2.1 Multiplicative Noise (Parameter Noise)……Page 37
2.2 Assumptions and Results — The Cubic Case……Page 39
2.2.1 Attractivity……Page 43
2.2.2 Residual……Page 44
2.2.3 Approximation……Page 45
2.3 A priori Estimates for u……Page 48
2.4 Results for Quadratic Nonlinearities……Page 52
2.4.1 Attractivity……Page 55
2.4.2 Residual……Page 56
2.4.4 Proofs……Page 58
2.5 Setting for Additive Noise (Thermal Noise)……Page 66
2.5.1 Assumptions……Page 67
2.5.2 Existence of Solutions……Page 69
2.5.3.1 Attractivity……Page 70
2.5.3.2 Approximation……Page 71
2.6 Quadratic Nonlinearities……Page 73
3. Applications — Some Examples……Page 78
3.1 Approximation of InvariantMeasures……Page 80
3.1.1 The Results……Page 83
3.2.1 Additive Noise……Page 86
3.2.2 Multiplicative Noise……Page 90
3.3 Approximative CentreManifold……Page 91
3.3.1 Random Fixed Points……Page 94
3.3.2 Random Set Attractors……Page 95
4.1 Introduction……Page 100
4.2 Setting……Page 101
4.3 Approximation of the Stochastic Convolution……Page 104
4.3.1 Noise……Page 105
4.3.2 Main Result……Page 106
4.3.3 Remarks……Page 107
4.3.4 The General Result……Page 108
4.4 Nonlinear Result……Page 111
A. Basic Inequalities……Page 114
A.1 Burkholder-Davis-Gundy Inequality……Page 115
A.2 Comparison Argument for ODEs……Page 117
B.1 LargeDeviation Estimate……Page 120
B.2 Moment Inequalities……Page 122
B.2.1 NegativeMoments……Page 124
Bibliography……Page 128
Index……Page 136