Amplitude Equations For Stochastic Partial Differential Equations

By: Dirk Blomker

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Hardcover | 25 April 2007

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136 Pages

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24.77 x 16.51 x 1.27

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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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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Stochastics Science Physics Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Introduction	p. 1
Formal Derivation of Amplitude Equations	p. 7
Cubic Nonlinearities	p. 8
Other Types of Nonlinearities	p. 10
Quadratic Nonlinearities	p. 11
Large or Unbounded Domains	p. 13
General Structure of the Approach	p. 17
Meta Theorems	p. 19
Examples of Equations	p. 21
Amplitude Equations on Bounded Domains	p. 25
Multiplicative Noise (Parameter Noise)	p. 26
Assumptions and Results - The Cubic Case	p. 28
Attractivity	p. 32
Residual	p. 33
Approximation	p. 34
A priori Estimates for u	p. 37
Results for Quadratic Nonlinearities	p. 41
Attractivity	p. 44
Residual	p. 45
Approximation	p. 47
Proofs	p. 47
Setting for Additive Noise (Thermal Noise)	p. 55
Assumptions	p. 56
Existence of Solutions	p. 58
Amplitude Equations - Main Results	p. 59
Attractivity	p. 59
Approximation	p. 60
Quadratic Nonlinearities	p. 62
Applications - Some Examples	p. 67
Approximation of Invariant Measures	p. 69
The Results	p. 72
Pattern Formation Below Criticality	p. 75
Additive Noise	p. 75
Multiplicative Noise	p. 79
Approximative Centre Manifold	p. 80
Random Fixed Points	p. 83
Random Set Attractors	p. 84
Amplitude Equations on Large Domains	p. 89
Introduction	p. 89
Setting	p. 90
Approximation of the Stochastic Convolution	p. 93
Noise	p. 94
Main Result	p. 95
Remarks	p. 96
The General Result	p. 97
Nonlinear Result	p. 100
Basic Inequalities	p. 103
Basic Inequalities	p. 103
Burkholder-Davis-Gundy Inequality	p. 104
Comparison Argument for ODEs	p. 106
Bounds for SDEs	p. 109
Bounds for SDEs	p. 109
Large Deviation Estimate	p. 109
Moment Inequalities	p. 111
Negative Moments	p. 113
Bibliography	p. 117
Index	p. 125
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Amplitude Equations For Stochastic Partial Differential Equations

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