Duan / Wang Effective Dynamics of Stochastic Partial Differential Equations

1. Auflage 2014
ISBN: 978-0-12-801269-7
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: 6 - ePub Watermark

E-Book, Englisch, 282 Seiten

Effective Dynamics of Stochastic Partial Differential Equations focuses on stochastic partial differential equations with slow and fast time scales, or large and small spatial scales. The authors have developed basic techniques, such as averaging, slow manifolds, and homogenization, to extract effective dynamics from these stochastic partial differential equations. The authors' experience both as researchers and teachers enable them to convert current research on extracting effective dynamics of stochastic partial differential equations into concise and comprehensive chapters. The book helps readers by providing an accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations. Each chapter also includes exercises and problems to enhance comprehension. - New techniques for extracting effective dynamics of infinite dimensional dynamical systems under uncertainty - Accessible introduction to probability tools in Hilbert space and basics of stochastic partial differential equations - Solutions or hints to all Exercises

Illinois Institute of Technology, Chicago IL

Autoren/Hrsg.

Duan, Jinqiao

Wang, Wei

Leseproben

Chapter 2

Deterministic Partial Differential Equations

Abstract

A few examples of deterministic partial differential equations (PDEs) together with their solutions by Fourier series or Fourier transforms are briefly presented. Then some equalities and inequalities useful for estimating solutions of both deterministic and stochastic, partial differential equations are recalled.

Keywords

Partial differential equations

Fourier series

Sobolev inequalities

Differential and integral inequalities

Examples of partial differential equations; Fourier methods and basic analytic tools for partial differential equations

In this chapter, we first briefly present a few examples of deterministic partial differential equations (PDEs) arising as mathematical models for time-dependent phenomena in engineering and science, together with their solutions by Fourier series or Fourier transforms. Then we recall some equalities that are useful for estimating solutions of both deterministic and stochastic partial differential equations.

For elementary topics on solution methods for linear partial differential equations, see [147,239,258]. More advanced topics, such as well-posedness and solution estimates, fordeterministic partial differential equations may be found in popular textbooks such as [121,176,231,264].

The basic setup and well-posedness for stochastic PDEs are discussed in Chapter 4.

2.1 Fourier Series in Hilbert Space

We recall some information about Fourier series in Hilbert space, which is related to Hilbert–Schmidt theory.

A vector space has two operations, addition and scalar multiplication, which have the usual properties we are familiar with in Euclidean space . A Hilbert space is a vector space with a scalar product , with the usual properties we are familiar with in ; see [198, p. 128] or [313, p. 40] for details. In fact, is a vector space and also a Hilbert space.

A separable Hilbert space has a countable orthonormal basis , where is the Kronecker delta function (i.e., it takes value 1 when , and 0 otherwise). Moreover, for any , we have Fourier series expansion

(2.1)

In the context of solving PDEs, we choose to work in a Hilbert space with a countable orthonormal basis. Such a Hilbert space is a separable Hilbert space. This is naturally possible with the help of the Hilbert–Schmidt theorem [316, p. 232].

The Hilbert–Schmidt theorem [316, p. 232] says that a linear compact symmetric operator on a separable Hilbert space has a set of eigenvectors that form a complete orthonormal basis for . Furthermore, all the eigenvalues of are real, each nonzero eigenvalue has finite multiplicity, and two eigenvectors that correspond to different eigenvalues are orthogonal.

This theorem applies to a strong self-adjoint elliptic differential operator ,

where the domain of definition of is an appropriate dense subspace of , depending on the boundary condition specified for .

When is invertible, let . If is not invertible, set for some such that exists. This may be necessary in order for the operator to be invertible, i.e., no zero eigenvalue, such as in the case of the Laplace operator with zero Neumann boundary conditions. Note that is a linear symmetric compact operator in a Hilbert space, e.g., , the space of square-integrable functions on .

By the Hilbert–Schmidt theorem, eigenvectors (also called eigenfunctions or eigenmodes in this context) of form an orthonormal basis for . Note that and share the same set of eigenfunctions. So, we can claim that the strong self-adjoint elliptic operator ’s eigenfunctions form an orthonormal basis for .

In the case of one spatial variable, the elliptic differential operator is the so-called Sturm–Liouville operator [316, p. 245],

where and are continuous on . This operator arises in solving linear (deterministic) partial differential equations by the method of separating variables. Due to the Hilbert–Schmidt theorem, eigenfunctions of the Sturm–Liouville operator form an orthonormal basis for .

2.2 Solving Linear Partial Differential Equations

We now consider a few linear partial differential equations and their solutions.

Example 2.1 Wave equation  Consider a vibrating string of length . The evolution of its displacement , at position and time , is modeled by the following wave equation:

(2.2)

(2.3)

(2.4)

where is a positive constant (wave speed), and are given initial data. By separating variables, , we arrive at an eigenvalue problem for the Laplacian with zero Dirichlet boundary conditions on , namely, . The eigenfunctions (which need to be nonzero by definition) are and the corresponding eigenvalues are for In fact, the set of normalized eigenfunctions (i.e., making each of them have norm ),

forms an orthonormal basis for the Hilbert space of square-integrable functions, with the usual scalar product . We construct the solution by the Fourier expansion or eigenfunction expansion

(2.5)

Inserting this expansion into the PDE , we obtain an infinite system of ordinary differential equations (ODEs)

(2.6)

For each , this second-order ordinary differential equation has general solution

(2.7)

where the constants and are determined by the initial conditions, to be

Thus, the final solution is

(2.8)

For Neumann boundary conditions, , the corresponding eigenvalue problem for the Laplacian is . The eigenfunctions are and the eigenvalues are , for The set of normalized eigenfunctions,

forms an orthonormal basis for the Hilbert space of square-integrable functions.

For mixed boundary conditions, , the corresponding eigenvalue problem for the Laplacian is . The eigenfunctions are and the eigenvalues are for Again, the set of normalized eigenfunctions,

forms an orthonormal basis for the Hilbert space .

Example 2.2 Heat equation  We now consider the heat equation for the temperature of a rod of length at position and time :

(2.9)

(2.10)

(2.11)

where is the thermal diffusivity and is the initial temperature. By separating variables, , we arrive at an eigenvalue problem for the Laplacian with zero Dirichlet boundary conditions on . Namely, . The eigenfunctions (which need to be nonzero) are and the corresponding eigenvalues are for The set of normalized eigenfunctions,

forms an orthonormal basis for the Hilbert space . We construct the solution by the Fourier expansion (or eigenfunction expansion)

(2.12)

Inserting this expansion into the above PDE (2.9), we get

(2.13)

This leads to the following system of ODEs:

(2.14)

whose solutions are, for each ,

(2.15)

where for Therefore, the final solution is

(2.16)

Introduce a semigroup of linear operators, , by

(2.17)

for every . Then (identity mapping in ) and , for . Thus the above solution can be written as ; see [253, Ch. 7] for more details.

Example 2.3 Heat equation on the real line  We consider the heat equation on the real line

(2.18)

(2.19)

where is the thermal diffusivity and is the initial temperature. Taking the Fourier transform of with respect to ,

(2.20)

we obtain an initial value problem for an ordinary differential equation

(2.21)

(2.22)

where is the Fourier transform for the initial temperature . The solution to (2.21) and (2.22) is . By the inverse Fourier transform,

and, finally [239, Ch. 12],

(2.23)

Usually, is called the heat kernel or Gaussian kernel.

2.3 Integral Equalities

In this and the next two sections we recall some equalities and inequalities useful for estimating solutions of SPDEs as well as PDEs.

Let us review some integral identities. For more details, see [1, Sec. 7.3], [17, Ch. 7] or [121, Appendix C].

Green’s theorem in : Normal form

where is a continuously differentiable vector field, is a piecewise smooth closed curve that encloses a bounded region in , and is the unit outward normal vector to . The curve is positively oriented (i.e., if you walk along in the positive orientation, the region is to your left).

Green’s theorem in : Tangential form

where is a continuously differentiable vector field, is a piecewise smooth closed curve that encloses a bounded region in , and is the unit tangential vector to . The curve is positively oriented (i.e., if you walk along in the positive orientation, the region is to your left).

Divergence theorem in

where is a continuously differentiable vector field, is a closed surface...