Stochastic porous media equations

Lecture Notes in Mathematics  2163

Viorel Barbu
Giuseppe Da Prato
Michael Röckner

Stochastic
Porous
Media
Equations


Lecture Notes in Mathematics
Editors-in-Chief:
J.-M. Morel, Cachan
B. Teissier, Paris
Advisory Board:
Camillo De Lellis, Zürich
Mario di Bernardo, Bristol
Alessio Figalli, Austin
Davar Khoshnevisan, Salt Lake City
Ioannis Kontoyiannis, Athens
Gábor Lugosi, Barcelona
Mark Podolskij, Aarhus
Sylvia Serfaty, Paris and New York
Anna Wienhard, Heidelberg

2163


More information about this series at />

Viorel Barbu • Giuseppe Da Prato •
Michael RRockner

Stochastic Porous Media
Equations

123


Viorel Barbu
Department of Mathematics
Al. I. Cuza University & Octav Mayer
Institute of Mathematics of
the Romanian Academy
Iasi, Romania

Giuseppe Da Prato
Classe di Scienze
Scuola Normale Superiore di Pisa
Pisa, Italy

Michael RRockner
Department of Mathematics
University of Bielefeld
Bielefeld, Germany

ISSN 0075-8434
Lecture Notes in Mathematics
ISBN 978-3-319-41068-5

DOI 10.1007/978-3-319-41069-2

ISSN 1617-9692 (electronic)
ISBN 978-3-319-41069-2 (eBook)

Library of Congress Control Number: 2016954369
Mathematics Subject Classification (2010): 60H15, 35K55, 76S99, 76M30, 76M35
© Springer International Publishing Switzerland 2016
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The registered company is Springer International Publishing AG Switzerland


Preface

This book is concerned with stochastic porous media equations with main emphasis
on existence theory, asymptotic behaviour and ergodic properties of the associated
transition semigroup. The general form of the porous media equation is

dX

ˇ.X/dt D .X/dW;

(1)

where ˇ W R ! R is a monotonically increasing function (possibly multivalued)
and W is a cylindrical Wiener process.
P in stochastic porous media equation
Stochastic perturbations of the form .X/W
were already considered by physicists but until recently no rigorous mathematical
existence result was known. In specific models the noise arises from physical
fluctuations of the media which in a first approximation can be taken of the form
P
.a C bX/W.
The porous media equation driven by a Gaussian noise, besides their relevance
in the mathematical description of nonlinear diffusion dynamics perturbed by noise,
has an intrinsic mathematical interest as a highly nonlinear partial differential
equation, which is not well posed in standard spaces of regular functions. In fact
the basic functional space for studying this equation is the distributional Sobolev
space H 1 and this is due to the fact that the porous media operator y !
ˇ.y/
is m-accretive in the spaces H 1 and L1 only. Since the Hilbertian structure of the
space is essential for getting energetic estimates via Itô’s formula, H 1 was chosen
as an appropriate space for this equation.
Compared with the deterministic porous media equation which benefits from
the theory of nonlinear semigroups of contractions in both the spaces L1 and
H 1 , the existence theory of the corresponding stochastic equations is not a direct
consequence of general theory of the nonlinear Cauchy problem in Banach spaces.
In fact, a nonlinear stochastic equation with additive noise (or with special linear

noise) is formally equivalent with a nonlinear random differential equation with nonsmooth time-dependent coefficients, which precludes the use of standard existence
result for the deterministic Cauchy problem. However, the existence theory for

v


vi

Preface

stochastic infinite dimensional equations uses many techniques of nonlinear Cauchy
problems associated with deterministic m-accretive nonlinear operators.
This book is organized into seven chapters. Chapter 1 is devoted to some standard
topics from stochastic and nonlinear analysis mainly included without proof because
they represent a necessary basic background for the subsequent topics.
Chapter 2 is devoted to existence theory for stochastic porous media equations
with Lipschitz nonlinearity and may also be viewed as a background for the theory
developed in Chap. 3, which is the core of the book. This chapter treats existence
theory for equations with maximal monotone nonlinearities which have at most
polynomial growths. The principal model described by this class of equations is the
slow and fast diffusion processes. Besides existence, the extinction in finite time for
fast diffusions and finite speed of propagation for slow diffusions are also studied.
Chapter 4 is devoted to the so-called variational approach to stochastic porous
media equations. In a few words, the idea is to represent the equation as an infinite
dimensional stochastic equation associated with a monotone and demi-continuous
operator from a reflexive Banach space V to its dual V 0 and apply the standard
existence theory developed in the early 1970s by E. Pardoux, N. Krylov and B.
Rozovskii.
Chapter 5 is devoted to stochastic porous media equations with nonpolynomial
growth to ˙1, for the diffusivity ˇ, a situation which was excluded from the

previous H 1 approach and which uses an L1 treatment based on weak compactness
arguments. The solution obtained in this way is weaker than in the previous case but
applies to a larger class of functions ˇ.
Chapter 6 is concerned with stochastic porous media equations in the whole Rd .
Chapter 7 is devoted to existence and uniqueness of invariant measures for the
transition semigroup associated with stochastic porous media equations.
These lecture notes have grown out of joint works and collaborations of authors
in the last decade. They were written during their visits to Scuola Normale Superiore
di Pisa and Bielefeld University.
Iasi, Romania
Pisa, Italy
Bielefeld, Germany

Viorel Barbu
Giuseppe Da Prato
Michael Röckner


Contents

1

Introduction .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 Stochastic Porous Media Equations and Nonlinear Diffusions .. . . . .
1.1.1 The Stochastic Stefan Two Phase Problem.. . . . . . . . . . . . . . . . . .
1.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2.1 Functional Spaces and Notation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2.2 The Gaussian Noise . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2.3 Stochastic Processes. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2.4 Monotone Operators . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1
1
5
6
6
8
9
12

2 Equations with Lipschitz Nonlinearities . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 Introduction and Setting of the Problem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1.1 The Definition of Solutions . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 The Uniqueness of Solutions .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 The Approximating Problem .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Convergence of fX g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4.1 Estimates for kXR .t/k2 1 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
t
2.4.2 Estimates for E 0 kF .X .s//k2 1 ds . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4.3 Additional Estimates in Lp . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.5 The Solution to Problem (2.1) . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.6 Positivity of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.7 Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

19
19
23
24
24
28

29
31
33
37
41
45

3 Equations with Maximal Monotone Nonlinearities .. . . . . . . . . . . . . . . . . . . .
3.1 Introduction and Setting of the Problem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Uniqueness .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3 The Approximating Problem .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.1 Estimating EkX .t/k2 1 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.3.2 Estimating EjX .t/jpp . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.4 Solution to Problem (3.1) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5 Slow Diffusions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.5.1 The Uniqueness.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.6 The Rescaling Approach to Porous Media Equations . . . . . . . . . . . . . . .

49
49
51
53
54
56
56
60
61
63
vii

viii

Contents

3.7

Extinction in Finite Time for Fast Diffusions and Self
Organized Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.8 The Asymptotic Extinction of Solutions to Self
Organized Criticality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.9 Localization of Solutions to Stochastic Slow Diffusion
Equations: Finite Speed of Propagation . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.9.1 Proof of Theorem 3.9.1 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.10 The Logarithmic Diffusion Equation . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.11 Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

65
70
78
80
88
93

4 Variational Approach to Stochastic Porous Media
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95
4.1 The General Existence Theory .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95
4.2 An Application to Stochastic Porous Media Equations .. . . . . . . . . . . . . 98
4.3 Stochastic Porous Media Equations in Orlicz Spaces. . . . . . . . . . . . . . . . 99
4.4 Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105

5 L1 -Based Approach to Existence Theory for Stochastic
Porous Media Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 Introduction and Setting of the Problem .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Proof of Theorem 5.1.4 .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.1 A-Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.2 Convergence for ! 0 .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2.3 Completion of Proof of Theorem 5.1.4.. .. . . . . . . . . . . . . . . . . . . .
5.2.4 Proof of Theorem 5.1.4 . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

107
107
111
112
115
123
129
131

6 The Stochastic Porous Media Equations in Rd . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Equation (6.2) with a Lipschitzian ˇ . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.4 Equation (6.2) for Maximal Monotone Functions ˇ
with Polynomial Growth . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.5 The Finite Time Extinction for Fast Diffusions . .. . . . . . . . . . . . . . . . . . . .
6.6 Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . .

147
162

165

7 Transition Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1 Introduction and Preliminaries .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.1.1 The Infinitesimal Generator of Pt . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.2 Invariant Measures for the Slow Diffusions Semigroup Pt . . . . . . . . . .
7.3 Invariant Measure for the Stefan Problem .. . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4 Invariant Measures for Fast Diffusions . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.1 Existence .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.4.2 Uniqueness.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
7.5 Invariant Measure for Self Organized Criticality Equation .. . . . . . . . .

167
167
169
170
175
178
178
180
186

133
133
134
137


Contents

7.6
7.7

ix

The Full Support of Invariant Measures
and Irreducibility of Transition Semigroups.. . . . .. . . . . . . . . . . . . . . . . . . . 187
Comments and Bibliographical Remarks. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195

References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 197
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 201


Chapter 1

Introduction

This is an introductory chapter mainly devoted to the formulation of problems,
models and some preliminaries on convex and infinite dimensional analysis,
indispensable for understanding the sequel.

1.1 Stochastic Porous Media Equations and Nonlinear
Diffusions
In this book we study nonlinear stochastic differential equations of the form
8
ˆ
dX.t/
ˇ.X.t//dt D .X.t//dW.t/;
ˆ
ˆ

ˆ
ˆ
<
ˇ.X.t// D 0; on Œ0; C1/ @O;
ˆ
ˆ
ˆ
ˆ
ˆ
: X.0/ D x; in O;

in Œ0; C1/

O;
(1.1)

where O is an open (bounded) domain of Rd ; d
1; with boundary @O, ˇ is a
(multivalued) maximal monotone function from R to itself, W is a Wiener process
and D .X/ is a suitable function to be made precise later on.
The deterministic equation
8
@
ˆ
ˆ
X.t/
ˆ
ˆ
@t
ˆ

ˆ
<

ˇ.X.t//dt D f .t; /;

ˇ.X.t// D 0; on Œ0; C1/
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
X.0/ D x; in O;

in Œ0; C1/

@O;

© Springer International Publishing Switzerland 2016
V. Barbu et al., Stochastic Porous Media Equations, Lecture Notes
in Mathematics 2163, DOI 10.1007/978-3-319-41069-2_1

O;
(1.2)

1


2

1 Introduction

is referred to in the literature as the porous media equation because the first model
described by (1.2) was the dynamics of the flow in a porous medium. (In this case
ˇ.X/ D X m ; m > 1). The stochastic equation (1.1) can be seen as an extension of
(1.2) when the forcing term is replaced by a noise term.
It should be said that equations of the form (1.1) describe a large class of
nonlinear diffusion mathematical models we briefly describe below.
If we formally represent (1.1) as
dX.t/

div .ˇ 0 .X.t//rX.t//dt D .X.t//dW.t/;

(1.3)

then we Rrecognize in (1.3) the classical diffusion equation with diffusion coefficient
r
j.r/ D 0 ˇ.s/ds. In all diffusion models X.t/ represents the mass concentration
and so (1.3) can be viewed as the mathematical model for the dynamics of diffusion
flows driven by a stochastic perturbation .X.t//dW.t/.
As mentioned earlier, the standard model of diffusion of a gas in porous media
is that where ˇ.r/ D jrjm 1 r with m > 1 which is also called the slow diffusion
model. More generally, we can consider the case where ˇ is a continuous monotone
function satisfying
jrjmC1 Ä rˇ.r/ Ä b1 jrjqC1 C b2 r;

8 r 2 R;

(1.4)

for m > 1 and q > m, ; b1 > 0.
The case m D 2 describes the flow of an ideal gas in porous media while m 2
that of a diffusion of a compressible fluid through porous media. There are other
situations such as thermal propagation in plasma (m D 6) or plasma radiation (m D
4/ which are modelled by the same equation. Some models in population dynamics
are represented by (1.3) for ˇ.r/ D ar2 .
The case when 0 < m < 1 is that of fast diffusion and is relevant in the
description of plasma physics, the kinetic theory of gas or fluid transportation
in porous media (see [28, 29, 42, 89]). As a matter of fact, these equations are
associated with superdiffusive processes in which the time growth of the mean
square displacement hX 2 .t/i has a nonlinear growth in time as t ! 1.
The singular case 1 < m < 0 or ˇ.r/ D log r in the limiting case m D 0 in the
equation
dX.t/

div .um 1 ru/dt D .X.t//dW.t/

(1.5)

models the superfast diffusion arising in the description of dynamics of plasma in
magnetic fields as well as in the central limit approximation to the Carleman model
of the Boltzmann equation [28, 29, 88]. In 2-D the corresponding deterministic
equation describes the evolution of a surface by the Ricci curvature flow (see e.g.
[90].)
A general feature of the fast and superfast diffusion models is that they model
diffusion processes with a fast speed of mass transportation and this is one reason
for which, as we shall see later on, the process terminates within finite time with
positive probability.

1.1 Stochastic Porous Media Equations and Nonlinear Diffusions

3

The self-organized criticality (SOC) equation is the special case of (1.1) where
ˇ.r/ D sign r C .r/;
where

> 0;

8 r 2 R;

is a maximal monotone graph in R

sign r D

8
r
ˆ
< jrj

(1.6)

R and

for r ¤ 0;

ˆ
: fr 2 R W jrj D 1g for r D 0:

(1.7)

In this case Eq. (1.1), that is
dX.t/

.sign X.t//dt C .X.t//dt 3 .X.t//dW.t/;

(1.8)

is used as a mathematical model for the standard self-organized criticality process,
called the sand-pile model or the Bak–Tang–Wiesenfeld (BTW) model [4, 5] and
which can be formalized via the cellular automaton model briefly presented below.
Consider an N N square matrix representing a spacial discrete region O D
fXi;j g; i; j D 1; : : : ; N. To each site .i; j/ is assigned at moment t a nonnegative (inte2
ger) variable Xij .t/. The dynamics of the RNC -valued variable X.t/ D fXi;j .t/g; i; j D
1; : : : ; N; is described by the equation
ij

Xi;j .t C 1/ ! Xi;j .t/
where ij D f.i C 1; j/; .i; j C 1/; .i
neighbors of .i; j/ and
ij
Zkl

Zkl

for .k; l/ 2

1; j/; .i; j

ij ;

(1.9)

1/g is the set of all four nearest

8
< 4 if i D k; j D l;
D
1 if .k; l/ 2 ij ;
:
0 if .k; l/ … ij :

The algebraic law (1.9) describes rigorously what happens with the “activated” site
.i; j/ (i.e. a site which has attained or is over the critical height Xc /: it looses four
grains of sands which move to nearest neighbors in the interval of time .t; t C 1/.
This is a small “avalanche” which leads to a new configuration of the sand-pile. This
transition from X.t/ to X.t C 1/ can be written as
Xi;j .t C 1/

Xi;j .t/ D Zij H.Xi;j .t/

Xc /;

i; j D 1; : : : ; N;

where H is the Heaviside function
H.r/ D

1
0

if r > 0;
if r < 0

(1.10)


4

1 Introduction

and Zij D Zijkl ; k; l 2 ij . We assume here that the boundary sites .i; j/ are in the
subcritical case, that is Xi;j .X /i;j Ä 0:
The exact meaning of (1.10) is that the transfer dynamics (1.9) works in the
critical or supercritical region only i.e., in an activated site .i; j/, where Xij > Xc .
Otherwise, we can consider that Xij remains unchanged. We can represent it as
X.t C 1/

X.t/ D

ZH.X.t/

Xc /;

(1.11)

where Z D Zij ; i; j D 1; : : : ; N. It can be seen that Zij is the second order difference
operator in the spacial domain O, i.e.

Zij .Yij / D YiC1;j C Yi

1;j

C Yi;jC1

4Yi;j C Yi;j

1

8 i; j D 1; : : : ; N

(1.12)

and so, Eq. (1.12) is the discrete version of the partial differential equation of
parabolic type
8
@
ˆ
ˆ
X.t/ D
ˆ
ˆ
@t
ˆ
ˆ
<

H.X.t/

Xc /

in O;

X.t/ Xc D 0 on @O;
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
X.0/ D x in O:
where is the Laplace operator.
Therefore, if we replace the finite region O by a continuous domain in 2dimensional space (for instance O D .0; 1/ .0; 1/) and the site location .i; j/
by a point in O the above model reduces to a nonlinear diffusion equation on the
spatial domain O R2 .
In the literature there are several versions of (SOC) equations modelling sandpile
processes. One of this is
8
@
ˆ
ˆ
.X.t/ D
ˆ
ˆ
@t
ˆ
ˆ
<

XH.X.t/

Xc //

in O;

X.t/ Xc D 0 on @O;
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
:
X.0/ D x in O:
If one perturbs this spontaneous process by a stochastic process of the form
P which, roughly speaking, means that one adds grains of sand to
.X.t//W.t/
Gaussian random locations one gets an equation of type (1.8).
It must be said that there are other SOC models described by superfast diffusion
equations of the form (1.1), (1.2) where ˇ.r/ D rm , 2 < m < 0 (see [28, 29]).
But the mathematical treatment of such a problem remains to be done.


1.1 Stochastic Porous Media Equations and Nonlinear Diffusions

5

It should be emphasized also that the self-organized Eq. (1.8) can be viewed itself

as a fast diffusion equation if we take into account that at least formally it can be
written as
dX.t/

div .ı.X.t//rX.t//dt C .X.t//dt 3 .X.t//dW.t/;

(1.13)

where ı is the Dirac measure concentrated in the origin. This reveals the complexity
and high singularity of Eq. (1.8).

1.1.1 The Stochastic Stefan Two Phase Problem
In the special case
8
< ar for r < 0;
ˇ.r/ D 0 for 0 Ä r Ä ;
:
b.r
/ for r > ;

(1.14)

a; b; > 0, .r/ D ˇ.r/, Eq. (1.1) that is
8
ˆ
dX.t/
ˇ.X.t//dt D ˇ.X.t//dW.t/;
ˆ
ˆ
ˆ

ˆ
<
ˇ.X.t// D 0; on Œ0; C1/ @O;
ˆ
ˆ
ˆ
ˆ
ˆ
:
X.0/ D x; in O;

in Œ0; C1/

O;
(1.15)

reduces to the two phase transition Stefan problem perturbed by a Gaussian noise.
More precisely
8
ˆ
d a Âdt D ÂdW.t/ in f.t; / W Â.t; / < 0g;
ˆ
ˆ
ˆ
ˆ
<
d b Âdt D ÂdW.t/ in f.t; / W Â.t; / > 0g;
ˆ
ˆ
ˆ

ˆ
ˆ
:
.ar C br / r` D
in f.t; / W Â.t; / D 0g;

(1.16)

where
f.t; / W Â.t; / D 0g D f.t; / W t D `. /g:
Here  D Â.t; / is the temperature and X D ˇ 1 .Â/ is the enthalpy of the
system (see e.g. [55]). This mathematical model describes the situation where the
P
melting or solidification process is driven by a stochastic heat flow  W.t//
which is
proportional to the temperature. We note that (1.16) is a stochastic partial differential


6

1 Introduction

equation with free (moving) boundary f.t; / W Â.t; / D 0g which involves a
transmission condition on the free boundary. Other phase transition diffusion models
(for instance the oxygen diffusion in an absorbing tissue) are described by similar
equations.
The Stefan one phase problem of the form (1.16) was intensively studied in last
years in 1-D (see for instance [69, 70] and the references given there). It should be
mentioned however that such a problem is a stochastic variational inequality which
can not be represented in the form (1.15) and so its treatment is beyond the scope of

this work.

1.2 Preliminaries
1.2.1 Functional Spaces and Notation
Let O be a bounded open subset of Rd . We assume that its boundary @O is
sufficiently regular (at least of class C2 ) in order to apply the paper [66].
The following spaces will be considered in what follows.
• Lp .O/ D Lp ; p 2 Œ1; 1; is the Banach space of all p-summable (equivalence
classes of) functions on O with the usual norm j jp . The inner product in the
Hilbert space L2 .O/ will be denoted by h ; i2.
• H k .O/ D H k ; k 2 N; is the Sobolev space of all functions in L2 whose
distributional derivatives of order lesser than k belong to L2 . H01 .O/ D H01 is
the set of all functions of H 1 that vanish on the boundary of O. The norm in H01
is denoted by k k1 and given by
ÂZ
kuk1 WD

Rd

jruj2Rd

Ã1=2
d

:

• D 0 .O/ denotes the space of Schwartz distributions on O.
• H 1 .O/ D H 1 is the dual of H01 .O/: Its norm will be denoted by k k 1 and the
inner product by h ; i 1 :
• Given a Banach space Z and 1 Ä p Ä 1, W 1;p .Œ0; TI Z/ denotes the space of all

absolutely continuous functions u W Œ0; T ! Z which are a.e. differentiable on
p
Œ0; T and du
dt 2 L .0; TI Z/.
It is well known that the linear operator A W H01 .O/ ! H
Ax D

x;

8 x 2 H01 .O/;

1

.O/
(1.17)

is continuous and positive definite, while its restriction to L2 .O/ with domain
D.A/ D H01 .O/ \ H 2 .O/ is self-adjoint. Moreover, for all x; y 2 H 1 .O/ we have
hx; yi

1

D hA

1=2

x; A

1=2

yi2 :


1.2 Preliminaries

7

If y 2 L2 then,
hx; yi

1

D hA 1 x; yi2 :

The operator A W D.A/
L2 .O/ ! L2 .O/ where D.A/ D H 2 .O/ \ H01 .O/, is
symmetric and possesses a complete orthonormal basis of eigenfunctions feh gh2N .
We denote by .˛h /h2N the corresponding set of eigenvalues,
eh D

˛h eh ;

8 h 2 N:

(1.18)

We note that, setting fh D .˛h /1=2 eh ; h 2 N, then ffh gh2N is a complete orthonormal
basis of H 1 .
Example 1.2.1 Let O D .0; 1/d then we have
eh . / D .2= /

d
2

fh . / D .2= /

d
2

sin.h1

1/

jhj sin.h1

d /;

sin.hd
1/

sin.hd

h1 ; : : : ; hd 2 N;
d //;

h1 ; : : : ; hd 2 N;

and
˛h D
where jhj2 D h21 C

2

jhj2 ;

C h2d .

t
u

It is useful to specify the asymptotic behavior of eh ; fh and ˛h as h ! 1, when
.0; 1/d is replaced by an arbitrary open set O Rd with smooth boundary. First we
note that there exists aO > 0 such that
2

2

˛0 1 h d Ä ah Ä aO h d ;

8 h 2 N;

(1.19)

where the enumeration is chosen in increasing order counting multiplicity. (see e.g.
[73, 91], [67, Vol. 3, Corollary 17.8.5] and the references therein.)
Moreover, there exists bO > 0 such that
d 1

jeh . /j Ä bO ˛h 2 ;

d

jfh . /j Ä bO ˛h2 ;

8 h 2 N;

2 O:

(1.20)

(see [66].) This estimate is optimal for general sets O. Obviously in the particular
case where O D .0; 1/d we have a better result
jeh . /j Ä .2= /

d
2

;

jfh . /j Ä .2= /

d
2

jhj;

8 h 2 Nd ;

2 O:

(1.21)


8

1 Introduction

1.2.2 The Gaussian Noise
Let H be a separable Hilbert space, fgh g an orthonormal basis of H and let .Wh /h2N
be a sequence of independent real Ft -Brownian motions on a filtered probability
space .˝; F ; P; .Ft /t 0 / for some normal filtration .Ft /t 0 . This means that Wh is
Ft -adapted and Wh .t C a/ Wh .a/ is independent of Ft for any h 2 N and any
t; a 0. Then we define the cylindrical Wiener process in H as the formal series
W.t/ WD

X

Wh .t/gh :

k2N

Though for any t > 0 this series is not convergent in L2 .˝; F ; P/; it is easily seen
that the series
X
BW.t/ WD
Wh .t/Bgh ;
k2N

is convergent in L2 .˝; F ; PI C.Œ0; TI H// if and only if B 2 L2 .H/. Here L2 .H/
is the space of all Hilbert–Schmidt operators in H endowed with the norm

kTkL2 .H/ D .Tr ŒTT /1=2 ;

T 2 L2 .H/

and Tr ŒTT  represents the trace of TT ,
Tr ŒTT  D Tr ŒT T D

X

kTgk k2 1 ;

k2N

for one (and consequently every) orthonormal basis fgk g of H.
Given a progressively measurable process F W Œ0; T ˝ ! L2 .H/ such that
RT
E 0 kF.s/k2L2 .H/ ds < 1, the Itô integral
Z

T

F.s/dW.s/ D
0

1 Z
X
kD1

T

F.s/gk dWk .s/

0

is a well defined random variable in L2 .˝; F ; PI H/ (see e.g. [51, 82]) and we have
Z
E

Z

2

T

0

T

D

F.s/dW.s/
1

0

EkF.s/k2L2 .H/ ds

(1.22)

1.2 Preliminaries

9

1.2.3 Stochastic Processes
Let again H be a separable Hilbert space and p; q 2 Œ1; C1. We consider the
following spaces of stochastic processes X W Œ0; T ˝ ! H.
q

• LW .0; TI Lp .˝; H// is the space of all progressively measurable processes such
that
Z

T

0

ŒE.jX.t/jp /q dt < 1:

• CW .Œ0; TI Lp .˝; H// is the space of all H-valued progressively measurable
processes which are p mean square continuous on Œ0; T, that is such that
sup E.jX.t/jp / < 1:
t2Œ0;T

• L2W .˝I C.Œ0; TI H// is the space of all H-valued progressively measurable
processes which are continuous on Œ0; T and such that
E sup jX.t/j2 < 1:
t2Œ0;T

It is well known that there is a natural imbedding of L2W .˝I C.Œ0; TI H// into

CW .Œ0; TI Lp .˝; H//.
Let F 2 L2W .Œ0; TI L2 .˝; L2 .H// and set
Z

t

X.t/ WD

F.s/dW.s/;
0

t 2 Œ0; T:

(Everywhere in the following the stochastic integral is considered in the sense of
Itô.) Then X belongs to CW .Œ0; TI L2 .˝; H// and possesses a version which belongs
to L2W .˝I C.Œ0; TI H//. Moreover, X is a martingale and the following result holds:
Proposition 1.2.2 (Burkholder–Davis–Gundy) For arbitrary p > 0 there exists a
constant cp > 0 such that
ˇZ t
ˇp Ã
Â
ÄZ
ˇ
ˇ
ˇ
ˇ
E sup ˇ F.s/dW.s/ˇ Ä cp E
tÄT

0

T
0

p=2

kF.s/k2L2 .H/ ds

!
:

(1.23)

(see e.g. [51, Theorem 4.36]).
By Doob’s maximal inequality we have
ˇZ t
ˇ2
Z
ˇ
ˇ
ˇ
ˇ
E sup ˇ F.s/dW.s/ˇ Ä 4
t2Œ0;T

0

0

T

EkF.s/k2L2 .H/ ds:

(1.24)


10

1 Introduction

An Itô’s process with values in H is a stochastic process X.t/; t 2 Œ0; T; of the
form
Z t
Z t
X.t/ D x C
b.s/ds C
.s/dW.s/;
(1.25)
0

0

where x 2 H; b 2 L1W .Œ0; TI L1 .˝; H// and 2 L2W .Œ0; TI L2 .˝; L2 .H//. Then
if ' 2 C2 .H/ (i.e. twice continuously Fréchet differentiable on H) the Itô formula
holds (see e.g. [38, Theorem 2.4], [51]),
Z
'.X.t// D '.x/ C
1
C
2

Z

hb.s/; D'.X.s//ids

0

t

0

Z

t

Tr Œ

.s/D2 '.X.s// .s/ds

(1.26)

t

C
0

hD'.X.s//; .s/dW.s/i;

where D' and D2 ' represent the first and second derivatives of ', respectively.
Identity (1.26) holds in L2W .˝I C.Œ0; TI H//, P-a.s.

We note for further use that
Tr Œ

.s/D2 '.X.s// .s/ D

1
X

hD2 '.X.s// .s/gk ; .s/gk i;

(1.27)

kD1

where fgk g is an orthonormal basis of H.
1.2.3.1 Itô’s Formula for the Lp Norm
Here we present a result on Itô’s formula for the Lp norm due to Krylov [71]. Let
.˝; F ; Ft ; P/ be a filtered probability space as before and let denote by P the
corresponding predictable -algebra in ˝ Œ0; C1/.
We consider processes
uW˝

Œ0; C1/ ! Lp .Rd /;

which satisfy the equation
du.t/ D f .t/dt C gj .t/dWj .t/;

(1.28)

where f ; gj ; 1 Ä j < 1 are Lp -valued processes. (Here we have used the summation

convention over repeated indices.)


1.2 Preliminaries

If

11

is a stopping time we set for a separable Banach space E
f Lp .Rd ; E//;
Lp . ; E/ D Lp .Œ0; ; P;

f is the completion of P with respect to P.d!/
where P
Lemma 5.1].

dt. We have [71,

Proposition 1.2.3 Let 2 Ä p < 1, f 2 Lp . ; R/, g D fgk g 2 Lp . ; `2 /, and
let u be a progressively measurable map on ˝ Œ0; 1/ with values in the space
of distributions on Rd such that for any ' 2 C01 .Rd / with probability one for all
t 2 .0; C1/ we have
Z
hu.t ^ /; 'i2 D hu0 ; 'i2 C

C

1 Z
X

1sÄ hf .s/; 'i2 ds

0

(1.29)

t

hgk .s/; 'i2 1sÄ dWk .s/;

0

kD1

t

where u0 2 Lp .˝; F0 ; Lp /.1 Then there is ˝ 0

F0 of full probability such that

(i) u.t ^ /1˝ 0 is an Lp -valued adapted continuous process on Œ0; C1/.
(ii) For all t 2 Œ0; C1/ and ! 2 ˝ 0 , we have
Z
ju.t ^ /jpp D ju0 jpp C
C 12 p. p
Z

t^

p
0

Z
1/

Z

Cp
0

Z

t^

Rd

ju.s/jp 2 u.s/f .s/dx

ju.s/jp

2

Rd

1
X

Á
jgk .s/j2 dx ds

(1.30)

kD1

ju.s/jp 2 u.s/

Rd

1
X

gk .s/dx dWk .s/:

kD1

We note also that Proposition 1.2.3 remains true on domains O
Rd by
replacing u with u1O :
The proof of Proposition 1.2.3 is given in the above quoted paper by Krylov [71].
Here we confine only to point out the main steps of the proof.
First it turns out that we may replace u in (1.29) by a function measurable with
respect to
F

1

B.Œ0; s/

B.Rd /

Here we use h ; i2 also to denote the duality between C01 .Rd / and the space of distributions.


12

1 Introduction

such that for each ; u.t; / is Ft -adapted, u.t; ; !/ is continuous in t 2 .0; 1/ for
each .!; / 2 ˝ Rd and u.t; ; !/ as a function of .t; !/ is Lp -valued, Ft -adapted
and continuous in t for each !.
Next consider a mollifier . / D d . / and set
f .t/ D .f

/.t/;

u .t/ D .u

/.t/;

0:

t

We obtain the equation
Z
u .t; / D .u0 / . / C

t
0

Z
f .s; /ds C

t
0

.gj / .s; /dWj .s/:

Since u is regular we may apply Itô’s formula and get P-a.s.
Z
ju

.t/jpp

D j.u0 /
Z

t

C
0

jpp

t

Cp
0

ju .s/jp 2 u .s/.gj / .s/dWj .s/

pju .s/jp 2 u .s/f .s/ds
Z

1
2

C p. p

1/

t
0

ju .s/jp 2 u.s/

1
X

j.gk / .s/jds:

kD1

Then formula (1.30) follows after some a priori estimates involving the stochastic
Fubini theorem and letting ! 0.
We shall apply the following version of a martingale convergence result (see
e.g.[75, p. 139]).
Lemma 1.2.4 Let Z be a nonnegative semimartingale with E.Z.t// < 1; 8 t
and let I be a nondecreasing continuous process such that

Z.t/ C I.t/ D Z.0/ C I1 .t/ C M.t/;
where M is a local martingale. Then if lim I1 .t/ < 1;
t!1

lim Z.t/ C I.1/ < 1;

t!1

8t

0;

0

(1.31)

P-a.s.; we have

P-a.s.

(1.32)

1.2.4 Monotone Operators
Let H be a real Hilbert space with the scalar product h ; i and norm j j. A multivalued
mapping G W D.G/ H ! 2H is called monotone if
hu

v; x

yi

0;

8 x 2 G.u/; y 2 G.v/:


1.2 Preliminaries

13

A monotone mapping G is called maximal monotone if 1 C ˛G is surjective for all
˛ > 0 (Equivalently for some ˛ > 0.) If G is maximal monotone we set
J˛ .x/ D .1 C ˛G/ 1 .x/;

˛ > 0; x 2 H:

(Here 1 is the identity operator in H.)
Lemma 1.2.5 J˛ is Lipschitzian and
jJ˛ .x/

J˛ .y/j Ä jx

yj;

8 x; y 2 H:

(1.33)

Proof Set x˛ D J˛ .x/; y˛ D J˛ .y/; so that
x D x˛ C ˛G.x˛ /;

y D y˛ C ˛G.y˛ /:

Then
y˛ C ˛.G.x˛ /

x˛

G.y˛ // D x

y:

Multiplying both sides by x˛ y˛ and taking into account the accretivity of G, yields
jx˛

y˛ j2 Ä hx˛

y˛ ; x

yi
t
u

and the conclusion follows easily.
We define the Yosida approximations G˛ W H ! H of G setting for any ˛ > 0.
G˛ D

1
.1
˛

J˛ /:

(1.34)

Since J˛ is 1-Lipschitz, it follows that G˛ is ˛2 -Lipschitz on H.
Proposition 1.2.6 Let ˛ > 0. Then we have
(i) G˛ .x/ 2 G.J˛ .x//;
(ii) jG˛ .x/j Ä jG0 .x/j;

8 x 2 H:
8 x 2 D.G/; where G0 .x/ is the minimal section of G.x/.

Proof
(i) Let ˛ > 0 and x 2 H. Then
G˛ .x/ D

1
Œ.1
˛

G˛ .x/ D

1
ŒJ˛ .x/.x
˛

˛G/.J˛ .x//

J˛ .x/ D G.J˛ .x//:

(ii) We have
˛y/

J˛ .x/;

8 y 2 G.x/:


14

1 Introduction

Since J˛ is 1-Lipschitz, it follows that
jG˛ .x/j Ä

1
jx
˛

.x

˛y/j D jyj;

8 y 2 G.x/
t
u

and (ii) follows.

If ' W H ! R WD . 1; C1 is a convex and lower semicontinuous function we
denote its subdifferential by @', that is
@'.x/ D fy 2 H W '.x/ Ä '.u/ C hy; x

ui; 8 u 2 Hg:

(1.35)

Then @' W H ! H is maximal monotone and its Yosida approximation .@'/˛ is
monotone, Lipschitz and it is given by
.@'/˛ .x/ D r'˛ .x/;

8 x 2 H;

(1.36)

where the convex function '˛ W H ! R, defined by
'˛ .x/ D inf '.y/ C

jx

yj2
W y2H
2˛

˛
D '..1 C ˛@'/ x/ C jx
2
1

(1.37)
1

2

.1 C ˛@'/ xj ;

is the Moreau regularization of '.
0
More generally, if X is a Banach space with dual X 0 , the operator G W X ! 2X is
said to be maximal monotone if it is monotone, that is
.u

v; x

y/

0;

8 u 2 G.x/; v 2 G.y/;

and ˛F C G W X ! X 0 is surjective for all ˛ > 0. Here F W X ! X 0 is the duality
mapping of X and . ; / is the duality pairing of X; X 0 . A maximal monotone operator
G W X ! X 0 is strongly–weakly closed, that is if yn 2 G.xn / and xn ! x strongly in
X and yn ! y weakly in X 0 then y 2 G.x/. Also G is weakly–strongly closed. We
note that if G W X ! X 0 is monotone and demicontinuous (that is strongly–weakly
continuous) it is maximal monotone (Minty–Browder theorem).
An important example of a maximal monotone operator is the subdifferential @' W
0
N D . 1; C1,

X ! 2X of a convex lower semicontinuous function, ' W X ! R
that is
@'.x/ D fy 2 X 0 W '.x/ Ä '.u/ C .y; x

u/;

8 u 2 Xg:

If ˇ W R ! 2R is a maximal monotone mapping (graph), that is
.u

v/.x

y/

0;

8 u 2 ˇ.x/; v 2 ˇ.y/;


1.2 Preliminaries

15

and .1 C ˇ/.R/ D R, then there is a unique convex lower semicontinuous function
j W R ! R such that @J D ˇ. This function j is called the potential of ˇ. (The
uniqueness of j is up to additive constants.)
Let j denote the conjugate of j (the Legendre transform of j), that is
j . p/ D supfpy
We recall that .@j/ D @j

1

j.y/ W y 2 Rg

(see e.g. [14]),

j.y/ C j . p/ D py if and only if p 2 @j.y/

(1.38)

and
j.y/ C j . p/

py

for all j; p 2 R:

(1.39)

If ˇ˛ ; ˛ > 0; is the Yosida approximation of ˇ we set
Z
j˛ .u/ D

u
0

ˇ˛ .r/dr;

u2R

and note that j˛ is just the Moreau approximation of j, that is
j˛ .u/ D min j.v/ C

1
ju
2˛

vj2 ;

v2R :

(1.40)

We have
j˛ .u/ D .1 C ˛ˇ/ 1 .u/ C

1
ju
2˛

.1 C ˛ˇ/ 1 .u/j2 :

(1.41)

Below we present a few examples of maximal monotone operators (see [6, 35])
Example 1.2.7 Let g W R ! . 1; C1 be a lower semicontinuous convex
e ! R be defined by
function and let ' W Lp .O/
8Z

ˆ
ˆ g.y. // .d /;
<
e
O
'.y/ D
ˆ
ˆ
:
C1 otherwise;

e
if g.y/ 2 L1 .O/
(1.42)

e is a measure space endowed with a -finite measure
where O
e and
Then ' is convex, lower semicontinuous on Lp .O/
e W z. / 2 @g.y. //;
@'.y/ D fz 2 Lq .O/

-a.e.

e
2 Og;

and 1 Ä p < 1.

(1.43)