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Differential Inclusion approach in Nonlinear PDEs
Swarnendu Sil
M.Phil Thesis TalkThesis Advisor: Prof. M. Vanninathan
Tata Institute of Fundamental ResearchCentre for Applicable
Mathematics
Bangalore, India
20th July, 2012
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 1 / 49
-
Introduction Brief outline
Introduction
In this talk, oscillations in nonlinear partial differential
equations is, in asense, our main theme.
More particularly, the framework of compensated compactness,
introduced byTartar (see [Tartar, 1979]), to analyze oscillations
in sequences of approximate orexact solutions to nonlinear partial
differential equations (henceforth PDEs) andits connection and
interrelation with methods to solve differential inclusionsand the
application of these ideas in nonlinear PDEs.
So our main focus will be on
Compensated compactness
Differential Inclusions
Casting nonlinear PDEs as differential inclusions and using the
aboveframework to construct solutions
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 2 / 49
-
Introduction Brief outline
Introduction
In this talk, oscillations in nonlinear partial differential
equations is, in asense, our main theme.
More particularly, the framework of compensated compactness,
introduced byTartar (see [Tartar, 1979]), to analyze oscillations
in sequences of approximate orexact solutions to nonlinear partial
differential equations (henceforth PDEs) andits connection and
interrelation with methods to solve differential inclusionsand the
application of these ideas in nonlinear PDEs.
So our main focus will be on
Compensated compactness
Differential Inclusions
Casting nonlinear PDEs as differential inclusions and using the
aboveframework to construct solutions
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 2 / 49
-
Introduction Brief outline
Introduction
In this talk, oscillations in nonlinear partial differential
equations is, in asense, our main theme.
More particularly, the framework of compensated compactness,
introduced byTartar (see [Tartar, 1979]), to analyze oscillations
in sequences of approximate orexact solutions to nonlinear partial
differential equations (henceforth PDEs) andits connection and
interrelation with methods to solve differential inclusionsand the
application of these ideas in nonlinear PDEs.
So our main focus will be on
Compensated compactness
Differential Inclusions
Casting nonlinear PDEs as differential inclusions and using the
aboveframework to construct solutions
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 2 / 49
-
Introduction Preliminary settings
Casting nonlinear PDEs in the framework
Many nonlinear partial differential equations can be written as
a system of linearPDEs together with a nonlinear pointwise
constitutive relations.
Motivation
The motivation for the idea comes from physics, since most PDEs
in physics areindeed derived using this very principle in the first
place. One figures out theconstitutive relations directly from the
physical law and one has the balanceequations for certain
quantities. One then substitutes the constitutive relationsinto the
balance equations and obtain the final form of the PDE. Since
mostequations in physics are derived in this way in the first
place, there can be littledoubt that these nonlinear PDEs can
obviously be cast into that form.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 3 / 49
-
Introduction Preliminary settings
Casting nonlinear PDEs in the framework
Many nonlinear partial differential equations can be written as
a system of linearPDEs together with a nonlinear pointwise
constitutive relations.
Motivation
The motivation for the idea comes from physics, since most PDEs
in physics areindeed derived using this very principle in the first
place. One figures out theconstitutive relations directly from the
physical law and one has the balanceequations for certain
quantities. One then substitutes the constitutive relationsinto the
balance equations and obtain the final form of the PDE. Since
mostequations in physics are derived in this way in the first
place, there can be littledoubt that these nonlinear PDEs can
obviously be cast into that form.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 3 / 49
-
Introduction Preliminary settings
Framework
Idea
We want to disentangle the role of the linear differential
structure and thealgebraic nonlinear structure of the nonlinear
operator.
Setting
We consider nonlinear PDEs that can be expressed as a system of
linear PDEs,called (balance laws)
m∑i=1
Ai∂iz = 0 (1)
coupled with a pointwise nonlinear constraint (constitutive
relations)
z(x) ∈ K ⊂ Rd a.e. x ∈ Ω (2)
where z : Ω ⊂ Rm → Rd is the unknown state variable.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 4 / 49
-
Introduction Preliminary settings
Framework
Idea
We want to disentangle the role of the linear differential
structure and thealgebraic nonlinear structure of the nonlinear
operator.
Setting
We consider nonlinear PDEs that can be expressed as a system of
linear PDEs,called (balance laws)
m∑i=1
Ai∂iz = 0 (1)
coupled with a pointwise nonlinear constraint (constitutive
relations)
z(x) ∈ K ⊂ Rd a.e. x ∈ Ω (2)
where z : Ω ⊂ Rm → Rd is the unknown state variable.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 4 / 49
-
Introduction Preliminary settings
General Problem
The general problem of the compensated compactness framework can
now bestated as follows:
Describe the oscillations in a weakly convergent sequence
offunctions z� : Ω ⊂ Rm → Rd subject to linear differential
constraints, calledbalance laws, of the form
m∑i=1
Ai∂iz� = φ� (3)
and nonlinear algebraic constraints, called constitutive
relations, of the form
{z�(y)} ⊂ M (4)
for almost all y in Ω. Here Ai denotes a constant s × d matrix
with s arbitraryand fixed. M is a subset of the state space Rd and
is usually a manifold.
In compensated compactness theory, one is particularly
interested in determininghow differential and algebraic constraints
collaborate to suppress oscillations in z�.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 5 / 49
-
Introduction Preliminary settings
General Problem
The general problem of the compensated compactness framework can
now bestated as follows: Describe the oscillations in a weakly
convergent sequence offunctions z� : Ω ⊂ Rm → Rd subject to linear
differential constraints, calledbalance laws, of the form
m∑i=1
Ai∂iz� = φ� (3)
and nonlinear algebraic constraints, called constitutive
relations, of the form
{z�(y)} ⊂ M (4)
for almost all y in Ω.
Here Ai denotes a constant s × d matrix with s arbitraryand
fixed. M is a subset of the state space Rd and is usually a
manifold.
In compensated compactness theory, one is particularly
interested in determininghow differential and algebraic constraints
collaborate to suppress oscillations in z�.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 5 / 49
-
Introduction Preliminary settings
General Problem
The general problem of the compensated compactness framework can
now bestated as follows: Describe the oscillations in a weakly
convergent sequence offunctions z� : Ω ⊂ Rm → Rd subject to linear
differential constraints, calledbalance laws, of the form
m∑i=1
Ai∂iz� = φ� (3)
and nonlinear algebraic constraints, called constitutive
relations, of the form
{z�(y)} ⊂ M (4)
for almost all y in Ω. Here Ai denotes a constant s × d matrix
with s arbitraryand fixed. M is a subset of the state space Rd and
is usually a manifold.
In compensated compactness theory, one is particularly
interested in determininghow differential and algebraic constraints
collaborate to suppress oscillations in z�.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 5 / 49
-
Introduction Preliminary settings
General Problem
The general problem of the compensated compactness framework can
now bestated as follows: Describe the oscillations in a weakly
convergent sequence offunctions z� : Ω ⊂ Rm → Rd subject to linear
differential constraints, calledbalance laws, of the form
m∑i=1
Ai∂iz� = φ� (3)
and nonlinear algebraic constraints, called constitutive
relations, of the form
{z�(y)} ⊂ M (4)
for almost all y in Ω. Here Ai denotes a constant s × d matrix
with s arbitraryand fixed. M is a subset of the state space Rd and
is usually a manifold.
In compensated compactness theory, one is particularly
interested in determininghow differential and algebraic constraints
collaborate to suppress oscillations in z�.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 5 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not suppress.
Instead we want the differential structure to allow many types
of oscillation, sothat the balance laws (1) has a large, infinite
family of oscillatory solutions.We want this family to be so large
that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not
suppress.Instead we want the differential structure to allow many
types of oscillation, sothat the balance laws (1) has a large,
infinite family of oscillatory solutions.
We want this family to be so large that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not
suppress.Instead we want the differential structure to allow many
types of oscillation, sothat the balance laws (1) has a large,
infinite family of oscillatory solutions.We want this family to be
so large that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not
suppress.Instead we want the differential structure to allow many
types of oscillation, sothat the balance laws (1) has a large,
infinite family of oscillatory solutions.We want this family to be
so large that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not
suppress.Instead we want the differential structure to allow many
types of oscillation, sothat the balance laws (1) has a large,
infinite family of oscillatory solutions.We want this family to be
so large that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
Our Target
We do not want to ascertain that the differential structure
kills all oscillationsthat the nonlinearity can not
suppress.Instead we want the differential structure to allow many
types of oscillation, sothat the balance laws (1) has a large,
infinite family of oscillatory solutions.We want this family to be
so large that
It is possible to ensure that there are rich supply of solutions
to (1),which takes values in some convex hulls of K ,
We can perturb these solutions of (1) by adding controlled
oscillationto generate a sequence in such a way that each term of
the sequencestill remains a solution of (1), but the image of each
term approachesthe constitutive set K .
Moreover, we want to be able to add controlled oscillations
allowed by thedifferential structure in such a way such that the
resulting sequence of solutions to(1), apriori only weakly
convergent, actually converges strongly and the stronglimit of this
sequence takes its values in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 6 / 49
-
Introduction Preliminary settings
The strong convergence is possible precisely because of our
ability to addcontrolled oscillation and is proved either by convex
integration or Bairecategory arguments, two already familiar
techniques in the theory of differentialinclusions.
The theory of differential inclusions have a long history though
their application tononlinear PDEs are rather new. Cellina
considered ordinary differential inclusionsrelated to problems in
optimal control (Cf. [Cellina, 1980],[Aubin and Cellina, 1984]).
Gromov studied extensively partial differentialinclusions in his
classic ‘Partial differential relations’ [Gromov, 1986]. An
excellentreference for both differential inclusions and their
applications to nonlinear PDEscan be found in [Dacorogna and
Marcellini, 1999],[Dacorogna and Marcellini, 1997]. We shall
illustrate the basic idea of solving adifferential inclusion by
using the most well-studied example of partial
differentialinclusions, namely the gradient inclusion. We will also
show how these ideastogether with compensated compactness framework
gives rise to a mechanism forgenerating oscillatory solutions to
nonlinear PDEs.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 7 / 49
-
Introduction Preliminary settings
The strong convergence is possible precisely because of our
ability to addcontrolled oscillation and is proved either by convex
integration or Bairecategory arguments, two already familiar
techniques in the theory of differentialinclusions.
The theory of differential inclusions have a long history though
their application tononlinear PDEs are rather new. Cellina
considered ordinary differential inclusionsrelated to problems in
optimal control (Cf. [Cellina, 1980],[Aubin and Cellina, 1984]).
Gromov studied extensively partial differentialinclusions in his
classic ‘Partial differential relations’ [Gromov, 1986]. An
excellentreference for both differential inclusions and their
applications to nonlinear PDEscan be found in [Dacorogna and
Marcellini, 1999],[Dacorogna and Marcellini, 1997]. We shall
illustrate the basic idea of solving adifferential inclusion by
using the most well-studied example of partial
differentialinclusions, namely the gradient inclusion. We will also
show how these ideastogether with compensated compactness framework
gives rise to a mechanism forgenerating oscillatory solutions to
nonlinear PDEs.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 7 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Introduction Preliminary settings
General Strategy
Cast nonlinear PDEs in the framework of Compensated
Compactness.
Reduce it to a differential inclusion.
Solve the inclusion.
We solve the inclusion by
Understanding the possible oscillations of sequences of
approximate solutions
Finding a way to add controlled oscillation to a sequence of
approximatesolutions
Obtaining strong convergence by iterating this procedure.
Passing to the limit, that is, by showing the strong limit
solves the PDE
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 8 / 49
-
Compensated Compactness General setting
Compensated Compactness
Let zn : Ω ⊂ Rm → Rd be a sequence of functions such that zn ⇀ z
in (L∞(Ω))dweak ∗. Also {zn} satisfies a system of linear
differential constraints, calledbalance laws, of the form
m∑i=1
Ai∂izn = φn (5)
and nonlinear algebraic constraints, called constitutive
relations, of the form
zn(x) ∈ K (6)
for all n for almost all x in Ω. Here Ai denotes a constant s ×
d matrix with sarbitrary and fixed. Ω is a bounded open subset of
Rm and K is a subset of thestate space Rd . Assume also that φn → φ
in some suitable topology on a suitablefunction space.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 9 / 49
-
Compensated Compactness General setting
We want to determine when it is possible to assert that
m∑i=1
Ai∂iz = φ (7)
andz(x) ∈ K (8)
for a.e. x in Ω.
Note that these questions are not the ones we will be pursuing
for long. But weare more interested in the framework than the
theory. Understanding the effect oflinear differential structure
and the nonlinear algebraic structure seperately on thepossible
oscillations of a weakly convergent sequence of approximate
solutions isour aim. The principle aim of this talk is show that
this knowledge might alsohelps us to solve problems which are in a
sense, outside what was initially thoughtto be the target of
compensated compactness theory.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 10 / 49
-
Compensated Compactness General setting
We want to determine when it is possible to assert that
m∑i=1
Ai∂iz = φ (7)
andz(x) ∈ K (8)
for a.e. x in Ω.
Note that these questions are not the ones we will be pursuing
for long. But weare more interested in the framework than the
theory. Understanding the effect oflinear differential structure
and the nonlinear algebraic structure seperately on thepossible
oscillations of a weakly convergent sequence of approximate
solutions isour aim. The principle aim of this talk is show that
this knowledge might alsohelps us to solve problems which are in a
sense, outside what was initially thoughtto be the target of
compensated compactness theory.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 10 / 49
-
Compensated Compactness Without differential constraint
Without Differential Constraints
The first question we can ask is whether we can assert that z(y)
∈ K for a.e y inΩ without any information on the derivatives of zn,
that is, without (5). Moreprecisely, we are asking if the
informations zn ⇀ z in (L
∞(Ω))d weak ∗ andzn(x) ∈ K for a.e. x in Ω is enough to imply
z(y) ∈ K for a.e y in Ω. Thefollowing theorem answers this question
in the negative.
Theorem
(1) Let zn : Ω→ Rd be such that zn ⇀ z in (L∞(Ω))d weak ∗ and
zn(x) ∈ K a.e.Then z(x) ∈ conv(K ) a.e (where conv(K ) is the
closed convex hull of K ).(2) Conversely, let z ∈ (L∞(Ω))d and z(x)
∈ conv(K ) a.e, then there exists asequence {zn} such that zn ⇀ z
in (L∞(Ω))d weak ∗ and zn(x) ∈ K a.e. for all n.
The following lemma is crucial in the proof of the theorem..
Lemma
If f is a convex continuous function on Rd , if zn ⇀ z in
(L∞(Ω))d weak ∗, and iff (zn) ⇀ l then l ≥ f (z) a.e.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 11 / 49
-
Compensated Compactness Without differential constraint
Without Differential Constraints
The first question we can ask is whether we can assert that z(y)
∈ K for a.e y inΩ without any information on the derivatives of zn,
that is, without (5). Moreprecisely, we are asking if the
informations zn ⇀ z in (L
∞(Ω))d weak ∗ andzn(x) ∈ K for a.e. x in Ω is enough to imply
z(y) ∈ K for a.e y in Ω. Thefollowing theorem answers this question
in the negative.
Theorem
(1) Let zn : Ω→ Rd be such that zn ⇀ z in (L∞(Ω))d weak ∗ and
zn(x) ∈ K a.e.Then z(x) ∈ conv(K ) a.e (where conv(K ) is the
closed convex hull of K ).(2) Conversely, let z ∈ (L∞(Ω))d and z(x)
∈ conv(K ) a.e, then there exists asequence {zn} such that zn ⇀ z
in (L∞(Ω))d weak ∗ and zn(x) ∈ K a.e. for all n.
The following lemma is crucial in the proof of the theorem..
Lemma
If f is a convex continuous function on Rd , if zn ⇀ z in
(L∞(Ω))d weak ∗, and iff (zn) ⇀ l then l ≥ f (z) a.e.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 11 / 49
-
Compensated Compactness Without differential constraint
Lemma 2 immediately suggests the following questions:
Q1. For which functions f do we have f (zn) ⇀ f (z), that is, f
is sequentiallyweakly continuous ?
Q2. For which functions f do we have if f (zn) ⇀ l , then l ≥ f
(z), that is, f issequentially weakly lower semicontinuous?
Theorem
Let zn : Ω→ Rd be such that zn ⇀ z in (L∞(Ω))d weak ∗ and zn(x)
∈ K a.e..Then we have,
(1) z(x) ∈ K if and only if K is closed and convex.(2) f (zn) ⇀
f (z) in (L∞)N weak ∗ if and only if f is affine on conv(K ).(3) If
f (zn) ⇀ l in (L∞)N weak ∗, then l ≥ f (z) if and only if f is
convex on
conv(K ).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 12 / 49
-
Compensated Compactness Without differential constraint
Lemma 2 immediately suggests the following questions:
Q1. For which functions f do we have f (zn) ⇀ f (z), that is, f
is sequentiallyweakly continuous ?
Q2. For which functions f do we have if f (zn) ⇀ l , then l ≥ f
(z), that is, f issequentially weakly lower semicontinuous?
Theorem
Let zn : Ω→ Rd be such that zn ⇀ z in (L∞(Ω))d weak ∗ and zn(x)
∈ K a.e..Then we have,
(1) z(x) ∈ K if and only if K is closed and convex.(2) f (zn) ⇀
f (z) in (L∞)N weak ∗ if and only if f is affine on conv(K ).(3) If
f (zn) ⇀ l in (L∞)N weak ∗, then l ≥ f (z) if and only if f is
convex on
conv(K ).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 12 / 49
-
Compensated Compactness With differential constraints
With Differential Constraints
We state the problem in the same way as before.Ω and K are both
assumedbounded.
u�1, . . . , u�d ⇀ u1, . . . , ud in L
∞(Ω) weak ∗ (9)u� ∈ K a.e. (10)∑
j,k
aijk∂u�j∂xk∈ bounded set in L∞ (11)
for i = 1, . . . , q where the aijk are real constants.
The questions are the same as before, that is,
Q1. Does u(x) ∈ K a.e.?Q2. For which functions f do we have f
(u�) ⇀ f (u), that is, f is sequentially
weakly continuous ?
Q3. For which functions f do we have if f (u�) ⇀ l , then l ≥ f
(u), that is, f issequentially weakly lower semicontinuous?
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 13 / 49
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Compensated Compactness With differential constraints
With Differential Constraints
We state the problem in the same way as before.Ω and K are both
assumedbounded.
u�1, . . . , u�d ⇀ u1, . . . , ud in L
∞(Ω) weak ∗ (9)u� ∈ K a.e. (10)∑
j,k
aijk∂u�j∂xk∈ bounded set in L∞ (11)
for i = 1, . . . , q where the aijk are real constants.
The questions are the same as before, that is,
Q1. Does u(x) ∈ K a.e.?Q2. For which functions f do we have f
(u�) ⇀ f (u), that is, f is sequentially
weakly continuous ?
Q3. For which functions f do we have if f (u�) ⇀ l , then l ≥ f
(u), that is, f issequentially weakly lower semicontinuous?
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 13 / 49
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Compensated Compactness With differential constraints
The partial answer to these questions will rely heavily on the
following definitions:
Definition (Oscillation Variety:)
The oscillation variety corresponding to (11) is the set V,
defined by,
V :=
(λ, ξ) ∈ Rd × Rm\{0} : ∑j,k
aijkλjξk = 0 for i = 1, . . . , q
. (12)
Definition (Wave Cone:)
The wave cone corresponding to (11) is the projection of V on
the physical space,defined by,
Λ :=
λ ∈ Rd : ∃ξ ∈ Rm\{0} with ∑j,k
aijkλjξk = 0 for i = 1, . . . , q
. (13)
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 14 / 49
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Compensated Compactness With differential constraints
The partial answer to these questions will rely heavily on the
following definitions:
Definition (Oscillation Variety:)
The oscillation variety corresponding to (11) is the set V,
defined by,
V :=
(λ, ξ) ∈ Rd × Rm\{0} : ∑j,k
aijkλjξk = 0 for i = 1, . . . , q
. (12)
Definition (Wave Cone:)
The wave cone corresponding to (11) is the projection of V on
the physical space,defined by,
Λ :=
λ ∈ Rd : ∃ξ ∈ Rm\{0} with ∑j,k
aijkλjξk = 0 for i = 1, . . . , q
. (13)
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 14 / 49
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Compensated Compactness With differential constraints
Remark
(1) In the case without derivatives, we have V = Rd × Rm\{0} and
Λ = Rd . IfΛ = Rd , this means (11) contains very little
information.
(2) In the compactness case, we have Λ = {0}, and this happens
if the list (11)
contains all the derivatives∂u�j∂xk
seperately , which are thus all bounded.
Theorem
If (9), (10) and (11) imply u(x) ∈ K a.e., then K must satisfy
the followingconditions:
NC0: K is closed.
NC1: If a, b ∈ K = K and b − a ∈ Λ, then the segment [a, b] is
in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 15 / 49
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Compensated Compactness With differential constraints
Remark
(1) In the case without derivatives, we have V = Rd × Rm\{0} and
Λ = Rd . IfΛ = Rd , this means (11) contains very little
information.
(2) In the compactness case, we have Λ = {0}, and this happens
if the list (11)
contains all the derivatives∂u�j∂xk
seperately , which are thus all bounded.
Theorem
If (9), (10) and (11) imply u(x) ∈ K a.e., then K must satisfy
the followingconditions:
NC0: K is closed.
NC1: If a, b ∈ K = K and b − a ∈ Λ, then the segment [a, b] is
in K .
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 15 / 49
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Compensated Compactness With differential constraints
Corollary
A necessary condition on K and f so that (Q2) can be answered
positively is:If a1, a2, . . . , ad ∈ K , ξ 6= 0 with (aj − ak , ξ)
∈ V for all j , k, then f is affine onconv{a1, . . . , ad}.
Corollary
A necessary condition on K and f such that (Q3) can be answered
positively is:If a1, a2, . . . , ad ∈ K , ξ 6= 0 with (aj − ak , ξ)
∈ V for all j , k, then f is convex onconv{a1, . . . , ad}.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 16 / 49
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Compensated Compactness With differential constraints
Corollary
A necessary condition on K and f so that (Q2) can be answered
positively is:If a1, a2, . . . , ad ∈ K , ξ 6= 0 with (aj − ak , ξ)
∈ V for all j , k, then f is affine onconv{a1, . . . , ad}.
Corollary
A necessary condition on K and f such that (Q3) can be answered
positively is:If a1, a2, . . . , ad ∈ K , ξ 6= 0 with (aj − ak , ξ)
∈ V for all j , k, then f is convex onconv{a1, . . . , ad}.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 16 / 49
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Compensated Compactness With differential constraints
Theorem
Suppose we have the following hypotheses
(H1′) u�i ⇀ ui in L2(Ω) weak for i = 1, . . . , d
(H3′)∑j,k
aijk∂u�j∂xk∈ a compact set (for the strong topology) of H−1loc
(Ω),
for i = 1, . . . , d.
Then if Q is quadratic and satisfies Q(λ) ≥ 0 for all λ ∈ Λ, and
if Q(u�) ⇀ l inthe sense of distributions ( l may be a measure),
then
l ≥ Q(u) (in the sense of measures).
Remark
(1) If Q is quadratic to say that Q is Λ-convex is equivalent to
saying Q(λ) ≥ 0,for all λ ∈ Λ.
(2) This theorem asserts that if f is quadratic then the
necessary conditionstated above in Corollary is also
sufficient.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 17 / 49
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Compensated Compactness With differential constraints
Corollary
If Q is quadratic and satisfies Q(λ) = 0 for all λ ∈ Λ, and if
{u�} satisfies (H1′)and (H3′), then Q(u�) ⇀ Q(u) in the sense of
distributions.
Corollary
Div-Curl Lemma: Let
v � ⇀ v in (L2(Ω))m weak , (14)
w � ⇀ w in (L2(Ω))m weak , (15)
div v � → div v in H−1(Ω) strong , (16)
curl w � → curl w in (H−1(Ω))m2
strong. (17)
Thenv � · w � ⇀ v · w in D′ (Ω) (18)
and this is the only nonlinear functional that is (sequentially)
weakly continuous.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 18 / 49
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Compensated Compactness With differential constraints
Corollary
If Q is quadratic and satisfies Q(λ) = 0 for all λ ∈ Λ, and if
{u�} satisfies (H1′)and (H3′), then Q(u�) ⇀ Q(u) in the sense of
distributions.
Corollary
Div-Curl Lemma: Let
v � ⇀ v in (L2(Ω))m weak , (14)
w � ⇀ w in (L2(Ω))m weak , (15)
div v � → div v in H−1(Ω) strong , (16)
curl w � → curl w in (H−1(Ω))m2
strong. (17)
Thenv � · w � ⇀ v · w in D′ (Ω) (18)
and this is the only nonlinear functional that is (sequentially)
weakly continuous.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 18 / 49
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Compensated Compactness Perspectives
Upto this point we asked the question:
Given a sequence of approximate solutions {zn} satisfying (5)
and (6), when is itpossible to assert that the weak limit z of the
sequence {zn} satisfies (7) and (8).
Now we will ask a question which is, in a sense reverse of the
previous one.
Given a sequence of approximate solutions {zn} satisfying
m∑i=1
Ai∂izn = φ (19)
andzn(x) ∈ Kn (20)
for all n for almost all x and Kn is a subset of the state space
Rd containing K forall n, when can we assert that there exists a z
satisfying (7) and (8).
This question however is a hopeless pursuit. Too little
information to concludeanything meaningful.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 19 / 49
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Compensated Compactness Perspectives
Upto this point we asked the question:
Given a sequence of approximate solutions {zn} satisfying (5)
and (6), when is itpossible to assert that the weak limit z of the
sequence {zn} satisfies (7) and (8).
Now we will ask a question which is, in a sense reverse of the
previous one.
Given a sequence of approximate solutions {zn} satisfying
m∑i=1
Ai∂izn = φ (19)
andzn(x) ∈ Kn (20)
for all n for almost all x and Kn is a subset of the state space
Rd containing K forall n, when can we assert that there exists a z
satisfying (7) and (8).
This question however is a hopeless pursuit. Too little
information to concludeanything meaningful.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 19 / 49
-
Compensated Compactness Perspectives
Upto this point we asked the question:
Given a sequence of approximate solutions {zn} satisfying (5)
and (6), when is itpossible to assert that the weak limit z of the
sequence {zn} satisfies (7) and (8).
Now we will ask a question which is, in a sense reverse of the
previous one.
Given a sequence of approximate solutions {zn} satisfying
m∑i=1
Ai∂izn = φ (19)
andzn(x) ∈ Kn (20)
for all n for almost all x and Kn is a subset of the state space
Rd containing K forall n, when can we assert that there exists a z
satisfying (7) and (8).
This question however is a hopeless pursuit. Too little
information to concludeanything meaningful.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 19 / 49
-
Compensated Compactness Perspectives
Upto this point we asked the question:
Given a sequence of approximate solutions {zn} satisfying (5)
and (6), when is itpossible to assert that the weak limit z of the
sequence {zn} satisfies (7) and (8).
Now we will ask a question which is, in a sense reverse of the
previous one.
Given a sequence of approximate solutions {zn} satisfying
m∑i=1
Ai∂izn = φ (19)
andzn(x) ∈ Kn (20)
for all n for almost all x and Kn is a subset of the state space
Rd containing K forall n, when can we assert that there exists a z
satisfying (7) and (8).
This question however is a hopeless pursuit. Too little
information to concludeanything meaningful.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 19 / 49
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Compensated Compactness Perspectives
With a lot of hindsight we rephrase and refine the question
if we know that there is set U such that it is possible to find
solutions of (19)taking values in U and it is possible to push the
distribution of its values inside Uwithout ever leaving the
solution set of (19), can we solve (19) and (20) ?
We will see that under appropriate assumptions, indeed we
can.
We already know there is something regarding convexity is
involved there. In thecase with differential constraints,
Λ-convexity plays a role. In the case withoutdifferential
constraints we saw if we start with solutions taking values in K ,
thelimit of those solutions will land up taking values in the
closed convex hull of K .Some information on the derivatives
enabled us to replace convexity bysome more general and weaker
notion of convexity. We now ask if we startwith solutions taking
values in the appropriate convex hulls of K , is it possibleto land
up, in the limit, in K ?
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 20 / 49
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Compensated Compactness Perspectives
With a lot of hindsight we rephrase and refine the question
if we know that there is set U such that it is possible to find
solutions of (19)taking values in U and it is possible to push the
distribution of its values inside Uwithout ever leaving the
solution set of (19), can we solve (19) and (20) ?
We will see that under appropriate assumptions, indeed we
can.
We already know there is something regarding convexity is
involved there. In thecase with differential constraints,
Λ-convexity plays a role. In the case withoutdifferential
constraints we saw if we start with solutions taking values in K ,
thelimit of those solutions will land up taking values in the
closed convex hull of K .Some information on the derivatives
enabled us to replace convexity bysome more general and weaker
notion of convexity. We now ask if we startwith solutions taking
values in the appropriate convex hulls of K , is it possibleto land
up, in the limit, in K ?
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 20 / 49
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Differential Inclusion Nonlinear PDEs as Differential
Inclusions
Nonlinear PDEs as Differential Inclusions
We consider nonlinear PDEs that can be expressed as a system of
linear PDEs(balance laws)
m∑i=1
Ai∂iz = 0 (21)
coupled with a pointwise nonlinear constraint (constitutive
relations)
z(x) ∈ K ⊂ Rd a.e (22)
where z : Ω ⊂ Rm → Rd is the unknown state variable.
In certain cases, this problem can be transformed into a
differentialinclusion problem.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 21 / 49
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Differential Inclusion Nonlinear PDEs as Differential
Inclusions
Nonlinear PDEs as Differential Inclusions
We consider nonlinear PDEs that can be expressed as a system of
linear PDEs(balance laws)
m∑i=1
Ai∂iz = 0 (21)
coupled with a pointwise nonlinear constraint (constitutive
relations)
z(x) ∈ K ⊂ Rd a.e (22)
where z : Ω ⊂ Rm → Rd is the unknown state variable.
In certain cases, this problem can be transformed into a
differentialinclusion problem.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 21 / 49
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Differential Inclusion Nonlinear PDEs as Differential
Inclusions
If there exists a potential E and a matrix of partial
differential operators P(D)such that z = P(D)E identically solves
(21), that is,
m∑i=1
Ai∂i (P(D)E ) = 0
then (21) and (22) together reduces to
P(D)E ∈ K ⊂ Rd a.e , (23)
which is a differential inclusion problem, where E is usually a
tensor and P(D) is amatrix of partial differential operators, that
is, each row of P(D) is a partialdifferential operator.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 22 / 49
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Differential Inclusion Gradient Inclusion
Gradient Inclusion
We shall now illustrate the basic idea of solving a differential
inclusion by usingthe most well-studied example of partial
differential inclusions, namely thegradient inclusion. The
presentation in the following section is influenced byworks of
Kirchheim and Sychev (Cf. [Sychev, 2001], [Sychev,
2006],[Kirchheim, 2003], [Kirchheim, 2001]).
Gradient inclusion
Consider the basic gradient inclusion problem with Dirichlet
boundary data, wherewe will be looking for Lipschitz solutions,
∇u(x) ∈ K for a.e. x in Ω (24)u(x) = f (x) for all x in ∂Ω
where u : Ω ⊂ Rn → Rm is Lipschitz, K ⊂Mm×n is closed and
bounded.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 23 / 49
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Differential Inclusion Gradient Inclusion
Gradient Inclusion
We shall now illustrate the basic idea of solving a differential
inclusion by usingthe most well-studied example of partial
differential inclusions, namely thegradient inclusion. The
presentation in the following section is influenced byworks of
Kirchheim and Sychev (Cf. [Sychev, 2001], [Sychev,
2006],[Kirchheim, 2003], [Kirchheim, 2001]).
Gradient inclusion
Consider the basic gradient inclusion problem with Dirichlet
boundary data, wherewe will be looking for Lipschitz solutions,
∇u(x) ∈ K for a.e. x in Ω (24)u(x) = f (x) for all x in ∂Ω
where u : Ω ⊂ Rn → Rm is Lipschitz, K ⊂Mm×n is closed and
bounded.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 23 / 49
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Differential Inclusion Gradient Inclusion
In general, it is difficult to write down a solution at one
strike. So we will beworking in an auxiliary ‘working space’ and
try to build the solution step by step.
We first choose a set U ∈ Mm×n of matrices such that(a) we can
realize the boundary data f with a “simple map” from Ω to Rm
having gradients in U almost everywhere.(b) “simple maps” with a
gradient in U allow a gradual and local modification of
their gradient distribution, which moves this distribution
inside U towards K .
The terminology “simple map” is a bit vague and it also slightly
differs in (a) and(b). In (b), it usually means an affine map,
whereas in (a), it usually means apiecewise affine map or almost
everywhere locally affine Lipschitz map.Once we are able to find
such a set U and the approximate solutions, we can solvethe
differential inclusion problem by using convex integration or Baire
Categoryarguments. Next we discuss these methods one by one and
show how they areused to produce solution of the gradient inclusion
problems.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 24 / 49
-
Differential Inclusion Gradient Inclusion
In general, it is difficult to write down a solution at one
strike. So we will beworking in an auxiliary ‘working space’ and
try to build the solution step by step.
We first choose a set U ∈ Mm×n of matrices such that(a) we can
realize the boundary data f with a “simple map” from Ω to Rm
having gradients in U almost everywhere.(b) “simple maps” with a
gradient in U allow a gradual and local modification of
their gradient distribution, which moves this distribution
inside U towards K .
The terminology “simple map” is a bit vague and it also slightly
differs in (a) and(b). In (b), it usually means an affine map,
whereas in (a), it usually means apiecewise affine map or almost
everywhere locally affine Lipschitz map.Once we are able to find
such a set U and the approximate solutions, we can solvethe
differential inclusion problem by using convex integration or Baire
Categoryarguments. Next we discuss these methods one by one and
show how they areused to produce solution of the gradient inclusion
problems.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 24 / 49
-
Differential Inclusion Gradient Inclusion
In general, it is difficult to write down a solution at one
strike. So we will beworking in an auxiliary ‘working space’ and
try to build the solution step by step.
We first choose a set U ∈ Mm×n of matrices such that(a) we can
realize the boundary data f with a “simple map” from Ω to Rm
having gradients in U almost everywhere.(b) “simple maps” with a
gradient in U allow a gradual and local modification of
their gradient distribution, which moves this distribution
inside U towards K .
The terminology “simple map” is a bit vague and it also slightly
differs in (a) and(b). In (b), it usually means an affine map,
whereas in (a), it usually means apiecewise affine map or almost
everywhere locally affine Lipschitz map.Once we are able to find
such a set U and the approximate solutions, we can solvethe
differential inclusion problem by using convex integration or Baire
Categoryarguments. Next we discuss these methods one by one and
show how they areused to produce solution of the gradient inclusion
problems.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 24 / 49
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Differential Inclusion Convex Integration
Convex Integration
We start with the convex integration method, first introduced by
Gromov in thefar reaching generalization of Nash’s work on
embedding problems. Müller andSverak adapted the method to the
framework of Lipschitz solutions. A typicalresult in the spirit of
convex integration is the following:
Theorem
Let S be a bounded subset of Mm×n and let K be a compact subset
of Mm×n.Assume that for every A ∈ S and every � > 0 there is a
piecewise affine functionφ ∈ lA + W 1,∞0 (Ω;Rm) with the
properties:
1) Dφ ∈ (S ∪ K ) a.e. in Ω;2) ‖dist(Dφ,K )‖L1(Ω) 6 � |(Ω)|.
Then for each piecewise affine function f ∈W 1,∞(Ω;Rm) with Df ∈
S ∪ K a.e.in Ω the problem (24) has a solution. Moreover, each
�-neighborhood of f inL∞(Ω;Rm) norm contains a solution of this
problem.
Here the notation lA + W1,∞0 (Ω;Rm) means the set of Lipschitz
functions u such
that u = lA on ∂Ω where lA is a linear function with gradient
equal to A.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 25 / 49
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Differential Inclusion Convex Integration
Idea of the proof
Let f be a piecewise affine function with Df ∈ (S ∪ K ) a.e. in
Ω. The main pointis to construct a sequence of piecewise affine
functions fj : Ω→ Rm such that
Dfj ∈ (S ∪ K ) a.e. in Ω, ‖dist(Dfj ,K )‖L1 → 0, (25)fj |∂Ω = f
|∂Ω, (26)
fj → f∞ in W 1,1(Ω,Rm). (27)
This is done via explicit construction which relies heavily on
Vitali-type coveringarguments and rescaling.
The key point in the proof is that Controlled L∞ convergence
gives strongW 1,1 convergence.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 26 / 49
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Differential Inclusion Convex Integration
Idea of the proof
Let f be a piecewise affine function with Df ∈ (S ∪ K ) a.e. in
Ω. The main pointis to construct a sequence of piecewise affine
functions fj : Ω→ Rm such that
Dfj ∈ (S ∪ K ) a.e. in Ω, ‖dist(Dfj ,K )‖L1 → 0, (25)fj |∂Ω = f
|∂Ω, (26)
fj → f∞ in W 1,1(Ω,Rm). (27)
This is done via explicit construction which relies heavily on
Vitali-type coveringarguments and rescaling.
The key point in the proof is that Controlled L∞ convergence
gives strongW 1,1 convergence.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 26 / 49
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Differential Inclusion Convex Integration
We now introduce a definition needed to obtain a corollary of
the above theorem.
Definition
A sequence of open sets Ui ∈Mm×n is an in-approximation of a set
K ∈Mm×nif
(1) the Ui are uniformly bounded;
(2) Ui ⊂ U lci+1;(3) Ui → K in the following sense: if Fi ∈ Ui
and Fi → F then F ∈ K .
The subscript lc means lamination convex hull. Similarly
rc-in-approximations canbe defined using rank-1-convex hulls. A
result similar to the corollary holds true inthat case too.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 27 / 49
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Differential Inclusion Convex Integration
Theorem
Suppose that {Ui} is an in-approximation of a compact set K
∈Mm×n and thatf : Ω ⊂ Rn → Rm is C 1(or piecewise C 1) and
satisfies f ∈ U1 a.e. Then theproblem (24) has a solution.
Theorem (14) and its rc-in-approximation version are actually
both particularcases of Theorem (12) where the set S is taken as
the union of the elements ofthe sets Ui s. The proof of this fact
relies on versions of the following lemma:
Lemma
Assume that c ∈ Rm and assume that b ∈ Rn. Let b1 = t1b, b2 =
t2b, wheret2 < 0 < t1, and let b1, . . . , bq be extreme
points of a compact convex set with0 ∈ int co{b1, . . . , bq}.
Define Bi := c ⊗ bi , i ∈ {1, . . . , q}. Then for each � >
0there exists a piecewise affine function φ ∈W 1,∞0 (Ω;Rm) such
that,
|{x ∈ Ω : Dφ(x) = B1 or Dφ(x) = B2}| ≥ |Ω)| − �, (28)Dφ ∈ {B1, .
. . ,Bq} a.e. in Ω, ‖φ‖C(Ω) ≤ �. (29)
The proof relies on pyramidal construction and translation,
covering and rescalingarguments.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 28 / 49
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Differential Inclusion Convex Integration
Theorem
Suppose that {Ui} is an in-approximation of a compact set K
∈Mm×n and thatf : Ω ⊂ Rn → Rm is C 1(or piecewise C 1) and
satisfies f ∈ U1 a.e. Then theproblem (24) has a solution.
Theorem (14) and its rc-in-approximation version are actually
both particularcases of Theorem (12) where the set S is taken as
the union of the elements ofthe sets Ui s. The proof of this fact
relies on versions of the following lemma:
Lemma
Assume that c ∈ Rm and assume that b ∈ Rn. Let b1 = t1b, b2 =
t2b, wheret2 < 0 < t1, and let b1, . . . , bq be extreme
points of a compact convex set with0 ∈ int co{b1, . . . , bq}.
Define Bi := c ⊗ bi , i ∈ {1, . . . , q}. Then for each � >
0there exists a piecewise affine function φ ∈W 1,∞0 (Ω;Rm) such
that,
|{x ∈ Ω : Dφ(x) = B1 or Dφ(x) = B2}| ≥ |Ω)| − �, (28)Dφ ∈ {B1, .
. . ,Bq} a.e. in Ω, ‖φ‖C(Ω) ≤ �. (29)
The proof relies on pyramidal construction and translation,
covering and rescalingarguments.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 28 / 49
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Differential Inclusion Baire Category
Baire Category
The core of the Baire Category method lies in the fundamental
observation thatthe gradient map u ∈ (W 1,∞, ‖.‖L∞)→ ∇u ∈ Lp, p
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Differential Inclusion Baire Category
Baire Category
The core of the Baire Category method lies in the fundamental
observation thatthe gradient map u ∈ (W 1,∞, ‖.‖L∞)→ ∇u ∈ Lp, p
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Differential Inclusion Baire Category
Definition
Given Ω ⊂ Rn bounded and open, U ⊂Mm×n bounded but not
necessarily open,we put
P(Ω,U) = {u ∈ Lip(Ω,Rm) : u is piecewise affine and ∇u ∈ U a.e.
in Ω}.
If there is in addition a map f from a superset of ∂Ω into Rm
given, then wedefine the space
P(Ω,U , f ) = {u ∈ P(Ω,U) : u(x) = f (x) for all x ∈ ∂Ω}.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
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Differential Inclusion Baire Category
Definition
Let U , K ⊂Mm×n be given. We say that gradients in U are stable
only near K ifU is bounded, K is closed and for each � > 0 there
is a δ = δ(�) > 0 such that forall A ∈ U with dist(A,K ) > �
there exists a piecewise affine φ ∈ Lip(Rn,Rm) withbounded support
which satisfies
A +∇φ(x) ∈ KA for a.e x ∈ Ω, where KA ⊂ U∫|∇φ(x)| > δ |(supp
φ)|
Theorem
Suppose that gradients in U are stable only near K . Given any Ω
⊂ Rn boundedopen set and f : Ω→ Rm piecewise affine, the typical u
∈ P(Ω,U , f )
∞( in the
sense of Baire category ) is a solution of (24).
Here P(Ω,U , f )∞
means the closure of the space P(Ω,U , f ) in
L∞-normtopology.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
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Application to Euler equation
Incompressible Euler equation
Consider the incompressible Euler equation in n-space
dimensions,
∂tv + div(v ⊗ v) +∇p = 0div v = 0
}(30)
Definition (Weak solutions)
A vectorfield v ∈ L2loc(Rn × (0,T )) is a weak solution of the
incompressible Eulerequations if ∫ T
0
∫Rn
(∂tφ· v +∇φ : v ⊗ v) dxdt = 0 (31)
for all φ ∈ C∞c (Rn × (0,T );Rn) with div φ = 0 and∫ T0
∫Rn
v · ∇ψ dxdt = 0 (32)
for all ψ ∈ C∞c (Rn × (0,T )).
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Application to Euler equation
Incompressible Euler equation
Consider the incompressible Euler equation in n-space
dimensions,
∂tv + div(v ⊗ v) +∇p = 0div v = 0
}(30)
Definition (Weak solutions)
A vectorfield v ∈ L2loc(Rn × (0,T )) is a weak solution of the
incompressible Eulerequations if ∫ T
0
∫Rn
(∂tφ· v +∇φ : v ⊗ v) dxdt = 0 (31)
for all φ ∈ C∞c (Rn × (0,T );Rn) with div φ = 0 and∫ T0
∫Rn
v · ∇ψ dxdt = 0 (32)
for all ψ ∈ C∞c (Rn × (0,T )).
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Application to Euler equation
Definition (Subsolutions)
Let e ∈ L1loc(Rn × (0,T )) with e ≥ 0. A subsolution to the
incompressible Eulerequation with given kinetic energy density e is
a triple
(v , u, q) : Rn × (0,T )→ Rn × Sn×n0 × R
with the following properties:
v ∈ L2loc , u ∈ L1loc , q is a distribution;{∂tv + div u +∇q =
0div v = 0,
in the sense of distributions; (33)
v ⊗ v − u ≤ 2n
eI a.e.. (34)
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Application to Euler equation
we define U =
(u + qIn v
v 0
)with q = p + 1n |v |
2, u = v ⊗ v − 1n |v |2In. We also
set y = (x , t). Then (30) is equivalent to ,
divyU = 0, (35)
along with the set of constraint K defined by the relations
between u and v and pand q.
We denote by M the set of symmetric (n + 1)× (n + 1) matrices A
such thatA(n+1)×(n+1) = 0. Now, in view of the linear
isomorphism
Rn × Sn0 × R 3 (v , u, q) 7→ U =
(u + qIn v
v 0
)∈M, (36)
the wave cone corresponding to (35) can be written as
Λ =
{(v , u, q) ∈ Rn × Sn0 × R : det
(u + qIn v
v 0
)= 0
}. (37)
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Application to Euler equation
we define U =
(u + qIn v
v 0
)with q = p + 1n |v |
2, u = v ⊗ v − 1n |v |2In. We also
set y = (x , t). Then (30) is equivalent to ,
divyU = 0, (35)
along with the set of constraint K defined by the relations
between u and v and pand q.
We denote by M the set of symmetric (n + 1)× (n + 1) matrices A
such thatA(n+1)×(n+1) = 0. Now, in view of the linear
isomorphism
Rn × Sn0 × R 3 (v , u, q) 7→ U =
(u + qIn v
v 0
)∈M, (36)
the wave cone corresponding to (35) can be written as
Λ =
{(v , u, q) ∈ Rn × Sn0 × R : det
(u + qIn v
v 0
)= 0
}. (37)
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
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Application to Euler equation
Now we will present two lemmas which are crucial for the
construction of thesubsolutions. The first one is about a symmetry
in the equation and the secondone, in effect, reduces the problem
to a differential inclusion problem.
Lemma (Galilean group symmetry)
Let G be the subgroup of GLn+1(R) defined by,
{A ∈ R(n+1)×(n+1) : detA 6= 0, Aen+1 = en+1}. (38)
For every divergence free map U : Rn+1 →M and every A ∈ G the
map
V (y) = At ·U(A−ty)·A (39)
is also a divergence free map V : Rn+1 →M
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Application to Euler equation
Now we will present two lemmas which are crucial for the
construction of thesubsolutions. The first one is about a symmetry
in the equation and the secondone, in effect, reduces the problem
to a differential inclusion problem.
Lemma (Galilean group symmetry)
Let G be the subgroup of GLn+1(R) defined by,
{A ∈ R(n+1)×(n+1) : detA 6= 0, Aen+1 = en+1}. (38)
For every divergence free map U : Rn+1 →M and every A ∈ G the
map
V (y) = At ·U(A−ty)·A (39)
is also a divergence free map V : Rn+1 →M
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Application to Euler equation
Proof.
It is easy to check that whenever B ∈M, then AtBA ∈M for all A ∈
G. Now letφ ∈ C∞c (Rn+1;Rn+1) be a compactly supported test
function and consider φ̃defined by
φ̃(x) = Aφ(Atx)
Then ∇φ̃(x) = A∇φ(Atx)At , and by a change of variables we
obtain∫tr(V (y)∇φ(y))dy =
∫tr(AtU(A−ty)A∇φ(y))dy
=
∫tr(U(A−ty)A∇φ(y)At)dy
=
∫tr(U(x)A∇φ(Atx)At)(detA)−1dx
= (detA)−1∫
tr(U(x)∇φ̃(x))dx = 0,
since U is divergence free. But this implies that V is also
divergence free.
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Application to Euler equation
Lemma (Potential)
Let E klij ∈ C∞(Rn+1) be functions for i , j , k , l = 1, . . .
, n + 1 so that the tensor Eis skew-symmetric in ij and kl, that
is
E klij = −E lkij = −E klji = E lkji . (40)
Then
Uij = L(E ) =1
2
∑k,l
∂2kl(Ekjil + E
kijl ) (41)
is symmetric and divergence free. If in addition
E(n+1)i(n+1)j = 0 for every i and j . (42)
then U takes values in M.
Proof.
Easy and direct calculation.
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Application to Euler equation
Using these two lemmas, we can prove the following, which is
essentially the coreof the construction of De Lellis and
Székelyhidi Jr. in[De Lellis and Székelyhidi, 2009]:
Proposition (Localized plane waves)
Let a = (v0, u0, q0) ∈ Λ with v0 6= 0 and denote by σ the line
segment joining thepoints a and −a. For every � > 0 there exists
a smooth solution (v , u, q) of (35)(in view of the linear
isomorphism (36)) with the properties:
the support of (v , u, q) is contained in B1(0) ⊂ Rnx × Rt ,the
image of (v , u, q) is contained in the �-neighborhood of σ,∫|v(x ,
t)|dxdt ≥ α|v0|,
where α > 0 is a dimensional constant.
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Application to Euler equation
Using these two lemmas, we can prove the following, which is
essentially the coreof the construction of De Lellis and
Székelyhidi Jr. in[De Lellis and Székelyhidi, 2009]:
Proposition (Localized plane waves)
Let a = (v0, u0, q0) ∈ Λ with v0 6= 0 and denote by σ the line
segment joining thepoints a and −a. For every � > 0 there exists
a smooth solution (v , u, q) of (35)(in view of the linear
isomorphism (36)) with the properties:
the support of (v , u, q) is contained in B1(0) ⊂ Rnx × Rt ,the
image of (v , u, q) is contained in the �-neighborhood of σ,∫|v(x ,
t)|dxdt ≥ α|v0|,
where α > 0 is a dimensional constant.
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Application to Euler equation
Proof.
Detailed proof can be found in [De Lellis and Székelyhidi,
2009]. We just sketchthe argument emphasizing the crucial points.
First note that using Lemma (22)and a standard covering/rescaling
argument, it is enough to prove the propositionfor the case where U
∈M is such that
Ue1 = 0, Uen+1 6= 0. (43)
Define
E i1j1 = −E 1ij1 = −E i11j = E 1i1j = U ijsin(Ny1)
N2. (44)
and all the other entries equal to 0. Now choose a smooth
cut-off function φ suchthat
|φ| ≤ 1,φ = 1 on B 1
2(0),
supp(φ) ⊂ B1(0),and consider the map
U = L(φE ). (45)
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Application to Euler equation
Clearly, U is smooth and supported in B1(0). Using Lemma (23), U
is M-valuedand divergence free. Also,
U(y) = U sin(Ny1) for y ∈ B 12(0),
and hence ∫|U(y)en+1|dy ≥ |Uen+1|
∫B 1
2(0)
| sin(Ny1)|dy ≥ 2α|Uen+1|, (46)
for large enough N. Also note that
‖U − φŨ‖∞ ≤C
N‖φ‖C 2 , (47)
where Ũ = L(E ). Hence by choosing N large enough, we can
easily obtain‖U − φŨ‖∞ ≤ �. Since |φ| ≤ 1 and consequently the
image of Ũ is contained inσ, this proves the proposition.
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Application to Euler equation
Theorem
Let Ω ⊂ Rnx × Rt be a bounded open domain. There exist (v , p) ∈
L∞(Rnx × Rt)solving the Euler equations
∂tv + div(v ⊗ v) +∇p = 0div v = 0
such that
|v(x , t)| = 1 for a.e. (x , t) ∈ Ωv(x , t) = 0 and p(x , t) = 0
for a.e. (x , t) ∈ Rnx × Rt\Ω.
Instead of spelling out every details, which can be found in[De
Lellis and Székelyhidi, 2009], we describe the geometric and
functionalanalytic setup and outline the general strategy,
specialized to the problem at hand.
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Application to Euler equation
The main idea is consider the sets
K = {(v , u) ∈ Rn × Sn0 : u = v ⊗ v −1
n|v |2In, |v | = 1}, (48)
andU = int (conv K × [−1, 1]). (49)
Clearly, a triple solving (35) (in the sense of the isomorphism)
and taking values inthe convex extreme points of U is our desired
solution. The plan is to prove0 ∈ U , showing the existence of
plane waves taking values in U and then add suchplane waves to get
an infinite sum
(v , u, q) =∞∑i=1
(vi , ui , qi ) (50)
with the properties that
the partial sums∑∞
i=1(vi , ui , qi ) take values in U ,(v , u, q) is supported in
Ω,
(v , u, q) takes values in the convex extreme points of U a.e.
in Ω,(v , u, q) solves the linear partial differential equations
(35)Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
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Application to Euler equation
The key functional analytic set up is the following:Let
X0 := {(v , u, q) ∈ C∞(Rnx × Rt) : (1), (2), (3) below hold }
(51)
(1) supp(v , u, q) ⊂ Ω,(2) (v , u, q) solves (35)(in the sense
of the isomorphism) in Rnx × Rt(3) (v(x , t), u(x , t), q(x , t)) ∈
U for all (x , t) ∈ Rnx × Rt .We equip X0 with the topology of
L
∞-weak-∗ convergence of (v , u, q) and wedefine X to be the
closure of X0 in this topology. Now the set X with thistopology is
a nonempty compact metrizable space. We fix a metric d∗∞
inducingthe weak-∗ topology of L∞ in X , so that (X , d∗∞) is a
complete metric space.Now, the identity map
I : (X , d∗∞)→ L2(Rnx × Rt) defined by (v , u, q) 7→ (v , u,
q)
is a Baire-1 map and therefore the set of points of continuity
is residual in(X , d∗∞).
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Application to Euler equation
Proving that 0 ∈ U is done by considering the linear map
T : C (Sn−1)→ Rn × Sn0 , φ 7→∫Sn−1
(v , v ⊗ v − In
n
)φ(v)dµ,
where µ is the Haar measure on Sn−1. Clearly, if
φ ≥ 0 and∫Sn−1
φdµ = 1,
then T (φ) ∈ conv K . Also note that T (1) = 0 = T (α) and
usingα = 1−
∫Sn−1
ψdµ and choosing a ψ such that ‖ψ‖C(Sn−1) < 12 , we obtainT
(B 1
2) ⊂ conv K . Now we use intelligent choices of φ and the
orthogonality in L2
with a dimension counting argument to conclude T is
surjective.
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Application to Euler equation
Using the fact that 0 ∈ U and the construction of localized
plane waves, we canshow that
Lemma
There exists a dimensional constant β > 0 with the following
property. Given(v , u, q) ∈ X0 there exists a sequence (vk , uk ,
qk) ∈ X0 such that
‖vk‖2L2(Ω) ≥ ‖v‖2L2(Ω) + β
(|Ω| − ‖v‖2L2(Ω)
)2, (52)
and(vk , uk , qk)
∗⇀ (v , u, q) in L∞(Ω). (53)
Now if (v , u, q) is a point of continuity of the identity map I
, passing to the limitin (52) we obtain |v | = 1 a.e in Ω,
concluding the proof of Theorem (24).
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References
References I
Aubin, J.-P. and Cellina, A. (1984).Differential inclusions,
volume 264 of Grundlehren der MathematischenWissenschaften
[Fundamental Principles of Mathematical Sciences].Springer-Verlag,
Berlin.Set-valued maps and viability theory.
Cellina, A. (1980).On the differential inclusion x ′ ∈ [−1,
+1].Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8),
69(1-2):1–6(1981).
Dacorogna, B. and Marcellini, P. (1997).General existence
theorems for Hamilton-Jacobi equations in the scalar andvectorial
cases.Acta Math., 178(1):1–37.
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Nonlinear PDEs 20th July, 2012 46 / 49
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References
References II
Dacorogna, B. and Marcellini, P. (1999).Implicit partial
differential equations.Progress in Nonlinear Differential Equations
and their Applications, 37.Birkhäuser Boston Inc., Boston, MA.
De Lellis, C. and Székelyhidi, J. L. (2009).The Euler equations
as a differential inclusion.Ann. of Math. (2),
170(3):1417–1436.
Gromov, M. (1986).Partial differential relations, volume 9 of
Ergebnisse der Mathematik und ihrerGrenzgebiete (3) [Results in
Mathematics and Related Areas (3)].Springer-Verlag, Berlin.
Kirchheim, B. (2001).Deformations with finitely many gradients
and stability of quasiconvex hulls.C. R. Acad. Sci. Paris Sér. I
Math., 332(3):289–294.
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Nonlinear PDEs 20th July, 2012 47 / 49
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References
References III
Kirchheim, B. (2003).Rigidity and geometry of
microstructures.Habilitation thesis.
Sychev, M. A. (2001).Comparing two methods of resolving
homogeneous differential inclusions.Calc. Var. Partial Differential
Equations, 13(2):213–229.
Sychev, M. A. (2006).A few remarks on differential
inclusions.Proc. Roy. Soc. Edinburgh Sect. A, 136(3):649–668.
Tartar, L. (1979).Compensated compactness and applications to
partial differential equations.In Nonlinear analysis and mechanics:
Heriot-Watt Symposium, Vol. IV,volume 39 of Res. Notes in Math.,
pages 136–212. Pitman, Boston, Mass.
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Nonlinear PDEs 20th July, 2012 48 / 49
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The End
Thank you
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in
Nonlinear PDEs 20th July, 2012 49 / 49
IntroductionBrief outlinePreliminary settings
Compensated CompactnessGeneral settingWithout differential
constraintWith differential constraintsPerspectives
Differential InclusionNonlinear PDEs as Differential
InclusionsGradient InclusionConvex IntegrationBaire Category
Application to Euler equationReferencesThe End