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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 Nonlinear PDEs 20th July, 2012 30 / 49
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 Nonlinear PDEs 20th July, 2012 31 / 49
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 )).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 32 / 49
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 )).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 32 / 49
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)
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 33 / 49
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 Nonlinear PDEs 20th July, 2012 34 / 49
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 Nonlinear PDEs 20th July, 2012 34 / 49
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
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 35 / 49
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
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 35 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 36 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 37 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 38 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 38 / 49
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)
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 39 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 40 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 41 / 49
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 Nonlinear PDEs 20th July, 2012 42 / 49
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∗∞).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 43 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 44 / 49
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).
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 45 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 46 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 47 / 49
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.
Swarnendu Sil (TIFR CAM) Differential Inclusion approach in Nonlinear PDEs 20th July, 2012 48 / 49
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