differential inclusion approach in nonlinear pdes · di erential inclusion approach in nonlinear...

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

Introduction

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.

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.

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.

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)

z(x) ∈ K ⊂ Rd a.e. x ∈ Ω (2)

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�.

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)

{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.

General Problem

m∑i=1

Ai∂iz� = φ� (3)

{z�(y)} ⊂ M (4)

General Problem

m∑i=1

Ai∂iz� = φ� (3)

{z�(y)} ⊂ M (4)

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 .

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

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

Our Target

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.

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.

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

General Strategy

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)

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.

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.

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.

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.

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.

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 ).

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 ).

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?

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?

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)

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)

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 .

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 .

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}.

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}.

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.

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.

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.

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.

m∑i=1

Ai∂izn = φ (19)

m∑i=1

Ai∂izn = φ (19)

m∑i=1

Ai∂izn = φ (19)

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 ?

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 ?

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)

z(x) ∈ K ⊂ Rd a.e (22)

In certain cases, this problem can be transformed into a differentialinclusion problem.

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)

z(x) ∈ K ⊂ Rd a.e (22)

In certain cases, this problem can be transformed into a differentialinclusion problem.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

Baire Category

The core of the Baire Category method lies in the fundamental observation thatthe gradient map u ∈ (W 1,∞, ‖.‖L∞)→ ∇u ∈ Lp, p

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}.

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.

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 )).

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 )).

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)

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)

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)

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

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

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.

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.

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.

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.

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)

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.

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.

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

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∗∞).

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.

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).

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.

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.

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.

The End

Thank you

IntroductionBrief outlinePreliminary settings

Compensated CompactnessGeneral settingWithout differential constraintWith differential constraintsPerspectives

Differential InclusionNonlinear PDEs as Differential InclusionsGradient InclusionConvex IntegrationBaire Category

Application to Euler equationReferencesThe End

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