Sulla buona positura di equazioni differenziali associate a campi debolmente differenziabili

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Facoltà di Scienze Matematiche, Fisiche e Naturali Corso di Laurea Magistrale in Matematica

Tesi di Laurea Magistrale

Sulla buona positura di equazioni differenziali

ordinarie associate a campi debolmente

differenziabili

Candidato: Relatore: Controrelatore:

Elia Brué

Prof. Luigi Ambrosio

Prof. Valentino Magnani

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Introduction

This thesis is devoted to the study of the ordinary differential equation

(

γ0(t) = b(t, γ(t))

γ(0) = x (ODE)

under various differentiability assumption on the time dependent vector field

b : [0, T ] × Rd→ Rd.

The classical Cauchy-Lipschitz theory ensures that, when the vector field is regular enough (Lipschitz in the spatial variable, uniformly in time) the problem admits, a unique classical solution. Moreover there exists a unique flow associated to the vector field b, that is a map

X : [0, T ] × Rd→ Rd, that solves      d dtX(t, x) = b(t, X(t, x)) X(0, x) = x. (0.0.1)

We are interested in the extension of these results to the case the vector field b is less regular, for instance when b has the Sobolev or BV spatial regularity. Such an extension, in addition to its theoretic importance, is crucial in the study of many non linear partial differential equations of mathematical physics.

However, it is well known that, when the vector field b is not Lipschitz in the spatial variable, uniqueness of solutions doesn’t hold. We mention the very classical example

  

γ0(t) =q|γ(t)|

γ(0) = x0

in which the vector field b(x) =p

|x|, smooth everywhere except at the origin, admits an infinite number of trajectories that pass through x = 0. As a consequence, in order to extend the Cauchy-Lipschitz theory to less regular vector fields, a weaker notion of flow is needed.

The first result in this direction was the seminal work [36] by Di Perna and Lions, in which was introduced a weak notion of flow in an axiomatic way: every distributional solution X of0.0.1is a flow whenever it is satisfied the semi-group property

X(t, X(s, x)) = X(t + s, x), Ld a.e. on Rd t, s, t + s ∈ [0, T ],

and the compressibility assumption 1

CL

d_{≤ X}

t#Ld≤ CLd, ∀ t ∈ [0, T ], iii

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where Xt#(E) :=Ld((Xt)−1(E)). Di Perna and Lions proved existence and uniqueness of this flow, when the vector field has the Sobolev regularity in the spatial variable. The following important step, in the development of the theory, was the work [6] by Ambrosio. The author introduces a more manageable notion of flow, the so-called regular Lagrangian flow.

We say that a map X : [0, T ] × Rd→ Rd_{is a regular Lagrangian flow associated to the} vector field b if the following two condition are satisfied:

1) for Ld _{a.e. x ∈ R}dthe curve t → Xt(x) satisfies

Xt(x) = x +

Z t

0

b(s, Xs(x)) ds,

2) there exists a constant L, called compressibility constant, such that

Xt#Ld≤ LLd ∀t ∈ [0, T ].

Ambrosio doesn’t ask for a semi-group property, actually it is a consequence of the other axioms, as soon as b has some regularity. Moreover, only an upper bound for Xt#Ld is required; this property is crucial in order to have uniqueness, indeed it singles out the "good" solutions by the property that trajectories do not concentrate, at any time, in

Lebesgue negligible sets.

Ambrosio also proved the existence and uniqueness of regular Lagrangian flows, when the vector field has the BV spatial regularity, and absolutely continuous and bounded divergence.

The works by Di Perna, Lions and Ambrosio are based on the so-called Eulerian approach. It means that the starting point is the transport equation

(

∂tut(x) + bt(x) · ∇ut(x) = 0

u0 = ¯u, (PDE)

in the smooth context, it is naturally linked to the ODE problem. Indeed a classical solution u ofPDE must be constant along every integral curve γ; more precisely there holds

d

dtu(t, γ(t)) =∂tu(t, γ(t)) + ∇xu(t, γ(t)) · γ 0_(t)

=∂tu(t, γ(t)) + b(t, γ(t)) · ∇xu(t, γ(t)) =0.

This implies that, in the classical context, the problemPDEcan be solved using the method

of characteristics, the unique solution u(t, x) is obtained transporting the smooth initial

data along the trajectories of b:

u(t, x) = ¯u((Xt)−1(x)).

The Eulerian approach to the ODEproblem, out of the smooth setting, is divided in two main parts. The first one is the study of the well-posedness of the problemPDE. A notion of weak solution is introduced, and the main purpose is to find regularity conditions on b in order to have existence, uniqueness and stability for weak solutions of the transport equation. The second step consists in the study of the links between thePDEproblem and theODEout of the smooth setting. The surprising result is that we are able to translate

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CONTENTS v all the well-posedness properties of the transport equation in results regarding existence, uniqueness and stability of regular Lagrangian flows. The general principle is that, when the problemPDEis well-posed in a sufficiently small class of bounded solutions, then there exists a unique regular Lagrangian flow associated to the vector field b.

The other important side of the theory is the so-called Lagrangian viewpoint. It consists in a direct study of theODE problem, by means of quantitative a priori estimates of the regular Lagrangian flow, without exploiting the link with the transport equation. A central role is played by functions of the form

Φ_δ(t) = Z log 1 +|X 1 t(x) − Xt2(x)| δ ! dx,

where X1 and X2 are regular Lagrangian flows associated to b, that give a quantitative measure of the distance between the two regular Lagrangian flows.

We finally illustrate the content of the various chapter of this thesis.

In Chapter 1 we present the theory of ordinary differential equations in the classical setting, stating the main properties regarding the classical flows. We also introduce the transport equation and the continuity equation, and using the theory of characteristics we prove the well-posedness in the smooth framework, underlining the connections with the ODEproblem.

In Chapter 2 we study the problemPDEout of the smooth setting, we start defining the notion of weak solutions and we investigate the well-posedness in two main step. The first one consists in the so called theory of the renormalized solutions, introduced in [36]. We say that a weak bounded solution u of the transport equation is renormalized if, for every

β ∈ C1(R; R), β(u) is also a weak solution. Notice that this property holds for smooth solutions, by an immediate application of the chain-rule. However, when the vector field is not smooth, we cannot expect any regularity of the solutions, so that the renormalization property is a nontrivial request. We say that a vector field b has the renormalization property if all bounded weak solutions ofPDE are renormalized. The central result which motivates this definition is that, if the renormalization property holds, then weak solutions of the transport equation are unique and stable. The second step consists in finding sufficient and reasonably general conditions for the validity of the renormalization property. This property is not satisfied by all vector fields: some regularity, tipically in terms of weak differentiability, is needed. The standard argument used to show the renormalization property, relies on the regularization scheme introduced in [36], and the regularity of the vector field comes into play when one shows that the error term in the approximation goes to zero. The two most significant results of renormalization are due to DiPerna and Lions [36] and to Ambrosio [6], who deal respectively with the Sobolev and the BV case. We conclude the chapter presenting the very elegant example by De Pauw, which provides an explicit vector field with the nonuniqueness property, that enjoys a regularity very close to the one needed in Ambrosio’s Theorem.

In Chapter 3 we illustrate the link between the problems ODE andPDE out of the smooth setting. We first define the notion of super-position solutions, that is a class of solutions ofPDE, obtained moving the initial data along every selection of trajectories of

b. Then, we prove that every bounded weak solution is actually a super-position solution.

This characterization could be seen as an extension of the method of characteristic to the non smoothing setting. Finally, using all this ingredients, we show how the uniqueness of bounded solutions of the continuity equation implies existence and uniqueness of the regular Lagrangian flows.

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In Chapter 4 we present the purely Lagrangian approach to the ODE problem. It consists in a series of a priori estimates, performed directly at the ODE formulation, without exploiting the link with the transport equation. Using these techniques, many results of the Di Perna-Lions theory are recovered in a more direct manner. We start studying Sobolev vector fields with summability exponent p > 1; we illustrate the work [30] by Crippa and De Lellis, in which the well-posedness of theODE problem and new compact and regularity results are proved. The theory could be extended to Sobolev vector fields with p = 1, loosing some quantitative estimate; but, until now, the Lagrangian view point is not able to recover Ambrosio’s result. We conclude the chapter presenting the work [22] by Crippa and Bouchut in which a new class of vector fields is studied: the authors consider therein vector fields whose derivative can be represented by convolution of an L1 function with a Calderon-Zygmund kernel. This class is very important in the applications, and it is not included in, nor it includes, BV .

In the Appendix, we state some basic results, widely used through the thesis.

We finally mention that the basic material for the preparation of the thesis have been the lectures notes [15] by Crippa and Ambrosio, about the theory of regular Lagrangian flows, and Crippa’s Phd thesis [28].

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Notations

Ω _{open set in R}d,

I interval in R,

x = (x1, ..., xd) point in Rd,

|x| _{euclidean norm of x ∈ R}d_, hx, yi = x · y euclidean scalar product,

Br(x) ball centered at x ∈ Rdwith radius r,

Br ball centered at the origin,

ωd measure of the unitary ball in Rd, ∂f

∂xi = ∂if partial derivative of f with respect to the variable xi

∇f gradient of f ,

Df distributional derivative of f ,

Dif partial distributional derivative of f with respect to the variable xi, supp(f ) support of f ,

Lip(f ) Lipschitz constant of f , Ld _{Lebesgue measure in R}d_,

E measurable set in Rd_,

ν µ absolutely continuity of ν with respect to µ (µ(E) = 0 ⇒ ν(E) = 0), |µ| total variation of µ,

supp(µ) support of µ,

χE characteristic function of E,

Lp(E) _{Lebesgue space of functions defined on E ⊂ R}d, with 1 ≤ p ≤ ∞

Ck(Ω) space of k-time continuously differentiability function on Ω,

C_c∞(Ω) ∩_kNCk(Ω),

Cb(Ω) space of continuous and bounded functions on Ω,

M (Ω) space of Radon measures on Ω, with finite total variation, P(Ω) space of probability measures on Ω,

P1(Ω) space of probability measures on Ω, with finite first moment, W1,p_(Ω) _{space of Sobolev functions, i.e.}

space of Lp(Ω) functions with distributional derivative in Lp(Ω),

BV (Ω) space of bounded variation functions, i.e.

space of L1(Ω) functions with distributional derivative in M (Ω),

AC(I; Rd) space of L1_{(I; R}d) functions such that there exists f0 ∈ L1_{(I; R}d₎ satisfying f (x) − f (y) =Ry

x f

0_{(t) dt, for every x, y ∈ I.}

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Chapter 1

Classical theory

In this chapter we illustrate the main results of the theory of ordinary differential equations in the smooth framework. In section1.1 we study existence and uniqueness of the solutions, we introduce the classical notion of flow and we illustrate its main properties. In section 1.2we introduce two partial differential equations, namely the transport equation and the continuity equation and we study the important links between these problems and the ordinary differential equations. Standard reference for this topics are [42], [19] and [39].

1.1 Ordinary differential equations

Let b : R × Rd_{→ R}d_{be a time-dependent vector field, we consider the Cauchy problem}

(

γ0(t) = b(t, γ(t))

γ(0) = x (ODE)

we say that γ solves the problem ODE _{in the interval I containing 0 if γ ∈ AC(I; R}d),

γ(0) = x and forL1 a.e t ∈ I the following identity holds

γ0(t) := b(t, γ(t)).

We are interested in existence and uniqueness of solutions of the problemODE.

In this section we assume the following hypotheses on the vector field b(t, x) = b_t(x): Hypotheses 1.1.1. The map t → bt(x) is measurable for all x ∈ Rd, the map x → bt(x) is Lipschitz for almost every t ∈ R and there holds

m(t) := Lip(bt) ∈ L1loc(R).

Observe that for every continuous curve γ the function t → b_t(γ(t)) is measurable. The most important result of existence and uniqueness of problemODEin this smooth framework is the Cauchy-Lipschitz theorem:

Theorem 1.1.2. Let b be a vector field with the hypotheses 1.1.1_{. Then for every x ∈ R}d

there exists ε > 0 such that in the interval (−ε, +ε) there exists a unique solution γ ∈ AC((−ε, ε); Rd) of the problem ODE with initial value x0.

Actually the classical Cauchy-Lipschitz theorem works under a stronger assumption on

b, i.e.

Lip bt≤ L ∀t ∈ (−δ, +δ), 1

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where L and δ are positive fixed constants. We refer to [39] for a more detailed overview of the subject and to [42] for a complete proof of Theorem1.1.2.

In general, a solution of ODE doesn’t exist for all time t ∈ R, this fact is not related to the regularity of b, it is related to the "growth" of b. A very classical example is

γ0(t) = |γ(t)|2,

where the vector field is b(x) = |x|2. The equation could be integrated and it is simple to check that every solution, different from the null one γ = 0, is defined only in a finite interval of time.

A natural assumption on the growth of b is the following

|b(t, x)| ≤ C(1 + |x|). (1.1.1)

Under the assumptions of Theorem 1.1.2 and the growth estimate (1.1.1) we have existence and uniqueness for all t ∈ R.

We now define the classical notion of flow.

Definition 1.1.3. Let I ⊂ R be an interval containing 0, we say that the map X : I ×Rd→ Rd is a classical flow of the vector field b if, for all x ∈ Rd, the curve t → Xt(x) ∈ AC(I) solves the problemODE.

Under the regularity assumptions 1.1.1and the growth condition (1.1.1) there exists a unique flow X : R × Rd→ Rd _{associated to b. Moreover the flow is Lipschitz with respect} to the spatial variable and the following inequality holds

Lip Xt≤ exp

Z t

0

Lip bsds. This property could be checked using the Gronwall lemma and

d

dt|Xt(x) − Xt(y)| ≤ |bt(Xt(x) − bt(Xt(y))| ≤ Lip bt|Xt(x) − Xt(y)|.

In general the regularity of the flow depends on the regularity of the vector field, for sake of completeness we give the precise statement in the C∞ context.

Theorem 1.1.4. For every b ∈ C∞_{(R × R}d_{; R}d_{) bounded vector field there exists a unique}

flow X ∈ C∞_{(R × R}d_{; R}d_{) associated to b, moreover for all t ∈ R the map} Xt: Rd→ Rd,

is a diffeomorphism. The Jacobian J (t, x) := det ∇X_t satisfies the equation d

dtJ (t, x) = (div bt)(Xt(x))J (t, x), (1.1.2) which also implies that J (t, x) > 0 for all t ∈ R.

1.2 Continuity and transport equation

The aim of this section is to introduce two important partial differential equations and illustrate the classical method to treat them, underlining the connection with the problem ODE. The first one is the continuity equation

∂tµt+ div(btµt) = 0. (PDE) We are interested in distributional measure valued solutions according to the following definition:

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1.2. CONTINUITY AND TRANSPORT EQUATION 3 Definition 1.2.1. We say that the measure valued curve t → µt∈ M+(Rd) is a solution

of the equationPDEin [0, T ] if

kb_tk_L1_(µ t)∈ L 1 loc([0, T ]), (1.2.1) and Z T 0 Z Rd ∂tϕ(t, x) + b(t, x) · ∇xϕ(t, x) dµt(x) dt = 0, (1.2.2) for all ϕ ∈ C_c∞_{((0, T ) × R}d_).

The continuity equation holds sense without any regularity assumption on b and µ_t, provided the integrability condition (1.2.1).

Proposition 1.2.2. Let µt be a solution of the transport equation, up to a modification in L1_{-negligible set of times, it is continuous from [0, T ] to M}

+(Rd) with respect to the weak star topology. Moreover for all ϕ ∈ C_c1_(Rd) the map

t →

Z

Rd

ϕ(x) dµt(x)

belongs to AC([0, T ]), and the following equality holds d dt Z Rd ϕ(x) dµt(x) = Z Rd

h∇ϕ(x), b_t(x)i dµ_t(x) for L1a.e. t ∈ [0, T ].

Proof. Taking a test function of the form ϕ(x)ψ(t) with ϕ ∈ C_c∞_(Rd) and ψ ∈ C_c∞((0, T )) and plugging it on (1.2.2) we have the identity

d dt Z Rd ϕ(x) dµt(x) = Z Rd h∇ϕ(x), bt(x)i dµt(x)

in the sense of distributions in [0, T ]; moreover it still holds when ϕ ∈ C_c1_(Rd). In particular for all t > s Lebesgue points of t →R

Rdϕ(x) dµt(x) we have Z Rd ϕ dµt− Z Rd ϕ dµs ≤ k∇ϕk_∞ Z t s kbrk_L1_(µ r) dr. (1.2.3)

Let D be a dense and countable subset of C_c∞_(Rd) with respect to the C1 norm, there exists J ⊂ [0, T ] such thatL1([0, T ] \ J ) = 0 and every t ∈ J is a Lebesgue point for every map t →R

ϕ dµt, ϕ ∈ D. The inequality (1.2.3) holds for all ϕ ∈ D and for all t, s ∈ J , this implies that t → µtis uniformly weak star continuous from S to M+(Rd), by density

it can be extended at the whole interval [0, T ].

Thanks to Proposition1.2.2we can assume that a solution µ_t of the continuity equation is defined for all t ∈ [0, T ] and so is well defined the Cauchy problem associated toPDE:

(

∂tµt+ div(btµt) = 0

µ0= ¯µ.

(1.2.4) The important link between the transport equation and the problemODEis described in the following Proposition.

Proposition 1.2.3. Let us assume ¯µ ∈P1(Rd), the regularity assumptions 1.1.1 and the growth condition (1.1.1) on b. Let X be the classical flow associated to b, we set µt:= Xt#µ.¯

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Proof. We verify that µt∈P1(Rd): Z Rd |y| dµt(y) = Z Rd |Xt(x)| dµ0(x) ≤ C(T ) Z Rd (1 + |x|) dµ0(x) < ∞,

where we have used

|Xt(x)| ≤ C(T )(1 + |x|),

that is a simple consequence of the growth assumption (1.1.1) on b.

Clearly t → µt is continuous with respect to weak star topology. We have to prove that t →R

ϕ dµt belongs to ACloc([0, T ]) for all ϕ ∈ Cc∞(Rd). Let us fix s < t in [0, T ] and

where we have used the growth condition (1.1.1). It’s now immediate to check that for every ϕ ∈ C_c∞_(Rd) we have d dt Z Rd ϕ(x) dµt(x) = Z Rd

h∇ϕ(x), bt(x)i dµt(x) forL1a.e. t ∈ [0, T ], that implies Z T 0 Z Rd ∂t(ϕ(t)ψ(x)) + b(t, x) · ∇x(ϕ(t)ψ(x)) dx dt = 0, (1.2.5) for every ϕ ∈ C_c∞_(Rd) and ψ ∈ C_c∞((0, T )). Recalling that the set of functions of the form

N

X

i=1

ϕiψi,

with ϕ ∈ C_c∞_(Rd) and ψ ∈ C_c∞((0, T )) is dense in C_c∞_{([0, T ] × R}d) with respect to the topology of test functions, we obtain that the identity (1.2.5) is enough to conclude the proof.

We now prove that the already found solution of the continuity equation is the only one. We use the so-called duality method, that consists in studying another partial differential equation, called transport equation

∂tut(x) + bt(x) · ∇ut(x) = ψt(x), (1.2.6) we find classical solutions of it using the method of characteristics and we prove that existence of a sufficiently nice and general solutions for the transport equation implies uniqueness for the continuity equation. More precisely we consider X, the flow associated to b, and ut one solution of the transport equation, we study the curve t → u(t, X(t, x)). We have d dtut(X(t, x)) =∂tut(X(t, x)) + (∇ut)(X(t, x)) d dtX(t, x) = − b_t(X(t, x))∇u_t(X(t, x)) + b_t(X(t, x))∇u(t, X(t, x)) + ψ_t(X(t, x)) =ψt(X(t, x)).

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1.2. CONTINUITY AND TRANSPORT EQUATION 5 Integrating in time and assuming u0= 0 we obtain

ut(X(t, x)) = Z t 0 ψs(X(s, x)) ds, and finally ut(y) = Z t 0 X(s, (Xt)−1(y)) ds.

The previous identity defines one solution of (1.2.6), as the following proposition points out:

Proposition 1.2.4. Let us assume the regularity assumption1.1.1and the growth condition

(1.1.1) on b, then for every ψ(t, x) ∈ C∞_{(R × R}d_{) source term we have that}

u(t, x) =

Z t

0

ψ(s, X(t, s, x)) ds,

provides a solution of (1.2.6). We have used the notation X(t, s, x) to indicate the solution

of the problem ODE at time t starting on x at time s. Proof. It follow by a simple computation.

In the same spirit of Proposition1.2.4the following formula

u(t, x) = ¯u(X(t, 0, x)) +

Z t

0

ψ(s, X(t, s, x)) ds, (1.2.7) provides a solution of the transport equation when u0 = ¯u.

We are ready to show the uniqueness property for the continuity equation.

Theorem 1.2.5. Let b be a vector field satisfying the regularity assumptions1.1.1 and the growth condition (1.1.1). Let X be the classical flow associated to b, if Xt∈ C1(Rd) for

every t ∈ [0, T ] then the formula µ_t= X_t#µ provides the unique measure valued solution¯

of continuity equation.

The regularity condition X_t∈ C1_(Rd_{) is satisfied, for example, when the vector field is} of class C1 with respect to the spatial variable for L1 a.e time. Moreover we can prove the uniqueness property of the continuity equation (1.2.4) in the class L∞_{([0, T ] × R}d) (smaller than the class of all measure valued curves) dropping the regularity condition

Xt∈ C1(Rd). With uniqueness property in L∞([0, T ] × Rd) we mean that for every initial data ¯µ ∈ L∞([0, T ] × Rd) if µ1_t, µ2_t ∈ L∞_{([0, T ] × R}d_{) are two solutions of (}_1.2.4_{) with}

µ1₀ = µ2₀ = ¯µ then µ1_t = µ2_t for every t ∈ [0, T ]. Where we have used the notation "µ ∈ L∞" to indicate that µ is absolutely continuous with respect toLd and its density belongs to

L∞.

Proof. Let us fix a finite interval of time I := [0, T ] and a function ψ ∈ C_c∞_{(R × R}d), we consider the backward transport equation with source term

(

∂tut(x) + bt(x) · ∇ut(x) = ψt(x)

uT = 0.

This equation admits solution that could be found considering a small modification of formula (1.2.4). We assume µ0 = 0 we prove that every solution µt of the continuity equation must be identically zero. We claim that

d dt Z Rd ut(x) dµt(x) = Z Rd ψt(x) dµt(x).

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We first illustrate how to achieve the thesis from the claim. Integrating between 0 and T we obtain

0 = Z Rd uT dµT − Z Rd u0dµ0= Z T 0 Z Rd ψtdµtdt,

it must be satisfied for every ψ ∈ C_c∞_{(R × R}d), this implies µt⊗ dt = 0 and therefore

µt= 0 for almost every t ∈ I.

In order to conclude the proof we have to check the claim. We observe that u_t∈ C1

c(Rd) for every t ∈ [0, T ] since Xt∈ C1(Rd) and (1.2.4), then we are allowed to use the identity

d dt

Z

ϕ(x) dµt(x) =

Z

h∇ϕ(x), b_t(x)i dµ_t(x) forL1a.e. t ∈ [0, T ], whit ϕ = ut. We compute d dt Z Rd utdµt= Z Rd d dtutdµt+ d dt Z Rd ϕ dµt |_ϕ=u_t = − Z Rd bt∇utdµt+ Z Rd ψtdµt+ Z Rd bt∇utdµt = Z Rd ψtdµt,

where we have used a "distributional" Leibniz rule that could be checked writing the difference quotient and passing to the limit as in the classical proof of Leibniz rule.

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Chapter 2

Eulerian viewpoint

In this chapter we investigate the well-posedness of the transport and continuity equations out of the smooth setting. In section2.1we point out the weak formulation of the transport equation and we give a general existence result. In section2.2we introduce the notion of renormalized solution, and in section2.3 we describe a general strategy, first introduced by Di Perna and Lions [36], useful to study the well-posedness of the transport and continuity equations. The well-posedness results when the vector field b is either Sobolev or BV are shown, following the works [36], [6] by Di Perna Lions and Ambrosio.

2.1 Weak formulation and existence

In this section we study the Cauchy problem associated to the transport equation

(

∂tut(x) + bt(x) · ∇ut(x) = 0

u0= ¯u. (2.1.1)

In order to specify the meaning of the problem (2.1.1) we first define the distributional solutions of the transport equation

∂tut(x) + bt(x) · ∇ut(x) = 0, (2.1.2) and we prove that every distributional solution u provides a continuous curve t → u_t with respect to a weak topology that we specify later. It follow in particular that ut is defined for all t, this gives sense to the Cauchy problem.

Definition 2.1.1. Let b ∈ L1_loc(I × Rd; Rd) be a vector field with div b ∈ L1_loc_{(I × R}d), where I ⊂ R is an open interval. We say that a locally bounded function u : I × Rd→ R is a weak solution of (2.1.2) if the following identity holds

Z I Z Rd u(t, x)[∂tϕ(t, x) + div b(t, x)ϕ(t, x) + b(t, x)∇ϕ(t, x)] dx dt = 0, (2.1.3) for all ϕ ∈ C_c∞_{(I × R}d).

The definition of weak solutions corresponds to the distributional one since ∂tu has a meaning as a distribution, and assuming that div b ∈ L1_loc_{(I × R}d) we can define the product bt(x) · ∇ut(x) as a distribution via the equality

hb · ∇u, ϕi := −hbu, ∇ϕi − hu div b, ϕi, for every ϕ ∈ C_c∞_{(I × R}d).

The weak solutions of (2.1.2) are (up to modification in aL1 negligible set of times) weak star continuous. In the same spirit of Proposition1.2.2we have the following:

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Proposition 2.1.2. For every u weak solution of (2.1.2), up to modification in a L1

negligible subset of I, the map

t → ut

is weak star continuous from ¯I (the topological closure of I) to L∞_loc_(Rd_{). Moreover for all}

ϕ ∈ C_c∞(Rd) the function t →R

ut(x)ϕ(x) dx is absolutely continuous in ¯I and

d dt Z Rd ut(x)ϕ(x) dx = Z Rd ut(x)(bt(x) · ∇ϕ(x) + ϕ(x) div bt(x)) dx.

The argument of the proof is the same of the one in Proposition1.2.2, thus we omit it. From now we assume that every weak solution is weak star continuous in time and defined for all times t ∈ ¯I, in particular taking I = (0, T ) we say that a weak solution u of

(2.1.2) is a solution of the Cauchy problem (2.1.1) if u0 = ¯u.

Now, using a simple density argument we prove the existence of weak solutions of the problem (2.1.1).

Theorem 2.1.3. (Existence of weak solutions) Let b : [0, T ] × Rd _{→ R}d _{be a bounded}

vectorfield with div b ∈ L1_loc_{([0, T ] × R}d). For every ¯u ∈ L∞(Rd) there exists a weak solution

of transport equation with initial data ¯u.

Proof. We proceed by approximation. Let ρ_ε be a standard family of mollifiers, we define the regular vector field bε := b ∗ ρε and the regular initial data ¯uε := ¯u ∗ ρε. Now we consider uε the strong solution of the problem

(

∂tu + bε· ∇ut= 0

u0 = ¯uε.

By (1.2.7) we have that the family uε is equi-bounded in L∞_{([0, T ] × R}d), and so admits limit points with respect to weak star topology. It is very simple to check that each limit point has to satisfy the weak formulation of the transport equation.

2.2 The theory of renormalized solutions

In this section we describe one important tool that can be used to show the well-posedness of the transport equation

(

∂tut(x) + bt(x) · ∇ut(x) = 0,

u0 = ¯u.

We refer to [36] for the original work of Di Perna and Lions, who first adopted this approach.

2.2.1 Heuristic idea and motivation

In order to illustrate the motivation for the concept of renormalized solution we present some formal computations.

Let us suppose that u_tis a solution of the transport equation, and that b is a divergence free vector field. We first observe that the mean of ut is preserved in time

d dt Z Rd utdx = Z Rd bt· ∇utdx = Z Rn div(utbt) dx = 0.

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2.2. THE THEORY OF RENORMALIZED SOLUTIONS 9 This fact is very natural in the smooth settings since using the formula (1.2.7) and the measure preserving property, of the flow associated to divergence free vector field, we have that every integral norm is preserved.

We also observe that, at least in the classical setting, if ut is a solution of (2.1.1) than

u2_t is a solution as well:

∂tu2t = 2ut∂tut= −2utbt· ∇ut= −bt· ∇u2t.

Taking into account the two observations we have that the L2 norm of u_t is preserved in time, consequently if u₀ = 0 than u_t= 0 for Ld _{a.e. x ∈ R}d, this implies uniqueness of the transport equation.

Clearly this argument is only formal, but it motivates the following definition:

Definition 2.2.1. Let b : I × Rd → Rd _{be a locally summable vector field such that} div b is locally summable, where I ⊂ R is a time interval. We say that a weak solution

u ∈ L∞(I × Rd) of the transport equation2.1.2_{is renormalized if for all β : R → R of class}

C1 the function β(u) is also a weak solution of the transport equation.

We say that the vector field b has the renormalization property if every weak solution of (2.1.2) relative to b is a renormalized solution.

2.2.2 Main theorem

The importance of the renormalization property is summarized in the following Theorem, which corresponds to the rough statement “renormalization implies well-posedness”.

Let us introduce the following notation: if b : [0, T ] × Rd → Rd _{is a vectorfield we} extend b to negative times setting

˜_b t=

(

b t ∈ [0, T ]

0 t ∈ (−∞, T )

Theorem 2.2.2. Let b be a bounded vector field with div b ∈ L1([0, T ]; L∞_(Rd_{; R}d)). If ¯b has the renormalization property then there exists a unique bounded weak solutions of (2.1.1).

Moreover the following stability property holds: Let (b_k) and (˜uk) be smooth approximation

sequences of b and ¯u in L1_loc([0, T ] × Rd) with (¯uk) equi-bounded in L∞(Rd). Then the

solution u_k of the corresponding transport equation converges strongly in L1_loc_{([0, T ] × R}d)

to the solution u of (2.1.1).

Before the proof we make some comment about the assumption on b. If we come back to subsection 2.2.1, we notice that, in the heuristic scheme of the proof of uniqueness, we implicitly assume that u2_t at time t = 0 coincides with ¯u2. It is not true in general for distributional solutions, because the curve t → ut is only weak star continuous. The extension of b for negative times is a way to force strong continuity of the solution at the initial data. Moreover this assumption is not restrictive for our aims, since we will work with vector fields that have some weak regularity in space, the extension in time doesn’t modify it.

Proof. Since the transport equation is linear, it is sufficient to show that the only weak

solution of (2.1.1) with ¯u = 0 is u = 0. Let u be a weak solution in [0, T ] of the transport

equation with null initial condition, we consider ˜_{u : (−∞, T ] × R}d_{→ R the function equal} to u in [0, T ] × Rdand identically zero for negative times. Clearly ˜u is a weak solution of

(

∂tu + ˜bt· ∇u = 0 t ∈ (−∞, T ] ˜

u0 = 0

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Our aim is to prove that ˜u = 0. Thanks to the renormalization property of ˜b we have that

˜

u2 is also weak solution of (2.2.1). Let us define

f (t) :=

Z

Rd

˜

u2(t, x)ϕ(t, x) dx,

where ϕ is a nonnegative smooth function with the following properties: 1) ∂tϕ ≤ − kbk∞|∇ϕ|, in [0, T ] × Rd

2) ϕ = 1 in [0, T − ε] × BR, with ε and R fixed parameters. Taking g ∈ C_c∞((−∞, T )) nonnegative function we compute

− Z T −∞ f (t)g0(t) dt = Z T −∞ Z Rd ˜ u2g(∂tϕ + ˜b∇ϕ) dx dt + Z T −∞ f g div ˜b dt ≤ Z T −∞ f g div ˜bt _L∞_(Rd₎ dt, we conclude that f0(t) ≤ div ˜bt _L∞_(Rd₎f (t),

and by Gronwall lemma we have the thesis.

We now prove the stability property. Arguing as in2.1.3 we deduce that u_k weak star converge in L∞_{([0, T ] × R}d) to u a weak solution of (2.1.1), possibly after the extraction of a sub-sequence. By the uniqueness part of this Theorem we have that the whole sequence converges to u, since it is the unique solution of the transport equation with fixed initial data. Since bk are smooth, u2k are also solutions of transport equation associated to bk and by the renormalization property of b they must converge, with respect to weak star topology of L∞_{([0, T ] × R}d), to u2_{. Using the following elementary Lemma, with E = [0, T ] × R}d, the thesis follows.

Lemma 2.2.3. Let fk be a sequence of functions in L∞(E) with E ⊂ Rd+1. If fk

∗ * f and f_k2 * f∗ 2 then f_k converge strongly to f in L1_loc(E).

Proof. Let us fix R > 0,

Z BR∩E (f_k− f )2_{dx =}Z BR∩E f2dx + Z BR∩E f_k2dx − 2 Z BR∩E f fkdx, it goes to zero when k → ∞, thus

lim

k→∞kf − fkkL1(BR∩E)= 0.

It’s not difficult to extend the previous results to equations of the form

∂tu + b · ∇u = cu, with

c ∈ L1([0, T ]; L∞_loc_(Rd_{; R}d)),

we just have to take into account another term in the Gronwall lemma. In particular, choosing c = − div b we are able to translate the well posedness results for the transport equation in well posedness results for the continuity equation.

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2.3. REGULARIZATION SCHEME 11

2.3 Regularization scheme

We now describe an abstract scheme useful to understand when a general vector field has the renormalization property. This argument proposed by Di Perna and Lions in [36] consists in a regularization procedure.

Let us fix an even convolution kernel ρ_ε _{in R}d, we consider a weak solution u_tof the transport equation (2.1.1) and we regularize it. Writing

uε:= u ∗ ρ_ε we have

∂tuεt+ bt∇uεt = bt∇uεt − (bt∇ut) ∗ ρε, (2.3.1) in the sense of distributions.

We call commutator the error term

rε(t, x) := bt(x)∇uεt(x) − (bt∇ut) ∗ ρε(x),

and we observe that rε goes to zero in the sense of distributions when ε goes to zero. Let us fix β ∈ C1_{(R; R) multiplying both terms of the equation (}2.3.1) by β0(uε) we have

∂tβ(uεt) + ∇β(uεt)bt= β0(uεt)rε. (2.3.2) When ε goes to zero, the left hand side of the equation (2.3.2) converges to

∂tβ(ut) + ∇β(ut)bt,

in the sense of distributions. Our aim is to find condition on b such that

β(uε)rε→ 0

goes to zero in the sense of distributions, in this way we have that b has the renormalization property. Unfortunately to this purpose it is not enough that rε → 0 in the sense of distributions, a stronger convergence is needed.

Di Perna and Lions in [36] proved that when the vector field has the Sobolev regularity in the spatial variable then rε→ 0 strongly in L1

loc([0, T ] × Rd). In [6] Ambrosio extends the theory to vector field with BV spatial regularity.

2.3.1 Vector fields with the Sobolev spatial regularity

In this subsection we illustrate the results of Di Perna and Lions. We deal with vector fields defined in I × Rd where I ⊂ Rd is a generic interval, since in order to treat the well-posedness problem for t ∈ [0, T ] we need to show the renormalization property for the extending vector field, defined on the time interval (−∞, T ].

Theorem 2.3.1. Every bounded vector field belonging to L1_loc(I; W_loc1,1_(Rd_{; R}d)) has the

renormalization property.

We recall a standard property of Sobolev function that in fact characterizes them. Proposition 2.3.2. For every f ∈ W_loc1,1(Rd) and for every z ∈ Rd we have that

f (x + εz) − f (x)

ε → ∇f (x)z,

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See [44] for a standard reference on the subject. We are ready to prove theorem2.3.1

Proof. We prove that the commutator goes to zero in L1_loc_{(I × R}d). Let us show the crucial equality rε(t, x) = Z Rd u(x − εz)bt(x) − bt(x − εz) ε ∇ρ(z) dz + (utdiv bt) ∗ ρε, (2.3.3)

recalling that ρ_ε(x) = ρ_ε(−x) we test

rε(t, x) = b_t(x) · ∇uε_t(x) − (b_t· ∇u_t) ∗ ρ_ε(x), against ϕ ∈ C_c∞_(Rd),

hrε_t, ϕi =hbt· ∇uεt, ϕi − hbt· ∇u, ϕ ∗ ρεi

=hbt· (∇ρε∗ u), ϕi + hbtut, ∇(ϕ ∗ ρε)i + hutdiv bt, ϕ ∗ ρεi =hbt· (∇ρε∗ u) − (btut) ∗ ∇ϕε+ (utdiv bt) ∗ ϕε, ϕi, where we have used the identity

hb · ∇u, ϕi := −hbu, ∇ϕi − hu div b, ϕi,

that defines b · ∇u as a distribution, and the oddness of ∇ρ. We finally show that

bt(x) · (∇ρε∗ u(x)) − (btut) ∗ ∇ϕε(x) = Z Rd u(x + εz)bt(x) − bt(x − εz) ε ∇ρ(z) dz, we compute bt(x) · (∇ρε∗ ut(x)) − (btut) ∗ ∇ρε(x) = Z Rd bt(x) · ∇ρε(x − z)u(z) dz − Z Rd bt(z) · ∇ρε(x − z)ut(z) dz = Z Rd ut(z)(bt(x) − bt(z)) · ∇ρε(x − z) dz = Z Rd ut(z) bt(x) − bt(z) ε · ∇ρ x−z_ε εd dz = Z Rd u(x − εz)bt(x) − bt(x − εz) ε ∇ρ(z) dz,

where in the last passage we have changed variables.

Applying Proposition2.3.2we have that rε_t converge strongly in L1_loc_(Rd) to

ut(x) Z Rd h∇bt(x)z, ∇ρ(z)i dz + div bt(x) , (2.3.4)

forL1 a.e. t ∈ I. Moreover we have the bound

Z K |r_tε(x)| dx ≤ kuk_L∞k∇ρk_L∞ Z K |∇bt(z)| dz ∈ L1loc(I), that implies rε→ u Z Rd h∇bz, ∇ρ(z)i dz + div b strongly in L1_loc_{(I × R}d).

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2.3. REGULARIZATION SCHEME 13 Actually (2.3.4) is zero, indeed

Z Rd h∇b_t(x)z, ∇ρ(z)i dz + div b_t(x) = Z Rd

divz(ρ(z)∇bt(x)z) − ρ(z) div bt(x) dz + div bt(x) =

Z

Rd

divz(ρ(z)∇bt(x)z) dz − div bt(x) + div bt(x) =0,

where in the last passage we have used that ρ is supported in a compact set. This conclude the proof.

2.3.2 Ambrosio’s theorem in the BV context

The goal of this section is to extend the results of section 2.3.1 to the case of vector fields

BV in the spatial variable. The precise statement by Ambrosio is the following:

Theorem 2.3.3. Let b be a bounded vector field belonging to L1_loc(I; BVloc(Rd; Rd)) with div btLd for L1 a.e. t ∈ I, then b has the renormalization property.

This theorem has been applied to the study of various nonlinear PDEs: for instance the Keyfitz and Kranzer system ([7] and [10]) and the semigeostrophic equation ([32], [33]). We refer to [6] for the original work by Ambrosio.

Let us start by introducing notations and recalling some basic facts about BV functions, general reference on this topic are [5], [44] and [37].

Definition 2.3.4. Let Ω ⊂ Rd be an open set, a function u ∈ L1_loc_{(Ω; R}k) has bounded

variation if the distributional derivative Du is a matrix valued measure with finite total

variation in Ω.

We recall that the total variation |µ| of a Rk×d-valued measure µ is defined by |µ|(E) := sup {X

i

| µ(Ei)| |Ei ⊂ E i ∈ N, Ei∩ Ej = ∅ i 6= j } ,

(choosing the Hilbert-Schmidt norm) and that if |µ|(Ω) is finite then µ admits the polar decomposition µ = M |µ| with |M (x)| = 1 for |µ| a.e. every x ∈ Ω.

We can decompose the derivative of a BV function u in a canonical way

Du := ∇uLd+ Dsu,

where |∇u(x)| = 1 forLda.e. x ∈ Ω and Dsu is a singular measure with respect toLd. We often use the notation Dau for the absolutely continuous part ∇uLd, and write

Dzu := M z|Du|, ∀z ∈ Rd.

The following result by Alberti points out the structure of the singular measure, we refer to [1] for the original work.

Theorem 2.3.5. (Alberti’s rank one theorem) For every u ∈ BV (Ω; Rk) there exist

ξ : Ω → Rd and η : Ω → Rk measurable functions such that |η(x)| = |ξ(x)| = 1 for Dsu a.e. x ∈ Ω and

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In other words the singular part of the distributional derivative of BV functions has rank one almost everywhere.

In the particular case d = k we can find the following formula for the divergence Div u = tr ∇u + hη, ξi|Dsu|,

in particular it is absolutely continuous with respect toLd if and only if hη, ξi = 0 for |Ds_{u|-a.e. x ∈ Ω.}

When the vector field b belongs to L1_loc(I; BV_loc_(Rd_{; R}d)) we define the measure |Db| by the formula Z I×Rdϕ(t, x) d|Db|(t, x) := Z I Z Rd ϕ(t, x) d|Dbt(x)| dt, in the same way we define |Dsb| and |Dab|.

The main difficult to deal with the commutator

rε(t, x) =

Z

Rd

u(x − εz)bt(x) − bt(x − εz)

ε ∇ρ(z) dz + (utdiv b) ∗ ρε,

in the BV context is that, in general, the difference quotient of BV functions doesn’t converge with respect to a strong topology. Roughly speaking, the problem is in the support of the singular measure of the derivative, since in this set we expect that the difference quotient convergence to a singular measure. As in the Sobolev setting we can control the L1 norm of the difference quotient using the distributional derivative:

Lemma 2.3.6. For every u ∈ BVloc(Rd) and for every compact set K ⊂ Rd the following

estimate holds

Z

K

|u(x + εz) − u(x)| dx ≤ |D_zu|(Kε),

where |z| ≤ ε and

Kε:= { x | d(x, K) ≤ ε } .

Proof. Let us assume that u ∈ C1_(Rd_{), fixing z ∈ R}d such that |z| ≤ ε we have

For general u ∈ BV_loc_(Rd) we proceed by approximation: let { ρ_η} be a standard family of convolution kernels, we set u_η = u ∗ ρ_η, it’s simple to see that

∇u_η = Du ∗ ρ_η,

thus for every compact set K ⊂ Rdthere holds

Z

K

|h∇u_η(x), zi| dx → |D_zu|(K),

when η goes to zero. Applying (2.3.5) on uη and passing to the limit we obtain the thesis.

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2.3. REGULARIZATION SCHEME 15 The following result about BV functions ensures that, away from the support of |Dsu|,

the difference quotient converges in a strong sense to the absolutely continuous part of the derivative, and in the support of the singular part of the measure the difference quotient is asymptotically controlled by the total variation of Dsu.

Proposition 2.3.7. For all u ∈ BVloc(Rd), and for all z ∈ Rd there exists a canonical

decomposition of the difference quotient, u(x + εz) − u(x)

ε = u

1

ε,z(x) + u2ε,z(x),

such that u1_ε,z → h∇u, zi in L1

loc and for every K ⊂ Rd compact set lim sup

ε→0

Z

K

|u2_ε,z(x)| dx ≤ |z||Dsu|(K). (2.3.6)

In addition we have the uniform bound

sup z∈K0 sup δ<ε Z K |u1_ε,z(x)| + |u2_ε,z(x)| dx ≤ sup z∈K0 |z||Du|(Kε), (2.3.7)

whenever K, K0 _{⊂ R}d are compact and ε > 0.

Proof. We first treat the case d = 1 and after we generalize the argument by slicing. For

one variable function of class BV the following formula holds (see [5] for details)

u(x + ε) − u(x) =Du([x, x + ε])

=

Z x+ε

x

u0(y) dy + Dsu([x, x + ε]),

forL1 _{a.e. x ∈ R; where we have used the standard decomposition}

Du := u0L + Dsu. Choosing u1_ε(x) := 1 ε Z x+ε x u0(y), and u2_ε(x) := 1 εD s_{u([x, x + ε])} we have that u1_ε → u0 _{in L}1

loc(R), indeed u1 can be written as a convolution of u0 with an

L1 kernel u1_ε(x) = u0∗ χ[0,ε] ε . Moreover we have Z K |u2_ε(x)| dx =1 ε Z K |Dsu|([x, x + ε]) dx =1 ε Z K Z R χ[x,x+ε](z) d|Dsu|(z) dx =1 ε Z R Z K χ_[x,x+ε](z) dx d|Dsu|(z) ≤|Dsu|(Kε),

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for every K ⊂ R compact set.

When d > 1 we assume without loss of generality that |z| = 1 and we write every

x ∈ Rdas x = tz + y where hz, yi = 0. Using the coordinates (t, y) we set u_y(t) = u(t, y). We have that uy ∈ BV (R) for Ld−1 a.e. y ∈ Rd−1 with derivative

Duy(·) = Dzu(·, y) (2.3.8)

(see [5] for more details), thus applying the results in the case d = 1 we find

u(t + ε, y) − u(t, y) ε = uy(t + ε) − uy(t) ε =1 εDuy([t, t + ε]) = Rt+ε t u 0 y(s) ds ε + Ds_u y([t, t + ε]) ε =u1_ε,z(t, y) + u2_ε,z(t, y). By (2.3.8) we have u1_ε,z(t, y) := 1 ε Z t+ε t h∇u(s, y), zi ds.

and a simple application of Fubini theorem implies that u1_ε,z → h∇u, zi in L1

loc(Rd). Using that

Z

K2

|Dsuy|(K1) dy = |Dzsu|(K1× K2) ≤ |Dsu|(K1× K2), (2.3.9)

forLd−1 _{a.e. y ∈ R}d−1 and for every K₁ _{⊂ R and K}₂ _{⊂ R compact sets, we finally have}

The important consequence of the previous Proposition is that the sequence of commu-tators is bounded in L1_loc_{([0, T ] × R}d) and every weak star limit must be concentrated in the support of |Dsb|.

Proposition 2.3.8. (Isotropic estimate) In the hypotheses of Theorem2.3.3, for all J ⊂ I and for all K ⊂ Rd compact sets the following estimate holds

lim sup ε→0 Z J Z K |rε| dx dt ≤ kuk_∞ Z Rd |z||∇ρ(z)| dz |Dsb|(J × K).

Proof. Applying the decomposition of Proposition2.3.7 at the difference quotient of bt we have |rε| = Z Rd u(x − εz)bt(x) − bt(x − εz) ε ∇ρ(z) dz + (div bt) ∗ ρε = Z Rd u(x − εz)(bt)1ε,−z(x)∇ρ(z) dz + (div bt) ∗ ρε+ Z Rd u(x − εz)(bt)2ε,−z(x)∇ρ(z) dz.

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2.3. REGULARIZATION SCHEME 17 The term

Z

Rd

u(x − εz)(bt)1ε,−z(x)∇ρ(z) dz + (div bt) ∗ ρε,

goes to zero as in the Sobolev setting, we have to estimate the last term. Using (2.3.6) and (2.3.7) we have lim sup ε→0 Z J Z K

|rε| dx dt ≤ kuk_∞lim sup ε→0 Z Rd Z J Z K |(bt)2ε,z(x)| dx dt |∇ρ(z)| dz ≤ kuk_∞ Z Rd |z||∇ρ(z)| dz |Dsb|(J × K).

We now fix β ∈ C1_{(R; R). There exists a measure σ, depending on β, such that the}

following equation holds in the sense of distributions

∂tβ(ut) + b∇β(ut) = σ. (2.3.10) Indeed we have the identity

β(uε)rε= ∂tβ(uε) + b · ∇β(uε), and

∂tβ(uε) + b · ∇β(uε) → ∂tβ(u) + b · ∇β(u),

in the sense of distributions, thus β(uε)rε converges in the sense of distributions and it is equi-bounded in L1_loc_{(I × R}d) by Proposition 2.3.8. Therefore the limit σ of β(uε)rε belongs to M (I × Rd) and (2.3.10) holds.

As simple consequence of Proposition2.3.8we have the isotropic estimate for σ |σ| ≤ β0 _∞kuk_∞ Z Rd |z||∇ρ(z)| dz |Dsb|,

In the sense of measures in I × Rd.

We can interpret the estimate by saying that we have proved the renormalization property up to a singular set. The main idea to deal with the singular part is to find another estimate of rε, more accurate in the set in which |Dsb| is concentrated. We give it

starting from the commutator and using Lemma2.3.6.

Proposition 2.3.9. (Anisotropic estimate) Under the assumptions of Theorem 2.3.3 for every J ⊂ I and K ⊂ Rd compact sets the following estimate holds

lim sup ε→0 Z J Z K |rε| dt dx ≤ Z J Z K Λ(Mt(x), ρ) d|Db|(t, x) + |Dab|(J × K), where Λ(M, ρ) := Z Rd |hM z, ∇ρ(z)i| dz. Moreover we have |σ| ≤ β0 _∞kuk_∞[Λ(M_t(.), ρ)|Db| + |Dab|], as measures in I × Rd.

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Proof. Applying Lemma2.3.6at the function f (x) = hbt(x), ∇ρ(z)i we have Z J Z K |rε_{| dt dx ≤} ≤ kuk_∞ Z Rd Z J Z K |hbt(x) − bt(x − εz), ∇ρ(z)i| ε dx dt dz + Z J Z K |(div bt) ∗ ρε(x)| dx dt ≤ kuk_∞ Z Rd Z J Z Kε |hMt(x)z, ∇ρ(z)i| d|Dbt|(x) dt dz + |Dab| ∗ ρε(J × K) = kuk_∞ Z J Z Kε Λ(M_t(x), ρ) d|Db|(t, x) dt + |Dab| ∗ ρε(J × K). The thesis follow passing to the limit to ε → 0.

The final estimate of σ is given combining the isotropic estimate (2.3.8) and the anisotropic one (2.3.9). The first one ensures that σ is concentrated in the same set of |Ds_{b| and the second one implies}

|σ| ≤ β0

_∞kuk_∞Λ(M_t(·), ρ)|Dsb|, (2.3.11)

as measures in I × Rd.

The next step consists in a local optimization of the convolution kernel in order to conclude that σ = 0. Writing

σ = f |Dsb|,

we have the estimate

|f (t, x)| ≤ β0

_∞kuk_∞Λ(M_t(x), ρ).

for |Ds_{b|-almost every (t, x) ∈ I × R}d, thanks to (2.3.11). Our goal is to improve the estimate of f , minimizing Λ(M_t(x), ρ) in a set of admissible ρ. To this aim we define the set of even convolution kernels

K := ρ ∈ C_c∞(Rd)| ρ ≥ 0, Z Rd ρ(x) dx = 1, ρ(x) = ρ(−x) ,

and we consider a subset K0 ⊂ K, countable and dense with respect to the W1,1 _topology.

We have |f (t, x)| ≤ β0 _∞kuk_∞ inf ρ∈K0Λ(Mt(x), ρ),

for |Ds_{b|-almost every point (t, x) ∈ I × R}d. Finally we observe that for every M fixed matrix, the map ρ → Λ(M, ρ) is continuous with respect to the W1,1 topology, therefore the infimum over K0 coincides with the infimum over K.

In order to study the minimization problem inf

ρ∈KΛ(M, ρ), M ∈ R d×d

we observe that we can assume

M = η ⊗ ξ, η, ξ ∈ Rd, hη, ξi = 0

thanks to Alberti’s rank one Theorem2.3.5and the assumption div btLdfor L1 a.e.

t ∈ I in Theorem2.3.3. In this setting we can assume that d = 2 since Λ(M, ρ) =

Z

Rd

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2.4. DE PAUW’S COUNTEREXAMPLE 19 depends only on the orthogonal directions ξ and η.

We now fix ε > 0, ρ₀∈ C_c∞_{(R) a positive and even convolution kernel and we set}

ρ(z) := 1 ερ0 _{hξ, zi} ε ρ0(hη, zi),

a convolution kernel in R2 that mollifies very fast in the ξ direction. We compute Λ(M, ρ) = Z R2 |hξ, zi||hη, ∇ρ(z)i| dz = Z R2 |hξ, zi| ε ρ0 _{hξ, zi} ε |ρ0₀(hη, zi)| dz ≤ Z R |ρ0₀(t)| dt Z R t ερ0 _|t| ε dt = ε Z R |ρ0₀(t)| dt Z R |t|ρ(t) dt,

that clearly goes to zero when ε → 0. This implies that σ = 0 and the proof of Theorem 2.3.3follows.

This anisotropic regularization procedure was first introduced by Bouchut [23] in the context of the Vlasov equation with BV vector field and subsequently exploited by Colombini and Lerner [27] to treat the transport equation with conormal BV vector field (vector fields, in which the derivative of b is a measure along one direction and an L1

function along the other d − 1 directions).

Actually we can avoid the use of rank one Theorem2.3.5 using the following lemma by Alberti

Lemma 2.3.10. For every M ∈ Rd×d the following equality holds

inf

ρ∈KΛ(M, ρ) = |tr M |.

The construction of the optimal (almost optimal) kernel is not simple, we refer to [9] for the detailed proof of the Alberti’s lemma.

2.4 De Pauw’s counterexample

In this section we present the counterexample to the well-posedness of the transport equation 2.1.1 found by De Pauw, it shows the sharpness of the result by Ambrosio (Theorem2.3.3) in the class of vector field with BV spatial regularity, since the vector field constructed is BV in the spatial variable, but not belongs to L1_loc_{(R; BV}_loc_(Rd_{; R}d)). We refer to [35] for the original work and we refer also to the work [4] by Alberti Crippa and Mazzucato where other examples of this type are constructed.

The example by Depauw consists of a planar divergence-free vector field b with two different weak solution of

(

∂tv + div(bv) = 0

v0 = 0. (2.4.1)

The heuristic idea is to find a vector field a that act as in the FIGURE 2, more precisely the flow associated to the vector field a at the time t = 1 transforms the chessboard of side 1/2 to the chessboard of side 1/4. Iterating this procedure we find a vector field b that splits in a half the side of the chessboard infinite times when t ∈ [0, 1), this property implies non uniqueness for the problem (2.4.2).

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We now illustrate the rigorous construction of the vector field. Let us define a : [−1/2, 1/2]2_{→ R}2 as a(x1, x2) :=        (0, 4x₁) if 0 < |x₂| < |x₁| < 1/4 (−4x2, 0) if 0 < |x1| < |x2| < 1/4 0 otherwise,

and we extend it periodically to R2.

Figure 2.1: Depauw’s vector field.

We consider the ODE problem associated to a

(

γ0(t) = a(t, γ(t))

γ(0) = x (2.4.2)

and we denote by X_ta(x) the solution of (2.4.2) at time t, Xa is well-defined because the vector field a is piecewise smooth. It is simple to see that X₁a acts as in FIGURE 2. More precisely, we define

u0(x1, x2) := sign (x1x2) x ∈ [1/2, 1/2]2,

and we extend it periodically in R2_{, we also define}

u1(x) := u0(2x), x ∈ R2,

notice that u0 is the characteristic function of the chessboard with side 1/2 while u1 is the

characteristic function of the chessboard with side 1/4. We have

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2.4. DE PAUW’S COUNTEREXAMPLE 21

Figure 2.2: The effect of the vector field a.

Now we iterate this construction an infinite number of times in the following way: dividing the interval (0, 1] in sub-intervals

Ij := (2−j−1, 2−j] j ∈ N, we define the vector field c : R × R2 → R2 _as

c(t, x) :=

(

0 if t ≤ 0 or t > 1 a(2jx) if t ∈ Ij.

It’s very simple to verify that c is bounded, divergence-free (since a is divergence-free) and ct∈ BVloc(R2; R2) for every t ∈ R. Let us consider the vector field b(t, x) := c(1 − t, x), the associated ODE problem

(

γ0(t) = b(t, γ(t))

γ(s) = x (2.4.3)

provides the map Xb(t, s, x) , that is the solution of (2.4.3) at the time t, with the following properties: denoting

uk(x) := u0(2kx) x ∈ R2,

we have

uk(Xb(1 − 2−k, 1 − 2−k−1, x)) = uk+1(x) x ∈ R2, by the semigroup property of Xb we conclude that

u0(Xb(0, 1 − 2−k, x)) = uk(x). Therefore the function

ut(x) := u0(Xb(0, t, x)) x ∈ R2, t ∈ [0, 1)

is a weak solution of the transport equation associated to the vector field b with initial data u₀.

The crucial observation is that the curve t → u_tis strongly continuous in [0, 1) and the strong limit t → 1 doesn’t exist for the oscillatory nature of ut, on the other hand there exists the weak star limit when t → 1 and it is zero. Setting

vt(x) :=

(

0 t = 0

u1−t(x) t ∈ [0, 1),

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Chapter 3

Connection between ODE and

PDE

In this section we describe the connection between the continuity equationPDE and the problemODE out of the smooth setting. In section 3.1we define the regular Lagrangian flow, that is the right notion of flow associated to weakly differentiable vector fields, introduced by Ambrosio in [6]. This notion of flow is inspired by the one presented in [36] by Di Perna and Lions. In section3.2 we study the existence and uniqueness properties for the regular Lagrangian flow in relation with the well-posedness of the continuity equation PDEin L∞.

3.1 The regular Lagrangian flow

Out of the smooth setting the Cauchy-Lipschitz notion of flow is not well-posed, since in general the solutions of the Cauchy problemODE_{are not unique. Let us fix b : [0, T ]×R}d→ Rd a bounded vector field and x ∈ Rd, we denote with T r(x) the set (possible empty) of the trajectories of b starting from x at the time 0, i.e.

T r(x) := { γ ∈ AC([0, T ]; Rd) | γ0(t) = b(t, γ(t)), γ(0) = x } .

When b is not Lipschitz, in general, the set T r(x) contains more than one trajectory, in order to define a notion of flow, we have to select one curve in T r(x) for all x ∈ Rd in a reasonable way. For example we can select trajectories by approximation: we consider a sequence of smooth vector fields that converges to b and we look at the limit points of the classical flows associated to these vector fields. The following simple example provides an illustration of the kind of phenomena that can occur.

Let us consider the autonomous ODE

   γ0(t) = q |γ(t)| γ(0) = x0

Then, solutions of the ODE are not unique for negative initial data. Indeed, they reach the origin in finite time, where they can stay for an arbitrary time T , then continuing as quadratic function of the time. For every T > 0 it’s possible to built Lipschitz

Sulla buona positura di equazioni differenziali associate a campi debolmente differenziabili