We present the theory of variational solutions for abstract evolution equations as- sociated to a coercive operator and we apply the theory to the case of uniformly elliptic parabolic equations.

CHAPTER 2

VARIATIONAL SOLUTION FOR PARABOLIC EQUATION

I write in blue color what has been teached during the classes.

We present the theory of variational solutions for abstract evolution equations as- sociated to a coercive operator and we apply the theory to the case of uniformly elliptic parabolic equations.

Contents

1. Introduction 1

2. A priori estimates 2

3. Variational solutions 4

4. Proof of the existence part of Theorem 3.2. 6

Appendix A. Exercises 8

Appendix B. The Bochner integral 11

Appendix C. Further results: C

-semigroup, evolution equation with

source term and Duhamel formula 13

Appendix D. References 15

1. Introduction

In this chapter we will focus on the question of existence (and uniqueness) of a solution f = f (t, x) to the (linear) evolution PDE of “parabolic type”

(1.1) ∂

f = Λ f on (0, ∞) × R

,

where Λ is the following integro-differential operator (1.2) (Λf )(x) = ∆f (x) + a(x) · ∇f (x) + c(x) f (x) +

Z

R^d

b(y, x) f (y) dy, that we complement with an initial condition

(1.3) f (0, x) = f

(x) in R

.

Here t ≥ 0 stands for the “time” variable, x ∈ R

stands for the “position” variable (for instance), d ∈ N

.

In order to develop the variational approach for the equation (1.1)-(1.2), we assume that

f

∈ L

( R

) =: H, which is an Hilbert space, and that the coefficients satisfy

a, c ∈ L

( R

), b ∈ L

( R

× R

).

1

The main result we will present in this chapter is the existence of a weak (varia- tional) solution (which sense will be specified below)

f ∈ X

:= C([0, T ); L

) ∩ L

(0, T ; H

) ∩ H

(0, T ; H

),

to the evolution equation (1.1)–(1.3). We mean variational solution because the space of “test functions” is the same as the space in which the solution lives. It also refers to the associated stationary problem which is of “variational type” (see [1, chapter VIII & IX]).

The existence of solutions issue is tackled by following a scheme of proof that we will repeat for all the other evolution equations that we will consider in the next chapters.

(1) We look for a priori estimates by performing (formal) differential and integral calculus.

(2) We deduce a possible natural functional space in which lives a solution and we propose a definition of a solution, that is a (weak) sense in which we may understand the evolution equation.

(3) We state and prove the associated existence theorem. For the existence proof we typically argue as follows: we introduce a “regularized problem” for which we are able to construct a solution and we are allowed to rigorously perform the calculus leading to the “a priori estimates”, and then we pass to the limit in the sequence of regularized solutions.

2. A priori estimates

We explain how we may obtain “a priori estimates” for solutions to the parabolic equation (1.1)-(1.2) and more general, but related, abstract “coercive+dissipative”

type equations. The term “a priori estimates” means that we do not seek in this first step to establish the estimates with full mathematical rigor but we rather try to perform formally some reasonable and usual computations (typically: derivation, integration, sommation and inversion of theses operations). This step is fundamen- tal in order to bring out what kind of information is reasonable to hope for. Of course, in some next steps, these bounds will have to be justified.

Define V = H

( R

) endowed with its usual norm. We first observe that for any nice function f and for any α ∈ (0, 1)

hΛf, f i = − Z

|∇f |

+ Z

af · ∇

f + Z

c f

+ Z Z

b(y, x) f (x)f (y) dxdy.

≤ −αkf k

+ α + 1

4α kak

+ kc

k

+ kbk

kf k

,

thanks to the Cauchy-Schwarz inequality in L

( R

) and L

( R

× R

) and to the Young inequality st ≤ αs

/2 + t

/(2α), ∀ s, t > 0.

Exercise 2.1. Prove that in the case div a ∈ L

, the following estimate holds hΛf, fi ≤ −kf k

+

1 + k(c − 1

2 div a)

k

+ kbk

kf k

.

We also observe that for any nice functions f, g

|hΛf, gi| ≤ k∇f k

k∇gk

+ kak

k∇f k

kgk

+ (kck

+ kbk

) kf k

kgk

≤ (1 + kak

+ kck

+ kbk

kf k

kgk

.

We easily deduce from the two preceding estimates that our parabolic operator falls into the following abstract variational framework.

Abstract variational framework. We consider a Hilbert space H endowed with the scalar product (·, ·) = (·, ·)

and the norm |·| = |·|

. We identify H with its dual H

= H. We consider another Banach space V endowed with a norm k · k = k · k

. We assume V reflexive and V ⊂ H with dense and bounded embedding. Observing that for any u ∈ H , the mapping

v ∈ V 7→ (u, v)

defines a linear and continuous form on V , we may identify u as an element of V

. In other words, we have

V ⊂ H ⊂ V

and hu, vi = (u, v), ∀ u ∈ H, v ∈ V, where we denote h., .i = h., .i

the duality product on V .

We consider a linear operator Λ : V → V

which is bounded (or continuous), which means

(i) ∃ M > 0 such that

|hΛg, hi| ≤ M kgk khk, ∀ g, h ∈ V ;

and such that Λ is “coercive+dissipative”

(or −Λ satisfies a “G˚ arding’s inequality”) in the sense

(ii) ∃ α > 0, b ∈ R such that

hΛg, gi ≤ −α kgk

+ b |g|

, ∀ g ∈ V.

We consider then the associated abstract evolution equation

(2.1) dg

dt = Λg on (0, T ), on a function g : [0, T ) → H with prescribed initial value

(2.2) g(0) = g

∈ H.

A priori bound in the abstract variational framework. With the above as- sumptions and notations, any solution g to the abstract evolution equation (2.1) (formally) satisfies the following estimate

(2.3) |g(T )|

+ 2α Z

0

kg(s)k

ds ≤ e

|g

|

∀ T.

Indeed, using just the coercivity+dissipativity assumption (ii), we have (at least formally)

d dt

|g(t)|

2 = hΛg, gi ≤ −αkg(t)k

+ b |g(t)|

, and we conclude to (2.3) thanks to the Gronwall lemma.

1We commonly say that (the bilinear form associated to)−Λ is coercive if (ii) holds withα >0 andb= 0, and that Λ−bis dissipative if (ii) holds withα= 0 andb∈R. Our assumption (ii) is then more general than a coercivity condition (on −Λ) but less general than a dissipativity condition (on Λ).

From the natural estimate (2.3) together with equation (2.1) and the continuity estimate (i) on Λ, we deduce

dg dt

= kΛgk

≤ M kgk

∈ L

(0, T ), and we conclude with

(2.4) g ∈ L

(0, T ; H) ∩ L

(0, T ; V ) ∩ H

(0, T ; V

).

The two first Lebesgue spaces are defined through the Bochner integral which main features are explained in Appendix B while the last Sobolev space is defined as explained below. It is worth emphasizing that (almost) everything in the Bochner integral for functions with values in Banach spaces holds similarly as for the usual Lebesgue integral for functions with values in R .

Definition 2.2. For a Banach space X and an exponent 1 ≤ p ≤ ∞, we say that g ∈ W

(0, T ; X ) if g ∈ L

(0, T ; X ) and there exists w ∈ L

(0, T ; X ) such that

− Z

0

g ϕ

dt = Z

0

w ϕ dt in X , ∀ ϕ ∈ D(0, T ).

We note w = g

or w =

g.

3. Variational solutions

We developp the theory of variational solution for evolution equation associated to coercive+dissipative operator and we state our main existence and uniqueness result.

Definition 3.1. For any given g

∈ H , T > 0, we say that

g = g(t) ∈ X

:= C([0, T ]; H) ∩ L

(0, T ; V ) ∩ H

(0, T ; V

)

is a variational solution to the Cauchy problem (2.1)–(2.2) on the time interval [0, T ] if it is a solution in the following weak sense

(3.1) (g(t), ϕ(t))

= (g

, ϕ(0))

+ Z

0

hΛg(s), ϕ(s)i

+ hϕ

(s), g(s)i

ds, for any ϕ ∈ X

and any 0 ≤ t ≤ T . We say that g is a global solution if it is a solution on [0, T ] for any T > 0. It is worth emphasizing that (3.1) is meaningful because of the assumptions made on g, ϕ and Λ.

Theorem 3.2 (J.-L. Lions). With the above notations and assumptions for any g

∈ H, there exists a unique global variational solution to the Cauchy problem (2.1), (2.2). As a consequence, any solution satisfies (2.3).

We start with some remarks and we postpone the proof of the existence part of Theorem 3.2 to the next section.

3.1. Parabolic equation. As a consequence of Theorem 3.2, for any f

∈ L

( R

) there exists a unique function

f = f (t) ∈ C([0, T ]; L

) ∩ L

(0, T ; H

) ∩ H

(0, T ; H

), ∀ T > 0,

which is a solution to the parabolic equation (1.1)-(1.2) in the variational sense.

3.2. About the functional space. The space obtained thanks to the a priori estimates established on g is nothing but X

as consequence of the following result.

Lemma 3.3. The following inclusion

L

(0, T ; V ) ∩ H

(0, T ; V

) ⊂ C([0, T ]; H )

holds true. Moreover, for any g ∈ L

(0, T ; V ) ∩ H

(0, T ; V

) there holds t 7→ |g(t)|

∈ W

(0, T )

and

(3.2) d

dt |g(t)|

= 2 hg

(t), g(t)i

a.e. on (0, T ).

Proof of Lemma 3.3. Step 1. We define ¯ g = g on [0, T ], ¯ g = 0 on R \[0, T ], and for a mollifier ρ with compact support included in (−1, −1/2), we define the approximation to the identity sequence (ρ

) by setting ρ

(t) := ε

ρ(ε

t) and then the sequence g

(t) := ¯ g ∗

ρ

where ∗ stands for the usual convolution operator on R . We observe that g

∈ C

( R ; H ), g

→ g a.e. on [0, T ] and in L

(0, T ; V ).

For a fixed τ ∈ (0, T ) and for any t ∈ (0, τ ) and any 0 < ε < T − τ, we have s 7→ ρ

(t − s) ∈ D(0, T ), since supp ρ

(t − ·) ⊂ [t + ε/2, t + ε] ⊂ [ε/2, τ + ε], and we compute

g

= Z

R

∂

ρ

(t − s) ¯ g(s) ds

= −

Z

0

(∂

ρ

(t − s)) g(s) ds

= Z

0

ρ

(t − s) g

(s) ds = ρ

∗ (g

).

As a consequence g

→ g

a.e. and in L

(0, τ ; V

).

Step 2. We fix τ ∈ (0, T ) and ε, ε

∈ (0, T − τ), and we compute d

dt |g

(t) − g

(t)|

= 2 hg

− g

, g

− g

i

, so that for any t

, t

∈ [0, τ]

(3.3) |g

(t

) − g

(t

)|

= |g

(t

) − g

(t

)|

+ 2 Z

t₁

hg

− g

, g

− g

ids.

Since g

→ g a.e. on [0, τ ] in H , we may fix t

∈ [0, τ] such that

(3.4) g

(t

) → g(t

) in H.

As a consequence of (3.3), (3.4) as well as g

→ g in L

(0, τ ; V ) and g

→ g

in L

(0, τ ; V

), we have

lim sup

ε,ε⁰→0

sup

[0,τ]

|g

(t) − g

(t)|

≤ lim

ε,ε⁰→0

Z

0

kg

− g

k

kg

− g

k

ds = 0,

so that (g

) is a Cauchy sequence in C([0, τ ]; H), and then g

converges in C([0, τ ]; H)

to a limit ˜ g ∈ C([0, τ]; H ). That proves g = ˜ g a.e. and thus g ∈ C([0, τ ]; H)

(up to modifying g on a set of zero Lebesgue measure). We prove similarly that

g ∈ C([τ, T]; H ) for any τ ∈ (0, T ) and thus g ∈ C([0, T ]; H ).

Step 3. Similarly as for (3.3), we have

|g

(t

)|

= |g

(t

)|

+ 2 Z

t₁

hg

, g

ids,

and passing to the limit ε → 0, we get

|g(t

)|

= |g(t

)|

+ 2 Z

t1

hg

, gids.

Using again that hg

, gi ∈ L

(0, T ), we easily deduce from the above identity the

two remaining claims of the Lemma.

3.3. A posteriori estimate and uniqueness. Taking ϕ = g ∈ X

as a test function in (3.1), we deduce from Lemma 3.3,

1

2 |g(t)|

− 1

2 |g

|

= |g(t)|

− |g

|

− Z

0

hg

(s), g(s)i ds

= Z

0

hΛg, gi ds

≤ Z

0

(−α kgk

+ b |g|

) ds,

where we have used (3.2) at the first line, the variational formulation (3.1) at the second line and the “coercive+dissipative” assumption on Λ at the last line. We then obtain (2.3) as an a posteriori estimate thanks to the Gronwall.

Let us prove now the uniqueness of the variational solution g associated to a given initial datum g

∈ H. In order to do so, we consider two variational solutions g and f associated to the same initial datum. Since the equation (2.1), (2.2) is linear, or more precisely, the variational formulation (3.1) is linear in the solution, the function g − f satisfies the same variational formulation (3.1) but associated to the initial datum g

− f

= 0. The a posteriori estimate (2.3) then holds for g − f and implies that g − f = 0.

4. Proof of the existence part of Theorem 3.2.

We first prove thanks to a compactness argument in step 1 to step 3 that there exists a function g ∈ L

(0, T ; V ) such that

(4.1) hg

, ϕ(0)i + Z

0

hΛg(s), ϕ(s)i

+ hϕ

(s), g(s)i

ds = 0

for any ϕ ∈ C

([0, T ); V ). We then deduce by some “regularization tricks” in step 4 and step 5 that the above weak solution is a variational solution.

Step 1. For a given g

∈ H and ε > 0, we seek g

∈ V such that

(4.2) g

− εΛg

= g

.

We introduce the bilinear form a : V × V → R defined by a(u, v) := (u, v) − ε hΛu, vi.

Thanks to the assumptions made on Λ, we have

|a(u, v)| ≤ |u| |v| + ε M kuk kvk,

and

a(u, u) ≥ |u|

+ ε α kuk

− ε b |u|

≥ ε α kuk

,

whenever ε b < 1, what we assume from now. On the other hand, the mapping v ∈ V 7→ (g

, v) is a linear and continuous form. We may thus apply the Lax- Milgram theorem which implies

∃! g

∈ V (g

, v) − εhΛg

, vi = (g

, v) ∀ v ∈ V.

Step 2. Fix ε > 0 as in the preceding step and build by induction the sequence (g

) in V ⊂ H defined by the family of equations

(4.3) ∀ k g

− g

ε = Λ g

. Observe that from the identity

(g

, g

) − ε hΛg

, g

i = (g

, g

), we deduce

|g

|

+ ε α kg

k

− ε b |g

|

≤ |g

| |g

| ≤ 1

2 |g

|

+ 1

2 |g

|

, and then

|g

|

+ 2εα kg

k

≤ (1 − 2ε b)

|g

|

, ∀ k ≥ 0.

Thanks to the discrete version of the Gronwall lemma, we deduce

|g

| + 2α

n

X

k=1

εkg

k

≤ (1 − 2ε b)

|g

| ≤ e

|g

|.

We now fix T > 0, n ∈ N

, and we define

ε := T /n, t

= k ε, g

(t) := g

on [t

, t

).

The last estimate writes then

(4.4) sup

[0,T]

|g

|

+ 2α Z

0

kg

k

dt ≤ e

|g

|

.

Step 3. Consider a test function ϕ ∈ C

([0, T ); V ) and define ϕ

:= ϕ(t

), so that ϕ

= ϕ(T ) = 0. Multiplying the equation (4.3) by ϕ

and summing up from k = 0 to k = n, we get

−(ϕ

, g

) −

n

X

k=1

hϕ

− ϕ

, g

i =

n

X

k=0

ε hΛg

, ϕ

i =

n

X

k=1

ε hΛg

, ϕ

i, where in the LHS we use the duality production h, i in V

× V instead of the scalar product (, ) in H thanks to the inclusions V ⊂ H = H

⊂ V

. Introducing the two functions ϕ

, ϕ

: [0, T ) → V defined by

ϕ

(t) := ϕ

and ϕ

(t) := t

− t

ε ϕ

+ t − t

ε ϕ

for t ∈ [t

, t

), in such a way that

ϕ

(t) = ϕ

− ϕ

ε for t ∈ (t

, t

), the above equation also writes

(4.5) − hϕ(0), g

i − Z

ε

hϕ

, g

i dt = Z

0

hΛg

, ϕ

i dt.

On the one hand, from (4.4) and Fact ?? about Bochner integral, we know that up to the extraction of a subsequence, there exists g ∈ X

such that g

* g weakly in L

(0, T ; V ). On the other hand, from the above construction, we have ϕ

→ ϕ

and ϕ

→ ϕ both strongly in L

(0, T ; V ). We may then pass to the limit as ε → 0 in (4.5) and we get (4.1).

Step 4. We prove that g ∈ X

. Taking ϕ := χ(t) ψ with χ ∈ C

((0, T )) and ψ ∈ V in equation (4.1), we get

D Z

0

gχ

dt, ψ E

= Z

0

hg, ψiχ

dt = − Z

0

hΛg, ψiχ dt = D

− Z

0

Λgχ dt, ψ E . This equation holding true for any ψ ∈ V , it is equivalent to

Z

0

gχ

dt = − Z

0

Λgχ dt in V

for any χ ∈ D(0, T ), or in other words

g

= Λg in the sense of distributions in V

.

Since g ∈ L

(0, T ; V ), we get that Λg ∈ L

(0, T ; V

) and the above relation precisely means that g ∈ H

(0, T ; V

). We conclude thanks to Lemma 3.3 that g ∈ X

. Step 5. Assume first ϕ ∈ C

([0, T ); H ) ∩ L

(0, T ; V ) ∩ H

(0, T ; V

). We define ϕ

(t) := ϕ ∗

ρ

for a mollifier (ρ

) with compact support included in (0, ∞) so that ϕ

∈ C

([0, T ); V ) for any ε > 0 small enough and

ϕ

→ ϕ in C([0, T ]; H ) ∩ L

(0, T ; V ) ∩ H

(0, T ; V

).

Writing the equation (4.1) for ϕ

and passing to the limit ε → 0 we get that (4.1) also holds true for ϕ.

Assume next that ϕ ∈ X

. We fix χ ∈ C

( R ) such that supp χ ⊂ (−∞, 0), χ

≤ 0, χ

∈ C

(] − 1, 0[) and R

−1

χ

= −1, and we define χ

(t) := χ((t − T )/ε) so that ϕ

:= ϕχ

∈ C

([0, T ); H ) and χ

→ 1

, χ

→ −δ

as ε → 0. Equation (4.1) for the test function ϕ

writes

−hg

, ϕ(0)i − Z

0

χ

hϕ, gi ds = Z

0

χ

hΛg, ϕi + hϕ

, gi ds,

and we obtain the variational formulation (3.1) for t

= 0 and t

= T by passing

to the limit ε → 0 in the above equation.

AppendixA. Exercises

Exercise A.1. Following Definition 2.2, we say thatg∈L²(0, T;V)is a solution to the abstract evolution equation(2.1)-(2.2)if

(A.1) −g0ϕ(0)−

ZT

0

g ϕ⁰dt= Z T

0

Λg ϕ dtinV⁰, ∀ϕ∈C_c¹([0, T)).

Prove that under the hypothesis of Theorem 3.2, a function g ∈ L²(0, T;V) is a solution to the abstract evolution equation (2.1)-(2.2)in the above sense if, and only if, it is a variational solution (so that in particularg∈XT). (Hint. Use some arguments presented in Lemma 3.3 and in steps 3, 4 and 5 of the proof of Theorem 3.2).

Exercise A.2. Prove thatf ≥0iff0 ≥0and b≥0for the solution of the parabolic equation (1.1). (Hint. Show that the sequence(gk)defined in step 2 of the proof of the existence part is such thatgk≥0for anyk∈N).

Exercise A.3. Prove the existence of a solutiong∈XT to the equation

(A.2) dg

dt = Λg+G in (0, T), g(0) =g0, for any initial datumg0∈H and any source termG∈L²(0, T;V⁰).

(Ind. Repeat the same proof as for the Theorem 3.2 where for the a priori bound one can use Z T

0

hg, Gidt≤α 2

Z T

0

kg(t)k²_Vdt+ 1 2α

Z T

0

kG(t)k²_V0dt, and for the approximation scheme one can define

ε⁻¹(gk+1−gk) = Λgk+1+Gk, Gk:=

Z t_k+1

t_k

G(s)ds).

Exercise A.4. Generalize the existence and uniqueness result to the PDE equation

(A.3) ∂tf=∂i(aij∂jf) +bi∂if+cf+ Z

k(t, x, y)f(t, y)dy+G

whereaij,bi,candkare times dependent coefficients and whereaij is uniformly elliptic in the sense that

(A.4) ∀t∈(0, T), ∀x∈R^d, ∀ξ∈R^d aij(t, x)ξiξj≥α|ξ|², α >0.

More precisely, establish the following result:

Theorem A.5(J.-L. Lions - the time dependent case). Assume that a, b, c∈L^∞((0, T)×R^d), k∈L^∞(0, T;L²(R^d×R^d)),

and thatasatisfies the uniformly elliptic condition(A.4). For anyg0∈L²(R^d)andG∈L²(0, T;

H⁻¹(R^d)), there exists a unique variational solution to the Cauchy problem associated to (A.3) in the sense that

f∈XT :=C([0, T);L²)∩L²(0, T;H¹)∩H¹(0, T;H⁻¹), such that for anyϕ∈XT and anyt∈(0, T)there holds

Z

R^d

g(t)ϕ(t)dx= Z

R^d

g0ϕ(0)dx+ Z t

0

Z

R^d

(Gϕ+g∂tϕ)dxds (A.5)

+ Z t

0

Z

R^d

{(b_i∂if+cf)ϕ−aij∂jf ∂iϕ}dxds+ Zt

0

Z

R^d×R^d

k(t, x, y)f(t, y)ϕ(s, x)dxdyds

Exercise A.6. We consider the nonlinear McKean-Vlasov equation (A.6) ∂tf= Λ[f] := ∆f+div(F[f]f), f(0) =f0, with

F[f] :=a∗f, a∈L^∞(R^d)^d. 1) Prove the a priori estimates

kf(t)k_L1 =kf₀k_L1 ∀t≥0, kf(t)k_L2 k

≤e^Ctkf₀k_L2 k

∀t≥0,

for anyk >0and a constantC:=C(k,kak_L∞,kf₀k_L1), where we define the weighted Lebesgue spaceL²_k by its normkfk_L2

k:=kfhxi^kk_L2,hxi:= (1 +|x|²)^1/2.

2) We setH :=L²_k,k > d/2, and V :=H_k¹, where we define the weighted Sobolev spaceH_k¹ by its normkfk²

H_k¹ :=kfk²

L²_k+k∇fk²

L²_k. Observe that for anyf ∈V the distributionΛ[f]is well defined inV⁰ thanks to the identity

hΛ[f], gi:=− Z

R^d

(∇f+ (a∗f)f)· ∇(ghxi^2k)dx ∀g∈V.

(Hint. Prove that L²_k ⊂ L¹). Write the variational formulation associated to the nonlinear McKean-Vlasov equation. Establish that if moreover the variational solution to the nonlinear McKean-Vlasov equation is nonnegative then it is mass preserving, that iskf(t)k_L1 =kf₀k_L1

for any t≥0. (Hint. TakeχMhxi^−2k as a test function in the variational formulation, with χM(x) :=χ(x/M),χ∈ D(R^d),1_B(0,1)≤χ≤1_B(0,2)).

3) Prove that for any 0≤f0 ∈H and g ∈C([0, T];H)there exists a unique mass preserving variational solution0≤f∈XT to the linear McKean-Vlasov equation

∂tf= ∆f+div(F[g]f), f(0) =f0.

Prove that the mappingg7→f is a contraction inC([0, T];H)forT >0small enough. Conclude to the existence and uniqueness of a global (in time) variational solution to the nonlinear McKean- Vlasov equation.

Exercise A.7. For a∈W^1,∞(R^d),c∈ L^∞(R^d), f0 ∈ L^p(R^d), 1 ≤p≤ ∞, we consider the linear parabolic equation

(A.7) ∂tf= Λf:= ∆f+a· ∇f+cf, f(0) =f0.

We introduce the usual notationsH:=L²,V :=H¹andXT the associated space for some given T >0.

1) Prove that for γ ∈ C¹(R), γ(0) = 0, γ⁰ ∈ L^∞, there holds γ(f) ∈ H for anyf ∈ H and γ(f)∈V for anyf∈V.

2) Prove thatf∈XT is a variational solution to(A.7)if and only if d

dtf= Λf inV⁰ a.e. on(0, T).

3) On the other hand, prove that for anyf∈XT and any functionβ∈C²(R),β(0) =β⁰(0) = 0, β⁰⁰∈L^∞, there holds

d dt

Z

R^d

β(f) =hd

dtf, β⁰(f)i_V0,V a.e. on(0, T).

(Hint. Considerfε=f∗tρε∈C¹([0, T];H¹)and pass to the limitε→0).

4) Consider a convex functionβ∈C²(R)such thatβ(0) =β⁰(0) = 0andβ⁰⁰∈L^∞. Prove that any variational solutionf∈XT to the above linear parabolic equation satisfies

Z

R^d

β(ft)dx≤ Z

R^d

β(f0)dx+ Zt

0

Z

R^d

{c f β⁰(f)−(diva)β(f)}dxds, for anyt≥0.

5) Assuming moreover that there exists a constantK∈(0,∞)such that0≤s β⁰(s)≤Kβ(s)for anys∈R, deduce that for some constantC:=C(a, c, K), there holds

Z

R^d

β(ft)dx≤e^Ct Z

R^d

β(f0)dx, ∀t≥0.

6) Prove that for anyp∈[1,2], for some constantC:=C(a, c)and for anyf0∈L²∩L^p, there holds

kf(t)k_Lp≤e^Ctkf0k_Lp, ∀t≥0.

(Hint. DefineβonR+and extend it toRby symmetry. More precisely, defineβ_α⁰⁰(s) = 2θ1s≤α+ p(p−1)s^p−21s>α, with 2θ=p(p−1)α^p−2 and then the primitives which vanish at the origin, which are thus defined byβ⁰_α(s) = 2θs1s≤α+ (ps^p−1+p(p−2)α^p−1)1s>α,βα(s) =θs²1s≤α+ (s^p+p(p−2)α^p−1s+Aα^p)1s>α,A:=p(p−1)/2−1−p(p−2). Observe thatsβ_α⁰(s)≤2βα(s) becausesβ_α⁰⁰(s)≤β⁰_α(s)andβα(s)≤β(s)becauseβ⁰⁰_α(s)≤β⁰⁰(s)).

7) Prove that for anyp∈[2,∞]and for some constantC:=C(a, c, p)there holds kf(t)k_Lp≤e^Ctkf0k_Lp, ∀t≥0.

(Hint. Define β_R⁰⁰(s) = p(p−1)s^p−21s≤R+ 2θ1s>R, with2θ = p(p−1)R^p−2, and then the primitives which vanish in the origin and which are thus defined by β_R⁰(s) = ps^p−11s≤R+ (pR^p−1+ 2θ(s−R))1s>R,βR(s) =s^p1s≤R+ (R^p+pR^p−1(s−R) +θ(s−R)²)1s>R. Observe thatsβ⁰_R(s)≤pβR(s)because sβ⁰⁰_R(s)≤(p−1)β⁰_R(s)and βR(s)≤β(s)becauseβ_R⁰⁰(s)≤β⁰⁰(s).

Pass to the limitp→ ∞in order to deal with the casep=∞).

8) Prove that for anyf0 ∈L^p(R^d),1≤p≤ ∞, there exists at least one weak (in the sense of distributions) solution to the linear parabolic equation (A.7). (Hint: Considerf0,n ∈L¹∩L^∞ such thatf0,n→f0inL^p,1≤p <∞, and prove that the associate variational solutionfn∈XT

is a Cauchy sequence inC([0, T];L^p). Conclude the proof by passing to the limitp→ ∞).

9) Establsih theL^p estimates with “optimal” constant C (that is the one given by the formal computations).

10) Extend the above result to an equation with an integral term and/or a source term.

11) Prove the existence of a weak solution to the McKean-Vlasov equation (A.6)for any initial datumf0∈L¹(R^d).

12) Recover the positivity result of exercice A.2. (Hint. Chooseβ(s) :=s−).

AppendixB. The Bochner integral

In this section we present several definitions and results (without proof) about the Bochner integral which generalizes the Lebegue integral for functions taking values in a general Banach space. We refer to [3, sections V.3, V.4] for details.

We considerX a Banach space endowed with the strong topology (associated to its norm) and (Ω,T, µ) a measured space.

We say thatf: Ω→ X is a simple function if f=X

i∈I

ai1A_i,

for some finite setI, some measurable setsAiand someai∈ X. For a simple function, we define the integral by

Z

Ω

f dµ:=X

i∈I

aiµ(Ai)∈ X.

We say that a functionf: Ω→ X ismeasurableif there exists a sequence (fn) of simple functions such that

a.e. on Ω, fn→f.

Similarly and for later reference, we say a functionf: Ω→ X isweakly measurableif there exists a sequence (fn) of simple functions such that

a.e. on Ω, fn* f weakly.

Here, weakly has to be understood in the sense of the weakσ(X,X⁰) topology or of the weak∗

σ(Y⁰,Y) when X = Y⁰ for some Banach spaceY. In both cases, we have kfk ≤ lim infkf_nk.

Observe that iff is (weakly) measurable thenkfkis measurable (because the norm is lsc for the weak topologies). We say that a measurable functionf is integrable (in the sense of Bochner) if there exists a sequence (fn) of simple functions such that

fn→f a.e. on Ω and Z

Ω

kfn−fkdµ→0.

Equivalently, a measurable functionf: Ω→ X is integrable ifkfkis integrable. For a integrable function, we define the (Bochner) integral by

Z

Ω

f dµ:= lim

n→∞

Z

Ω

fndµ.

We have then

Z

Ω

f dµ ≤

Z

Ω

kfkdµ.

In the sequel, we do not distinguish between two measurable functionsf, g such thatf =ga.e.

on Ω, and we just writef=gin that case (which as usually means thatfandgare in the same class of functions for the equivalence relation which is the a.e. equality). We define the Lebesgue spaces

L^p(Ω;X) :={f: Ω→ X measurable; kfkL^p<∞}

with

kfk_Lp:=

Z

Ω

kfk^pdµ 1/p

, p∈[1,∞), kfk_L∞:= inf{λ≥0; kfk ≤λa.e.}.

The fundamental exemple is the following. We consider two measurable spaces (E,A, µ) and (F,B, ν) as well as a measurable function f : E×F → R. Abusing notations, we have the equivalence betweenf∈L^p(E;L^q(F)) and

Z

E

Z

F

|f(x, y)|^qdν(y)p/q

dµ(x)<∞,

thanks to the Fubini-Tonelli theorem, and a similar equivalence holds whenporq=∞. Because of this observation, whenX =L^q(F), most of the results presented below can alternatively be proved using the usual Lebesgue integration theory and Fubini-Tonelli type results.

At this point, it is worth mentioning that another alternative to the Bochner integral is to define the integral by weak∗duality.

Fact B.1. Consider a functionf : Ω→ Y⁰ which is weakly∗ measurable and such thatkfk is integrable. Then, there exists a uniqueξ∈ Y⁰such that

∀y∈ Y, hξ, yi_Y0,Y= Z

Ω

hf(x), yi_Y0,Ydµ(x).

We say thatξis the weak∗integral off, and we write

ξ= Z

Ω

f(x)dµ(x).

Of course, whenX is reflexive the weak∗integral coincides with the Bochner integral. We do not pursue this point of view in the sequel.

Fact B.2. The Lebesgue spacesL^p(Ω;X)are Banach spaces for any1≤p≤ ∞, andCc(Ω;X) is dense inL^p(Ω;X)when1≤p <∞and(Ω,T)is the borelianσ-algebra associated to a locally compact and σ-compact topologic space Ω (for instance when Ω ⊂ Rendowed with the usual topology).

Fact B.3. If Ω = (0, T) and T is the Lebesgue σ-algebra, any weakly σ(X,X⁰) continuous functionf: Ω→ X is measurable.

Fact B.4. If (fn) is a sequence of measurable functions such thatfn* fa.e. inσ(X,X⁰), then f is measurable.

The two preceding facts are consequences of a more general and fundamental result.

Fact B.5. (Pettis) A function f : Ω → X is measurable iff it is weakly measurable (for the σ(X,X⁰) topology) and separately valued, which means that its range{f(x), x∈Ω}is a separable set.

Fact B.6. (Lebesgue dominated convergence Theorem) If (fn) is a sequence of integrable func- tions andg: Ω→R+ is an integrable function such that

fn→f a.e. and kfnk ≤ga.e., thenf is integrable andfn→finL¹(Ω,X).

Fact B.7. ConsiderX, Y two Banach spaces, 1≤p≤ ∞ andA∈ B(X, Y). Iff ∈L^p(Ω;X) thenAf∈L^p(Ω;Y). If furthermorep= 1 then

A Z

f dµ= Z

(Af)dµ.

In particular, we have D

ξ, Z

f(x)dµ(x)E

= Z

hξ, f(x)idµ(x), ∀ξ∈ X⁰.

If furthermore,X⊂Y, thenf∈L¹(Ω;X) impliesf∈L¹(Ω;Y) and the two integrals coincide.

Fact B.8. If f ∈ L^p(Ω;X), g ∈ L^p0(Ω,R), h ∈ L^p0(Ω,X⁰), we havef g ∈ L¹(Ω;X), hf, hi ∈ L¹(Ω,R). In the particular case Ω =R, we may also define the convolution productf∗g∈L¹(R;X) for anyf∈L¹(R;X) andg∈L¹(R;R). For a sequence of mollifiers (ρε) withρε∈C_c^k(R),ρε* δ0, we then havef∗ρε∈C^k(R;X) andf∗ρε→finL^p(R;X).

Fact B.9. If X is a reflexive space andp∈ (1,∞), thenL^p(Ω;X) is also a reflexive space, in particular if (fn) is a bounded sequence inL^p(Ω;X), there existsf∈L^p(Ω;X) such thatfn* f, which means

Z

Ω

hξ(x), fn(x)idµ(x) → Z

Ω

hξ, f(x)idµ(x),

for any ξ ∈ L^p0(Ω,X⁰). Similarly, if X = Y⁰ with Y separable, then L^∞(Ω;X) is weakly∗

sequentially compact, which means that if (fn) is a bounded sequence inL^∞(Ω;X), there exists f∈L^∞(Ω;X) such thatfn* f weakly∗, or in other words

Z

Ω

hfn(x), u(x)idµ(x) → Z

Ω

hf(x), u(x)idµ(x), for anyu∈L¹(Ω,Y).

Fact B.10. For 1≤p≤ ∞, if (fn) is a sequence ofL^p(Ω;X),f: Ω→ X is a function such that kf_nk_Lp(Ω;X)≤C, fn* f a.e.

thenf∈L^p(Ω;X) andkfk_Lp≤lim infkfnk_Lp.

Fact B.11. The spaceD((0, T);X) of smooth and with compact support functions with values inX is dense inL^p(0, T;X) for any 1≤p <∞, and weakly∗dense inL^∞(0, T;X).

Assume that Ω is an open set of R^d. A distribution is a linear and continuous mappingT : D(Ω)→ X, we noteT ∈ D⁰(Ω;X). For example, iff∈L¹_loc(Ω;X), we define

hT_f, ϕi:=

Z

Ω

f(x)ϕ(x)dx

and we observe thatTf ∈ D⁰(Ω;X). We haveTf = 0 implies f= 0 a.e. For T ∈ D⁰(Ω;X), we define∂T∈ D⁰(Ω;X) by

h∂T , ϕi:=−hT , ∂ϕi, ∀ϕ∈ D(Ω).

ForT∈(0,∞), we define

W^1,p((0, T);X) :={f∈L^p(0, T;X), f⁰∈L^p(0, T;X)}.

We haveW^1,p((0, T);X)⊂C([0, T];X).

AppendixC. Further results: C0-semigroup, evolution equation with source term and Duhamel formula

C.1. C0-semigroup. We explain how we may associate a C0-semigroup to the evolution equa- tion (2.1), (2.2) as a mere consequence of the linearity of the equation and of the existence and uniqueness result.

Definition C.1. ConsiderXa Banach space, and denote byB(X)the set of linear and bounded operators onX. We say thatS= (St)t≥0is a strongly continuous semigroup of linear operators onX, or just aC0-semigroup onX, we also writeS(t) =St, if

(i)∀t≥0, St∈B(X)(one parameter family of operators);

(ii)∀f∈X, t7→Stf∈C([0,∞), X)(continuous trajectories);

(iii)S0=I; ∀s, t≥0 St+s=StSs (semigroup property).

Proposition C.2. The operator Λgenerates a semigroup on H defined in the following way.

For anyg0∈H, we setStg:=g(t)whereg(t)is the unique variational solution associated tog0

and given by Theorem 3.2. We also denoteSΛ(t) =e^Λt=St for anyt≥0.

•Ssatisfies (i). By linearity of the equation and uniqueness of the solution, we clearly have St(g0+λf0) =g(t) +λf(t) =Stg0+λStf0

for anyg0, f0 ∈H,λ∈Randt≥0. Thanks to estimate (2.3) we also have|S_tg0| ≤e^bt|g₀|for anyg0∈Handt≥0. As a consequence,St∈B(H) for anyt≥0.

•Ssatisfies (ii). Thanks to lemma 3.3 we havet7→Stg0∈C(R+;H) for anyg0∈H.

•Ssatisfies (iii).Forg0∈Handt1, t2≥0 denoteg(t) =Stg0 and ˜g(t) :=g(t+t1). Making the difference of the two equations (3.1) written fort=t1 andt=t1+t2, we see that ˜gsatisfies

(˜g(t2),ϕ(t˜ 2)) = (g(t1+t2), ϕ(t1+t2))

= (g(t1), ϕ(t1)) + Zt1+t2

t₁

hΛg(s), ϕ(s)i+hϕ⁰(s), g(s)i ds

= (˜g(0),ϕ(0)) +˜ Zt₂

0

hΛ˜g(s),ϕ(s)i˜ +hϕ˜⁰(s),g(s)i˜ ds,

for anyϕ∈Xt₁+t₂with the notation ˜ϕ(t) :=ϕ(t+t1)∈Xt₂. Since the equation on the functions

˜

g and ˜ϕ is nothing but the variational formulation associated to the equation (2.1), (2.2) with initial datum ˜g(0), we obtain

St₁+t₂g0=g(t1+t2) = ˜g(t2) =St₂˜g(0) =St₂g(t1) =St₂St₁g0.

Exercise C.3. We denote byStthe semigroup inHgenerated by a coercive+dissipative operator Λ :V ⊂H→V⁰.

1) Prove that for anyg0∈H andϕ∈V the functiont7→(Stg0, ϕ)belongs toH¹(0, T)and d

dt(Stg0, ϕ) = (ΛStg0, ϕ) in H⁻¹(0, T).

2) Prove that for anyG∈C([0, T];H)andϕ∈V there holds d

dt Z t

0

(St−sG(s), ϕ)ds= (G(t), ϕ) + Z t

0

(ΛSt−sG(s), ϕ)ds in H⁻¹(0, T).

3) Establish the Duhamel formula, namely that forg0∈H andG∈C([0, T];H), the function g(t) :=Stg0+

Zt

0

St−sG(s)ds

is a weak (make precise the sense) solution to the evolution equation with source term dg

dt = Λg+G on [0,∞), g(0) =g0.

C.2. Evolution equation with source term and Duhamel formula. In that last section, we come back on Exercices A.3 and C.3.

•Forg0 ∈H andG∈L²(0, T;V⁰) a functiong∈XT is a variational solution to the evolution equation with source term

(C.1) dg

dt = Λg+G on [0, T], g(0) =g0, if that equation holds inV⁰, namely if for anyϕ∈V there holds

d

dt(g(t), ϕ) =hΛg(t), ϕi+hG(t), ϕi in the sense of D⁰(0, T;R).

That is equivalent to hd

dtg(t), ϕi=hΛg(t), ϕi+hG(t), ϕi a.e. t∈(0, T), or more explicitely

Z T

0

(g(t), ϕ)χ⁰dt−(g0, ϕ)χ(0) = ZT

0

{hΛg(t), ϕi+hG(t), ϕi}χ(t)dt

for anyχ∈C_c¹([0, T)) andϕ∈V. One can then deduce from the last formulation and by density of the separate variables functionsD(0, T)⊗V intoXT, or just by taking the next formulation as a definition of a variational solution, that for anyϕ∈XT

(C.2) [hg, ϕi]^T₀ −

Z T

0

hd

dtϕ, gidt= ZT

0

hΛg+G, ϕidt.

• Wheng0 ∈ H and G∈ C([0, T];H) one can define thanks to Proposition C.2 the following function

(C.3) g(t) :=e^Λtg0+

Zt

0

e^{Λ (t−s)}G(s)ds.

We see thatg∈C([0, T];H), and using the estimate (2.3) ZT

0

ke^Λtfk²_Vdt≤e^2bT 2α |f|²_H, we easily find

Z T

0

kg(t)k²_Vdt≤C1(T)|g₀|²_H+C2(T) Z T

0

|G(s)|²_Hds.

Finaly, since

t7→e^Λtf∈H¹(0, T;V⁰), k∂t(e^Λtf)k_L2(V⁰)≤C3(T)|f|²_H,

we deduce thatg∈H¹(0, T;V⁰) with explicit estimates. We may then compute inL²(0, T;V⁰)

∂tg= Λe^Λtg0+G(t) + Z t

0

Λe^{Λ (t−s)}G(s)ds= Λg+G(t),

(see also Exercice C.3) and we obtain thatg(t) is a variational solution to the evolution equation with source term (C.1).

•Wheng0∈H andG∈L²([0, T];V⁰) the sense of the Duhamel formula is less clear. One can however prove the existence of a variational solution by just repeating the proof used to tackle the sourceless evolution equation (1.1). More precisely, we consider the following discrete scheme:

we build (gk) iteratively by setting gk+1−gk

ε = Λg_k+1+G_k, G_k:=

Z t_k+1

t_k

G(s)ds.

We compute

|g_k+1|²(1−εb) +ε αkg_k+1k²_V ≤ |g_k| |g_k+1|+εkg_k+1k_VkG_kk_V0

≤ 1

2|gk|²+1

2|gk+1|²+εα

2kgk+1k²_V + ε

2αkGkk²_V0, and then

|g_k+1|²(1−2εb) +ε αkg_k+1k²_V ≤ |g_k|²+ ε

αkG_kk²_V0.

We get an estimate on|g_k+1|² which is uniform onkwhenk ε≤T, forT >0 fixed, by using a discrete version of the Gronwall lemma. We conclude as in the proof of Theorem 3.2.

•We may argue in a different way. When g0 ∈H andG∈ C([0, T];H) the Duhamel formula (C.3) gives a variational solution to the evolution equation with source term in the sense (C.2).

Making the choiceϕ=g, we get 1

2[|g|²]^T₀ = ZT

0

(hΛg, gi+hG, gi)dt

≤ ZT

0

{−αkgk²_V+b|g|²+kGk_V0kgk_V}dt

≤ −α 2

Z T

0

kgk²_Vdt+ b² 2α

ZT

0

kGk²_V0dt,

and thanks to the Gronwall lemma, we obtain

|g(T)|²+α Z T

0

kgk²_Vdt≤e^bT|g0|²_H+CT

ZT

0

kGk²_V0dt.

We conlcude to the existence by smoothing the source termG(what it is always possible in the explicit examplesH=L²,V =H¹) and by passing to the limit in the variational formulation.

Appendix D. References

Theorems 3.2 and A.5 are stated in [1, Th´ eor` eme X.9] where the quoted reference for a proof is [2].

• [1] Brezis, H. Analyse fonctionnelle. Collection Math´ ematiques Appliqu´ ees pour la Maˆıtrise. [Collection of Applied Mathematics for the Master’s De- gree]. Masson, Paris, 1983. Th´ eorie et applications. [Theory and applica- tions].

• [2] Lions, J.-L., and Magenes, E. Probl` emes aux limites non homog` enes et applications. (3 volumes). Travaux et Recherches Math´ ematiques. Dunod, Paris, 1968 & 1969.

• [3] Yosida, K. Functional analysis. Classics in Mathematics, Springer-

Verlag, Berlin, 1995.