Universit´e Paris-Dauphine 2020-2021

An introduction to evolution PDEs November 2020

Some exercises

Exercise 0.1 (Chapter 5) Prove that for any f ∈H^{1}(R^{d}), there holds
kρ_{ε}∗f−fk_{L}^{2} ≤C εk ∇fk_{L}^{2},

for a constant C > 0 which only depends on the function ρ ∈ P(R^{d})∩ D(R^{d}) used in the
definition of the mollifier (ρ_{ε}). Deduce the Nash inequality. (Hint. Write f = f −ρ_{ε}∗f +
ρ_{ε}∗f).

Exercise 0.2 (Chapter 6) Let S = SL be a strongly continuous semigroup on a Banach
space X ⊂L^{1}. Show that there is equivalence between

(a) SL is a stochastic semigroup;

(b) L^{∗}1 = 0 and L satisfies Kato’s inequality

(signf)Lf ≤ L|f|, ∀f ∈D(L).

(Hint. In order to prove(b)⇒(a), considerf ∈D(L^{2})and estimate|S_{t}f|−|f|by introducing
a telescopic sum and a Taylor expansion in the time variable).

Exercise 0.3 (Chapter 6) Consider SL^{∗} a (constant preserving) Markov semigroup and
Φ : R → R a concave function. Prove that L^{∗}Φ(m)≤ Φ^{0}(m)L^{∗}m. (Hint. Use that Φ(a) =
inf{`(a);` affine such that` ≥Φ}in order to prove S_{t}^{∗}(Φ(m))≤Φ(S_{t}^{∗}m)andΦ(b)−Φ(a)≥
Φ^{0}(a)(b−a)).

Exercise 0.4 We say that (S_{t})t≥0 is a dynamical system on a metric space (Z, d) if
(S1) ∀t ≥0, S_{t}∈C(Z,Z) (continuously defined onZ);

(S2) ∀x∈ Z, t 7→Stx∈C([0,∞),Z) (trajectories are continuous);

(S3) S_{0} =I; ∀s, t≥0, S_{t+s}=S_{t}S_{s} (semigroup property).

We say that z¯∈ Z is invariant (a steady state, a stationary point) if S_{t}z¯= ¯z for any t≥0.

Consider a bounded and convex subsetZ of a Banach spaceX which is sequentially compact
when it is endowed with the metric associated to the normk ·k_{X} (strong topology), to the weak
topology σ(X, X^{0}) or to the weak-? topology σ(X, Y), Y^{0} = X. Prove that any dynamical
system (S_{t})t≥0 on Z admits at least one steady state

(Hint. Observe taht for any dyadic number t > 0, there exists z_{t} ∈ Z such that S_{t}z_{t} = z_{t}
thanks to the Schauder point fixed theorem).

Exercise 0.5 (i) Prove that L^{p}∩L^{q}⊂L^{r} for any p≤r≤q and
kuk_{L}^{r} ≤ kuk^{θ}_{L}pkuk^{1−θ}_{L}q , 1

r = θ

p+ 1−θ

q , ∀u∈L^{p}∩L^{q}.
(ii) Prove that L^{p}^{1}(L^{p}^{2})∩L^{q}^{1}(L^{q}^{2})⊂L^{r}^{1}(L^{r}^{2}), and more precisely

kuk_{L}^{r}1(L^{r}^{2})≤ kuk^{θ}_{L}p1(L^{p}^{2})kuk^{1−θ}_{L}q1(L^{q}^{2}), 1
ri

= θ pi

+1−θ qi

for any u∈L^{p}^{1}(L^{p}^{2})∩L^{q}^{1}(L^{q}^{2}) for a same θ ∈(0,1).

(iii) Prove that H˙^{s}^{1} ∩H˙^{s}^{2} ⊂H˙^{s} for any s_{1} ≤s≤s_{2} and
kukH˙^{s} ≤ kuk^{θ}_{H}_{˙}s1kuk^{1−θ}_{˙}

H^{s}^{2}, s =θs_{1}+ (1−θ)s_{2}, ∀u∈H˙^{s}^{1} ∩H˙^{s}^{2}.
(iv) Prove that L^{p}( ˙H^{a})∩L^{q}( ˙H^{b})⊂L^{r}( ˙H^{c}), and more precisely

kuk_{L}r( ˙H^{c}) ≤ kuk^{θ}_{L}p( ˙H^{a})kuk^{1−θ}

L^{q}( ˙H^{b}), 1
r = θ

p+ 1−θ

q , c=θa+ (1−θ)b
for any u∈L^{p}( ˙H^{a})∩L^{q}( ˙H^{b}) for a same θ∈(0,1).

(v) Prove that H˙^{1/2}(R^{2}) ⊂ L^{4}(R^{2}), H˙^{1/2}(R^{3}) ⊂ L^{3}(R^{2}). (Hint. We will use the classical
Sobolev embedding and the following interpolation theorem in order to prove (at least when
s∈(0, d−1])

H˙^{s}(R^{d})⊂L^{p}(R^{d}), with 1
p = 1

2− s d, when s∈[0, d)).

Theorem 0.6 (Interpolation) Assume that T is a linear mapping such that T : W^{s}^{0}^{,p}^{0} →
W^{σ}^{0}^{,q}^{0} and T : W^{s}^{1}^{,p}^{1} → W^{σ}^{1}^{,q}^{1} are bounded, for some p_{0}, p_{1}, q_{0}, p_{1} ∈ [1,∞] and some
s_{0}, s_{1}, σ_{0}, σ_{1} ∈R. Then T :W^{s}^{θ}^{,p}^{θ} →W^{σ}^{θ}^{,q}^{θ} for any θ ∈[0,1], with

1 pθ

= 1−θ p0

+ θ p1

, 1 qθ

= 1−θ q0

+ θ q1

,

and

s_{θ} = (1−θ)s_{0}+θs_{1}, σ_{θ} = (1−θ)σ_{0}+θσ_{1},
when (p_{0} =p_{1} or s_{0} =s_{1}) and (q_{0} =q_{1} or σ_{0} =σ_{1}).

Exam 2015 - Three problems

Problem I - the fragmentation equation We consider the fragmentation equation

∂tf(t, x) = (Ff(t, .))(x)

on the density functionf =f(t, x)≥0,t, x >0, where the fragmentation operator is defined by

(Ff)(x) :=

Z ∞ x

b(y, x)f(y)dy−B(x)f(x).

We assume that the total fragmentation rateB and the fragmentation rate b satisfy
B(x) =x^{γ}, γ >0, b(x, y) = x^{γ−1}℘(y/x),

with

0< ℘∈C((0,1]), Z 1

0

z ℘(z)dz = 1, Z 1

0

z^{k}℘(z)dz <∞, ∀k <1.

1) Prove that for any f, ϕ:R+→R, the following identity Z ∞

0

(Ff)(x)ϕ(x)dx= Z ∞

0

f(x) Z x

0

b(x, y)

ϕ(y)− y xϕ(x)

dydx holds, whenever the two integrales at the RHS are absolutely convergent.

1) We define the moment function
M_{k}(f) =

Z ∞ 0

x^{k}f(x)dx, k∈R.

Prove that any solution f (∈C^{1}([0, T);L^{1}_{k})∩L^{1}(0, T;L^{1}_{k+γ}), ∀T > 0) to the fragmentation
equation formally (rigorously) satisfies

M_{1}(f(t)) = cst, M_{k}(f(t, .))% if k <1, M_{k}(f(t, .))& if k > 1.

Deduce that any solutionf ∈C([0,∞);L^{1}_{k+γ}) to the fragmentation equation asymptotically
satisfies

f(t, x)x * M_{1}(f(0, .))δ_{x=0} weakly in (C_{c}([0,∞))^{0} as t→ ∞.

2) For which values ofα, β ∈R, the function

f(t, x) =t^{α}G(t^{β}x)

is a (self-similar) solution of the fragmentation equation such that M1(f(t, .)) = cst. Prove then that the profile G satisfies the stationary equation

γFG=x ∂_{x}G+ 2G.

3) Prove that for any solutionf to the fragmentation equation, the rescalled density
g(t, x) :=e^{−2t}f e^{γ t}−1, x e^{−t}

solves the fragmentation equation in self-similar variable

∂_{t}g+x ∂_{x}g+ 2g =γFg.

Problem II - the discret Fokker-Planck equation

In all the problem, we consider the discrete Fokker-Planck equation

∂_{t}f =L_{ε}f := ∆_{ε}f+ div_{x}(xf) in (0,∞)×R, (0.1)
where

∆_{ε}f = 1

ε^{2}(k_{ε}∗f −f) =
Z

R

1

ε^{2}k_{ε}(x−y)(f(y)−f(x))dx,
and where k_{ε}(x) = 1/εk(x/ε), 0 ≤k ∈W^{1,1}(R)∩L^{1}_{3}(R),

Z

R

k(x)

1
x
x^{2}

dx =

1 0 1

,

that we complement with an initial condition

f(0, x) = ϕ(x) in R. (0.2)

Question 1

Establish the formula Z

(∆εf)β^{0}(f)m dx=
Z

β(f) ∆εm dx−

Z Z 1

ε^{2}kε(y−x)J(x, y)m(x)dydx
with

J(x, y) :=β(f(y))−β(f(x))−(f(y)−f(x))β^{0}(f(x)).

Formally prove that the discrete Fokker-Planck equation is mass conservative and satisfies
the (weak) maximum principle. Explain (quickly) why for ϕ∈L^{2}_{k}(R^{d}), k ≥0, the equation
(0.1)-(0.2) has a (unique) solution f(t) in some functional space to be specified. We recall
that

kgk_{L}^{p}

k :=kgh·i^{k}k_{L}^{p}, hxi:= (1 +|x|^{2})^{1/2}.
Establish that if L^{2}_{k}(R^{d})⊂L^{1}(R^{d}), then the solution satisfies

sup

t≥0

kf(t)k_{L}^{1} ≤ kϕk_{L}^{1}.

Question 2 We define

M_{k}(t) =M_{k}(f(t)), M_{k}(f) :=

Z

R

f(x)hxi^{k}dx.

Prove that if 0≤ϕ∈L^{2}_{k}(R^{d})⊂L^{1}(R^{d}), then the solution satisfies
M_{0}(t)≡M_{0}(ϕ) ∀t≥0.

Prove that whenL^{2}_{k}(R^{d})⊂L^{1}_{2}(R^{d}), then the solution satisfies
d

dtM_{2}(t) =M_{0}(0)−M_{2}(t).

Question 3 (discrete Nash inequality)

Prove that there exist θ, η ∈(0,1), ρ >0 such that

|k(r)|ˆ < θ ∀ |r|> ρ, 1−k(r)ˆ > ηr^{2} ∀ |r|< ρ.

Deduce that for any f ∈L^{1}∩L^{2} and any R >0, there holds
kk_{ε}∗fk^{2}_{L}2 ≤θkfk^{2}_{L}2+Rkfk^{2}_{L}1 + 1

ηR^{2} I_{ε}[f]
with

I_{ε}[f] :=

Z

R

1−ˆk(εξ)

ε^{2} |f|ˆ^{2}dξ.

Question 4

Prove that for any α >0, the solution satisfies d

dt Z

R

f^{2} = −(1
ε^{2} −α)

Z Z

k_{ε}(y−x) (f(y)−f(x))^{2}dxdy
+2α

Z

(k_{ε}∗f)f−2α
Z

f^{2}+
Z

f^{2},
and then

d dt

Z

R

f^{2} ≤ −(1−αε^{2})I_{ε}[f] +αkk_{ε}∗fk^{2}_{L}2 −(α−1)kfk^{2}_{L}2.
Deduce that for ε∈(0, ε_{0}), ε_{0} >0 small enough, the solution satisfies

d

dtkfk^{2}_{L}2 ≤ −1

2I_{ε}[f]−C_{1}kfk^{2}_{L}2 +C_{2}kfk^{2}_{L}1.
Also prove that

d dt

Z

R

f^{2}|x|^{2} ≤
Z

R

f^{2}−
Z

R

f^{2}|x|^{2}.

Question 5

Prove that there exists a constantC such that the set

Z :={f ∈L^{2}_{1}(R); f ≥0, kfk_{L}^{1} = 1, kfk_{L}^{2}

1 ≤C}

is invariant under the action of the semigroup associated to discrete Fokker-Planck equation.

Deduce that there exists a unique solution G_{ε} to

0< G_{ε} ∈L^{2}_{1}, ∆_{ε}G_{ε}+ div(xG_{ε}) = 0, M_{0}(G_{ε}) = 1.

Prove that for any ϕ∈L^{2}_{1}, M0(ϕ) = 1, the associated solution satisfies
f(t) * G as t→ ∞.

Problem III - the Fokker-Planck equation with weak confinement In all the problem, we consider the Fokker-Planck equation

∂_{t}f = Λf := ∆_{x}f + div_{x}(f∇a(x)) in (0,∞)×R^{d} (0.3)
for the confinement potential

a(x) = hxi^{γ}

γ , γ ∈(0,1), hxi^{2} := 1 +|x|^{2},
that we complement with an initial condition

f(0, x) =ϕ(x) in R^{d}. (0.4)

Question 1

Exhibit a stationary solution G∈P(R^{d}). Formally prove that this equation is mass conser-
vative and satisfies the (weak) maximum principle. Explain (quickly) why for ϕ∈ L^{p}_{k}(R^{d}),
p ∈ [1,∞], k ≥ 0, the equation (0.3)-(0.4) has a (unique) solution f(t) in some functional
space to be specified. Establish that if L^{p}_{k}(R^{d})⊂L^{1}(R^{d}) then the solution satisfies

sup

t≥0

kf(t)k_{L}^{1} ≤ kϕk_{L}^{1}.

Can we affirm that f(t)→G ast → ∞? and that convergence is exponentially fast?

Question 2 We define

Bf := Λf−M χ_{R}f

with χ_{R}(x) :=χ(x/R),χ∈ D(R^{d}), 0 ≤χ≤1, χ(x) = 1 for any |x| ≤1, and with M, R >0
to be fixed.

We denote by fB(t) = SB(t)ϕ the solution associated to the evolution PDE corresponding to the operatorB and the initial condition (0.4).

(1) Why such a solution is well defined (no more than one sentence of explanation)?

(2) Prove that there exists M, R >0 such that for any k ≥0 there holds d

dt Z

R^{d}

fB(t)hxi^{k}dx≤ −c_{k}
Z

R^{d}

fB(t)hxi^{k+γ−2} ≤0,
for some constant ck ≥0,ck >0 if k >0, and

kSB(t)k_{L}^{1}

k→L^{1}_{k} ≤1.

(3) Establish that if u∈C^{1}(R+) satisfies

u^{0} ≤ −c u^{1+1/α}, c, α >0,
there exists C =C(c, α, u(0)) such that

u(t)≤C/t^{α} ∀t >0.

(4) Prove that for any k_{1} < k < k_{2} there exists θ ∈(0,1) such that

∀f ≥0 M_{k} ≤M_{k}^{θ}_{1}M_{k}^{1−θ}

2 , M_{`} :=

Z

R^{d}

f(x)hxi^{`}dx
and write θ as a function of k_{1}, k and k_{2}.

(5) Prove that if ` > k >0 there exists α >0 such that
kSB(t)k_{L}^{1}

`→L^{1} ≤ kSB(t)k_{L}^{1}

`→L^{1}_{k} ≤C/hti^{α},
and that α >1 if ` is large enough (to be specified).

(6) Prove that

S_{L}=S_{B}+S_{B} ∗(AS_{L}),
and deduce that for k large enough (to be specified)

kSLk_{L}^{1}

k→L^{1}_{k} ≤C.

Remark. You have recovered (with a simpler and more general proof) a result established by Toscani and Villani in 2001.

Question 3 (difficult)

Establish that there exists κ >0 such that for any h∈ D(R^{d}) satisfying
hhi_{µ}:=

Z

R^{d}

h dµ= 0, µ(dx) := G(x)dx, there holds

Z

R^{d}

|∇h|^{2}dµ≥κ
Z

R^{d}

h^{2}hxi^{2(γ−1)}dµ.

Question 4

Establish that for anyϕ∈ D(R^{d}) and any convex function j ∈C^{2}(R) the associated solution
f(t) =S_{Λ}(t)ϕsatisfies

d dt

Z

R^{d}

j(f(t)/G)G dx=−Dj ≤0 and give the expression of the functional Dj. Deduce thgat

kf(t)/Gk|_{L}^{∞} ≤ kf_{0}/Gk|_{L}^{∞} ∀t≥0.

Question 5 (difficult)

Prove that for any ϕ∈ D(R^{d}) and for any α >0 there exists C such that
kf(t)− hϕiGk_{L}^{2} ≤C/t^{α}.

Exam 2016 - Two problems Problem I - Nash estimates

We consider the evolution PDE

∂_{t}f = div(A∇f), (0.1)

on the unknownf =f(t, x),t ≥0,x∈R^{d}, withA=A(x) a symmetric, uniformly bounded
and coercive matrix, in the sense that

ν|ξ|^{2} ≤ξ·A(x)ξ≤C|ξ|^{2}, ∀x, ξ∈R^{d}.

It is worth emphasizing that we do not make any regularity assumption on A. We comple- ment the equation with an initial condition

f(0, x) = f_{0}(x).

1) Existence. What strategy can be used in order to exhibit a semigroup S(t) in L^{p}(R^{d}),
p= 2, p = 1, which provides solutions to (0.1) for initial date in L^{p}(R^{d})? Is the semigroup
positive? mass conservative?

In the sequel we will not try to justify rigorously the a priori estimates we will establish, but we will carry on the proofs just as if there do exist nice (smooth and fast decaying) solutions.

We denote by C orC_{i} some constants which may differ from line to line.

2) Uniform estimate. a) Prove that any solutionf to (0.1) satisfies
kf(t)k_{L}^{2} ≤C t^{−d/4}kf_{0}k_{L}^{1}, ∀t >0.

b) We define the dual semigroup S^{∗}(t) by

hS^{∗}(t)g_{0}, f_{0}i=hg_{0}, S(t)f_{0}i, ∀t≥0, f_{0} ∈L^{p}, g_{0} ∈L^{p}^{0}.
IdentifyS^{∗}(t) and deduce that any solution f to (0.1) satisfies

kf(t)k_{L}^{∞} ≤C t^{−d/4}kf_{0}k_{L}^{2}, ∀t >0.

c) Conclude that any solution f to (0.1) satisfies

kf(t)k_{L}^{∞} ≤[C^{2}2^{d/2}]t^{−d/2}kf_{0}k_{L}^{1}, ∀t >0.

3) Entropy and first moment. For a given and (nice) probability measure f, we define the (mathematical) entropy and the first moment functional by

H :=

Z

R^{d}

flogf dx, M :=

Z

R^{d}

f|x|dx.

a) Prove that for any λ∈R, there holds

mins≥0{slogs+λ s}=−e^{−λ−1}.

Deduce that there exists a constant D=D(d) such that for any (nice) probability measure f and any a∈R+, b ∈R, there holds

H+aM +b ≥ −e^{−b−1}a^{−d}D.

b) Making the choice a:=d/M and e^{−b} := (e/D)a^{d}, deduce that

M ≥κ e^{−H/d}, (0.2)

for some κ=κ(d)>0.

From now on, we restrict ourselfto consider an initial datum which is a (nice) probability measure:

f_{0} ≥0,
Z

R^{d}

f_{0}dx= 1,

and we denote by f(t) the nonnegative and normalized solution to the evolution PDE (0.1)
corresponding to f_{0}. We also denote by H =H(t), M =M(t) the associated entropy and
first moment.

3) Dynamic estimate on the entropy. Deduce from 2) that f satisfies H(t)≤K− d

2logt, ∀t >0, (0.3)

for some constant K ∈R (independent of f_{0}).

4) Dynamic estimate on the entropy and the first moment. a) Prove that

d dtM(t)

≤C

Z

|∇f(t)|, for some positive constant C=C(A).

b) Deduce that there exists a constantθ =θ(C, ν)>0 such that

d dtM(t)

≤θ

−d

dtH(t)1/2

, ∀t >0. (0.4)

c) Prove that for the heat equation (whenA =I), we have d

dt Z

f|x|^{2}dx= 2,
and next

M(t)≤Chti^{1/2}, t ≥0. (0.5)

From now on, we will always restrict ourself to consider the Dirac mass initial datum
f_{0} =δ_{0}(dx),

and our goal is to establish a similar estimate as (0.5) (and in fact, a bit sharper estimate than (0.5)) for the corresponding solution.

5) Dynamic estimate on the first moment. a) Deduce from 1) that there exists (at least) one
functionf ∈C((0,∞);L^{1})∩L^{∞}_{loc}((0,∞);L^{∞}) which is a solution to the evolution PDE (0.1)
associated to the initial datum δ_{0}. Why does that solution satisfy the same above estimates
for positive times?

b) We define

R =R(t) :=K/d−H(t)/d− 1

2logt ≥0,

where K is defined in (0.3). Observing that M(0) = 0, deduce from the previous estimates that

C_{1}t^{1/2}e^{R} ≤M ≤C_{2}
Z t

0

1

2s +dR ds

1/2

ds, ∀t >0.

c) Observe that for a > 0 and a+b > 0, we have (a+b)^{1/2} ≤ a^{1/2}+b/(2a^{1/2}), and deduce
that

C_{1}e^{R}≤M t^{−1/2} ≤C_{2}(1 +R), ∀t >0.

d) Deduce from the above estimate thatR must be bounded above, and then
C_{1}t^{1/2} ≤M ≤C_{2}t^{1/2}, ∀t >0.

You have recovered one of the most crucial step of Nash’s article “Continuity of solutions of parabolic and elliptic equations”, Amer. J. Math. (1958).

Problem II - The fractional Fokker-Planck equation We consider the fractional Fokker-Planck equation

∂_{t}f =Lf :=I[f] + div_{x}(Ef) in (0,∞)×R, (0.1)
where

I[f](x) = Z

R

k(y−x) (f(y)−f(x))dy, k(z) = 1

|z|^{1+α}, α∈(0,1),
and where E is a smooth vectors field such that

∀ |x| ≥1, |E(x)| ≤Chxi, divE(x)≤C, x·E ≥ |x|^{2}.
We complement the equation with an initial condition

f(0, x) = ϕ(x) in R. (0.2)

We denote by F the Fourier transform operator, and next ˆf =Ff for a given functionf on the real line.

Question 1. Preliminary issues (if not proved, theses identities can be accepted). Here all the functions (f, ϕ,β) are assumed to be suitably nice so that all the calculations are licit.

a) - Establish the formula Z

(I[f])β^{0}(f)ϕ dx=
Z

β(f)I[ϕ]dx− Z Z

k(y−x)J(x, y)ϕ(x)dydx with

J(x, y) :=β(f(y))−β(f(x))−(f(y)−f(x))β^{0}(f(x)).

b) - Prove that there exists a positive constantC_{1} such that
F(I[f])(ξ) =

Z

R

f(x)e^{−ixξ}
nZ

R

cos(zξ)−1

|z|^{1+α} dz
o

dx=C1|ξ|^{α}fˆ(ξ), ∀ξ ∈R.
c) - Denotes :=α/2. Prove that

|||f|||^{2}_{H}_{˙}s :=

Z

R^{2}

|f(x)−f(y)|^{2}

|x−y|^{1+2s} dxdy =
Z

R

f(z+·)−f(·)

|z|^{s+1/2}

2
L^{2}(R)

dz,

and then that there exists a positive constant C_{2} such that

|||f|||^{2}_{H}_{˙}s =C_{2}
Z

R

|ξ|^{2s}|fˆ(ξ)|^{2}dξ =:C_{2}kfk^{2}_{H}_{˙}s.
d) - Prove that

Z

R

I[f]f dx=−1

2|||f|||^{2}_{H}_{˙}s.

From questions 2 to 4, we consider f (and g) a solution to the fractional Fokker-Planck equation (0.1) and we establish formal a priori estimates.

Question 2. Moment estimates. For anyk ≥0, we define Mk=Mk(t) =Mk(f(t)), with Mk =Mk(f) :=

Z

R

f(x)hxi^{k}dx.

Prove that the solution f satisfies

M0(t)≡M0(ϕ), ∀t≥0.

Prove that for any k∈(0, α), there exists C >0 such that
C^{−1}|hyi^{k}− hxi^{k}| ≤

|y|^{2}− |x|^{2}

k/2 ≤C |y−x|^{k/2}|x|^{k/2}+|y−x|^{k}
,
and deduce that there exist C_{1}, C_{2} >0 such that the solution f satisfies

d

dtM_{k} ≤C_{1}M_{k/2} −C_{2}M_{k}.

Conclude that there existsA_{k}=A_{k}(M_{0}(ϕ)) such that the solution f satisfies
sup

t≥0

M_{k}(t)≤max(M_{k}(0), A_{k}). (0.3)

Question 3. Fractional Nash inequality andL^{2} estimate. Prove that there exists a constant
C >0 such that

∀h ∈ D(R), khk_{L}^{2} ≤Ckhk

α 1+α

L^{1} khk

1 1+α

H˙^{s} .

Deduce that the square of the L^{2}-norm u:=kf(t)k^{2}_{L}2 of the solution f satisfies
d

dtu≤ −C_{1}u^{1+α}+C_{2},

for some constants C_{i} =C_{i}(M_{0}(ϕ)) >0. Conclude that there exists A_{2} =A_{2}(M_{0}(ϕ)) such
that

sup

t≥0

u(t)≤max(u(0), A_{2}). (0.4)

Question 4. Around generalized entropies and the L^{1}-norm. Consider a convex function β
and define the entropy H and the associated dissipation of entropy D by

H(f|g) :=

Z

R

β(X)g dx,

D(f|g) :=

Z

R

Z

R

k g∗{β(X∗)−β(X)−β^{0}(X)(X∗−X)}dxdx∗,

where k = k(x−x∗), g∗ = g(x∗), X = f(x)/g(x) and X∗ = f(x∗)/g(x∗). Why do two solutionsf and g satisfy

d

dtH(f(t)|g(t))≤ −D(f(t)|g(t))? (0.5)
Deduce that for β(s) = s_{+} and β(s) = |s|, any solution f satisfies

Z

β(f(t,·))dx≤ Z

β(ϕ)dx, ∀t≥0. (0.6)

Question 5. Well-posedness. Explain briefly how one can establish the existence of a weakly
continuous semigroup SL defined in the space X =L^{1}_{s} ∩L^{2} such that it is a contraction for
the L^{1} norm and such that for any ϕ ∈X the function f(t) :=SL(t)ϕ is a (weak) solution
to the fractional Fokker-Planck equation (0.1). Why is SL mass and positivity preserving
and why does any associated trajectory satisfy (0.3), (0.4) and (0.6)?

(Ind. One may consider the sequence of kernels k_{n}(z) :=k(z)_{n}^{−1}<|z|<n).

Question 6. Prove that there exists a constantC such that the set
Z :={f ∈L^{1}(R); f ≥0, kfk_{L}^{1} = 1, kfk_{X} ≤C}

is invariant under the action of SL. Deduce that there exists at least one function G ∈ X such that

IG+ divx(E G) = 0, G≥0, M0(G) = 1. (0.7) Question 7. We accept that for any convex function β, any nonnegative solution f(t) and any nonnegative stationary solution Gthe following inequality holds

H(f(t)|G) + Z t

0

D(f(s)|G)ds ≤ H(ϕ|G). (0.8)

Deduce that D(g|G) = 0 for any (other) stationary solutiong. (Ind. First consider the case
when β ∈W^{1,∞}(R)and next use an approximation argument). Deduce that

G(y)g(y)

G(y) − g(x) G(x)

2

= 0 for a.e. x, y ∈R,

and then that the solution to (0.7) is unique. Prove that for anyϕ∈X, there holds
SL(t)ϕ * M_{0}(ϕ)G weakly in X, as t → ∞.

Question 8. (a) Introducing the splitting

A:=λ I, λ∈R, B :=L − A, explain why

SL =SB +SBA ∗SL,

and next for any n≥1

S_{L} =S_{B}+...+ (S_{B}A)^{∗(n−1)}∗S_{L}+ (S_{B}A)^{∗n}∗S_{L},
where the convolution onR+ is defined by

(u∗v)(t) :=

Z t 0

u(t−s)v(s)ds,
and the itareted convolution by u^{∗1} =u, u^{∗k} =u^{∗(k−1)}∗u if k≥2.

(b) Prove that for λ >0, large enough, there holds

kSBkY→Y ≤e^{−t}, Y =L^{1}_{k}, k∈(0, α), Y =L^{2},
as well as

kSB(t)k_{L}^{1}→L^{2} ≤ C

t^{1/(2α)} e^{−t}, ∀t ≥0,

Z ∞ 0

kSB(s)k^{2}_{L}2→H˙^{s}ds≤C,
and deduce that for n large enough

Z ∞ 0

k(ASB)^{(∗n)}(s)k_{L}1→H˙^{s}ds≤C.

(c) Establish that for any ϕ∈L^{1}_{s}, k >0, the associated solution f(t) =S_{L}(t)ϕsplits as
f(t) =g(t) +h(t), kg(t)k_{L}^{1} ≤e^{−t}, kh(t)k_{L}1

k∩H˙^{s} ≤C(M_{0}(ϕ)).

(d) Conclude that

∀ϕ∈L^{1}(R), kSL(t)ϕ−M_{0}(ϕ)Gk_{L}^{1} →0 as t→ ∞.

Question 9. Justification of (0.8). We define the regularized operator
L_{ε,n}f :=ε∆_{x}f+I_{n}[f] + div_{x}(Ef)

with ε > 0, n ∈ N^{∗} and In associated to the kernel kn (introduced in question 5). Why is
there a (unique) solution Gε,n to the stationary problem

G_{ε,n} ∈X, L_{ε,n}G_{ε,n} = 0, G_{ε,n} ≥0, M_{0}(G_{ε,n}) = 1,

and why does a similar inequality as (0.8) hold? Prove that there exist G_{ε}, G ∈ X, ε > 0,
such that (up to the extraction of a subsequence) G_{n,ε} → G_{ε} and G_{ε} → G strongly in
L^{1}. Prove that a similar strong convergence result holds for the familly SL_{ε,n}(t)ϕ, ϕ ∈ X.

Conclude that (0.8) holds.

Exam 2017 - Two problems

Problem I - A general Fokker-Planck equation with strong confinement We consider the evolution PDE

∂_{t}f = ∆f+ div(Ef), (0.1)

on the unknown f =f(t, x),t ≥0, x∈R^{d}, withE =E(x) a given smooth force field which
satisfies for someγ ≥1

∀ |x| ≥1, |E(x)| ≤C|x|^{γ−1}, divE(x)≤C|x|^{γ−2}, x·E ≥ |x|^{γ}.
We complement the equation with an initial condition

f(0, x) = f_{0}(x).

Question 1. Which strategy can be used in order to exhibit a semigroup S(t) in L^{p}(R^{d}),
which provides solutions to (0.1) for initial data inL^{p}(R^{d})? Is the semigroup positive? mass
conservative? Explain briefly why there exists a functionG=G(x) such that

0≤G∈L^{2}(m), hGi:=

Z

G= 1, LG= 0.

We accept that G > 0. For any nice function f : R^{d} → R we denote h := f /G and,
reciprocally, for any nice function h:R^{d}→R we denotef :=Gh.

In the sequel we will not try to justify rigorously the a priori estimates we will establish, but we will carry on the proofs just as if there do exist nice (smooth and fast decaying) solutions.

We denote by C orC_{i} some constants which may differ from line to line.

Question 2. Prove that for any weight function m : R^{d} → [1,∞) and any nice function
f :R^{d} →R, there holds

Z

(Lf)f m=− Z

|∇f|^{2}m+ 1
2

Z

f^{2}L^{∗}m,
where we will make explicit the expression ofL^{∗}.

Question 3. Prove that there existw:R^{d} →[1,∞),α >0 and b, R0 ≥0 such that
L^{∗}w≤ −α w+b1_{B}_{R}

0.

Question 4. For some constant λ≥0 to be specified later, we define W :=w+λ. Deduce from the previous question that

Z

h^{2}w G≤ 1
α

Z

h^{2}(b1_{B}_{R}

0 − L^{∗}W)G,
for any nice function h:R^{d}→R.

Question 5. Take a nice function h : R^{d} → R such that hhGi = 0 and denote G(Ω) :=

hG1_{Ω}i. Prove that for any R ≥R_{0} there exists κ_{R} ∈(0,∞) such that
Z

h^{2}1_{B}_{R}G≤κ_{R}
Z

BR

|∇h|^{2}G+ 1
G(B_{R})

Z

B_{R}^{c}

h G2

and deduce that Z

h^{2}1_{B}_{R}G≤ κ_{R}
1 +λ

Z

|∇h|^{2}W G+ G(B_{R}^{c})
G(B_{R})

Z

h^{2}w G.

Question 6. Establish finally that there exist some constants λ, K_{1} ∈(0,∞) such that
2

K_{1}
Z

h^{2}wG≤
Z

W|∇h|^{2}− 1

2h^{2}L^{∗}W

G= Z

(−Lf)f G^{−1}W,
for all nice function h such that hhGi= 0.

Question 7. Consider a nice solution f to (0.1) associated to an initial datumf_{0} such that
hf_{0}i= 0. Establish thatf satisfies

1 2

d dt

Z

h^{2}_{t}W G=−
Z

|∇h|^{2}W G+1
2

Z

h^{2}GL^{∗}W.

Deduce that there exists K_{2} ∈(0,∞) such that f satisfies the decay estimate
Z

f_{t}^{2}W G^{−1}dx ≤e^{−K}^{2}^{t}
Z

f_{0}^{2}W G^{−1}dx, ∀t≥0.

Problem II - Estimates for the relaxation equation We consider the relaxation equation

∂_{t}f =Lf :=−v · ∇f +ρ_{f}M −f in (0,∞)×R^{2d}, (0.1)
on the unknown f =f(t, x, v),t≥0, x, v ∈R^{d}, with

ρ_{f}(t, x) =
Z

R^{d}

f(t, x, v)dv, M(v) := 1

(2π)^{d/2} exp(−|v|^{2}/2).

We complement the equation with an initial condition

f(0, x, v) = f_{0}(x, v) in R^{2d}. (0.2)
Question 1. A priori estimates and associated semigroup. We denote by f a nice solution
to the relaxation equation (0.1)–(0.2).

(a) Prove that f is mass conserving.

(b) Prove that

|ρ_{g}| ≤ kgk_{L}2

v(M^{−1/2}), ∀g =g(v)∈L^{2}_{v}(M^{−1/2})
and deduce that

kf(t,·)k_{L}2

xv(M^{−1/2})≤ kf_{0}k_{L}2

xv(M^{−1/2}).

(c) Consider m=hvi^{k}, k > d/2. Prove that there exists a constant C ∈(0,∞) such that

|ρ_{g}| ≤Ckgk_{L}^{p}_{v}_{(m)}, ∀g =g(v)∈L^{p}_{v}(m), p= 1,2,
and deduce that

kf(t,·)k_{L}^{p}_{xv}_{(m)} ≤e^{λt}kf_{0}k_{L}^{p}_{xv}_{(m)},
for a constant λ ∈[0,∞) that we will express in function of C.

(d) What strategy can be used in order to exhibit a semigroupS(t) inL^{p}_{xv}(m),p= 2,p= 1,
which provides solutions to (0.1) for initial date inL^{p}_{xv}(m)? Is the semigroup positive? mass
conservative? a contraction in some spaces?

The aim of the problem is to prove that the associated semigroup SL to (0.1) is
bounded in L^{p}(m), p= 1,2, without using the estimate proved in question (1b).

In the sequel we will not try to justify rigorously the a priori estimates we will establish, but we will carry on the proofs just as if there do exist nice (smooth and fast decaying) solutions.

We define

Af :=ρ_{f}M, Bf =Lf− Af.

Question 2. Prove that SB satisfies a growth estimate O(e^{−t}) in any L^{p}_{xv}(m) space. Using
the Duhamel formula

SL=SB+SBA ∗SL

prove thatSL is bounded in L^{1}_{xv}(m).

Question 3. Establish thatA:L^{1}_{xv}(m)→L^{1}_{x}L^{∞}_{v} (m) where
kgk_{L}^{1}

xL^{p}v(m) :=

Z

R^{d}

kg(x,)k_{L}^{p}_{(m)}dx.

Prove that d dt

Z Z
f^{p}dx

1/p

dv= Z Z

(∂tf)f^{p−1}dx
Z

f^{p}dx
1/p−1

dv.

Deduce thatSB satisfies a growth estimateO(e^{−t}) in anyL^{1}_{x}L^{p}_{v}(m) space forp∈(1,∞), and
then in L^{1}_{x}L^{∞}_{v} (m). Finally prove that SB(t)A is appropriately bounded in B(L^{1}, L^{1}_{x}L^{∞}_{v} (m))
and that S_{L} is bounded in L^{1}_{x}L^{∞}_{v} (m).

Question 4. We defineu(t) :=AS_{B}(t). Establish that
(u(t)f_{0})(x, v) = M(v)e^{−t}

Z

R^{d}

f_{0}(x−v∗t, v∗)dv∗.

Deduce that

ku(t)f_{0}k_{L}^{∞}_{xv}_{(m)} ≤Ce^{−t}

t^{d} kf_{0}k_{L}^{1}_{x}_{L}^{∞}_{v} _{(m)}.

Question 5. Establish that there exists some constantsn ≥1 andC ∈[1,∞) such that
ku^{(∗n)}(t)k_{L}^{1}_{xv}(m)→L^{∞}_{xv}(m) ≤C e^{−t/2}.

Deduce that SL is bounded in L^{∞}_{xv}(m).

Question 6. How to prove that S_{L} is bounded in L^{2}_{xv}(m) in a similar way? How to shorten
the proof of that last result by using question (1b)? Same question for the spaceL^{∞}_{xv}(m).

Exam 2018 - Local in time estimate (from Nash)

Consider a smooth and fast decaying initial datum f_{0}, the associated solution f = f(t, x),
t≥0, x∈R^{d}, to heat equation

∂_{t}f = 1

2 ∆f, f(0, .) =f_{0},
and for a givenα ∈R^{d}, define

g :=f e^{ψ}, ψ(x) :=α·x.

(1) Establish that

∂_{t}g = 1

2∆g−α· ∇g+ 1
2|α|^{2}g.

(2) Establish that k g(t, .)k_{L}^{1} ≤e^{α}^{2}^{t/2}kg_{0}k_{L}^{1} for any t ≥0.

(3) Establish that

k g(t)k^{2}_{L}2e^{−α}^{2}^{t}≤ k g0k^{2}_{L}1

(2/d C_{N}t)^{d/2}, ∀t >0.

(4) Denoting byT(t) the semigroup associated to the parabolic equation satisfies byg, prove successively that

T(t) :L^{1} →L^{2}, L^{2} →L^{∞}, L^{1} →L^{∞},
for some constants Ct^{−d/4}e^{α}^{2}^{t/2}, Ct^{−d/4}e^{α}^{2}^{t/2} and Ct^{−d/2}e^{α}^{2}^{t/2}.

(5) Denoting by S the heat semigroup and by F(t, x, y) := (S(t)δ_{x})(y) the fundamental
solution associated to the heat equation when starting from the Dirac function in x ∈ R^{d},
deduce

F(t, x, y)≤ C

t^{d/2} e^{α·(x−y)+α}^{2}^{t/2}, ∀t >0,∀x, y, α∈R^{d},
and then

F(t, x, y)≤ C

t^{d/2}e^{−}^{|x−y|}

2

2t , ∀t >0,∀x, y ∈R^{d}.

(6) May we prove a similar result for the parabolic equation

∂_{t}f = div_{x}(A(x)∇_{x}f), 0< ν ≤A∈L^{∞}?

Exam 2019 - two problems about subgeometric convergence Problem I - Subgeometric Harris estimate

In this part, we consider a Markov semigroup S =SL on L^{1}(R^{d}) which fulfills

(H1) there exist some weight functions m_{i} : R^{d} → [1,∞) satisfying m_{1} ≥ m_{0}, m_{0}(x) → ∞
as x→ ∞and there exists constant b >0 such that

L^{∗}m_{1} ≤ −m_{0}+b;

(H2) there exists a constant T >0 and for any R ≥R_{0} ≥ 0 there exists a positive and not
zero measureν such that

S_{T}f ≥ν
Z

BR

f, ∀f ∈L^{1}, f ≥0;

(H3) there exists m2 ≥m1 such that and for any λ >0 there exists ξλ such that
m_{1} ≤λm_{0}+ξ_{λ}m_{2}, ξ_{λ} →0 as λ → ∞.

(H4) We also assume that sup

t≥0

kS_{t}fk_{L}^{1}_{(m}_{i}_{)} ≤M_{i}kfk_{L}^{1}_{(m}_{i}_{)}, M_{i} ≥1, i= 1,2.

(1) Prove

kS_{T}f_{0}k_{L}^{1} ≤ kf_{0}k_{L}^{1}, ∀T >0, ∀f_{0} ∈L^{1}.

In the sequel, we fix f_{0} ∈L^{1}(m_{2}) such that hf_{0}i= 0 and we denote f_{t}:=S_{t}f_{0}.
(2) Prove that

d

dtkf_{t}k_{L}^{1}_{(m}_{1}_{)} ≤ −kf_{t}k_{L}^{1}_{(m}_{0}_{)}+bkf_{t}k_{L}^{1},
and deduce that

kS_{T}f_{0}k_{L}^{1}_{(m}_{1}_{)}+ T

M_{0}kS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)}≤ kf_{0}k_{L}^{1}_{(m}_{1}_{)}+Kkf_{0}k_{L}^{1}.
We define

kfk_{β} :=kfk_{L}^{1} +βkfk_{L}^{1}_{(m}_{1}_{)}, β >0.

We fix R ≥ R_{0} large enough such that A := m(R)/4 ≥ 3M_{0}/T, and we observe that the
following alternative holds

kf0k_{L}^{1}_{(m}_{0}_{)} ≤Akf0k_{L}^{1} (0.1)

or

kf_{0}k_{L}^{1}_{(m}_{0}_{)} > Akf_{0}k_{L}^{1}. (0.2)
(3) We assume that condition (0.1) holds. Prove that

kS_{T}f_{0}k_{L}^{1} ≤γ_{1}kf_{0}k_{L}^{1},
with γ_{1} ∈(0,1). Deduce that

kS_{T}f_{0}k_{β} ≤γ_{1}kf_{0}k_{L}^{1} − βT

M_{0}kS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)}+βkf_{0}k_{L}^{1}_{(m}_{1}_{)}+βKkf_{0}k_{L}^{1}
and next

kS_{T}f_{0}k_{β}+ βT
M0

kS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)} ≤ kf_{0}k_{β},
for β >0 small enough.

(4) We assume that condition (0.2) holds. Prove that
kS_{T}f_{0}k_{L}^{1}_{(m}_{1}_{)}+ T

M_{0}kS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)} ≤ kf_{0}k_{L}^{1}_{(m}_{1}_{)}+ T

3M_{0}kf_{0}k_{L}^{1}_{(m}_{0}_{)},
and deduce

kS_{T}f_{0}k_{β}+ βT

M_{0}kS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)} ≤ kf_{0}k_{β} + βT

3M_{0}kf_{0}k_{L}^{1}_{(m}_{0}_{)}.
(5) Observe that in both cases (0.1) and (0.2), there holds

kS_{T}f_{0}k_{β}+ 3αkS_{T}f_{0}k_{L}^{1}_{(m}_{0}_{)}≤ kf_{0}k_{β} +αkf_{0}k_{L}^{1}_{(m}_{0}_{)},
where from now β and α are fixed constants. Deduce that

Z(u_{1}+αv_{1})≤u_{0}+αv_{0}+ ξ_{λ}
λαw_{1},
with

u_{n}:=kS_{nT}f_{0}k_{β}, v_{n}:=kS_{nT}f_{0}k_{L}^{1}_{(m}_{0}_{)}, w_{n}:=kS_{nT}f_{0}k_{L}^{1}_{(m}_{2}_{)}
and for λ≥λ_{0} ≥1 large enough

Z := 1 + δ

λ ≤2, δ:= α 1 +β. Deduce that for any n ≥1, there holds

u_{n}≤Z^{−n}(u_{0}+αv_{0}) + Z
Z −1

ξ_{λ}α
λ sup

i≥1

w_{i},

and next

kSnTf0kβ ≤

e^{−}^{nT}^{λ} ^{2T}^{δ} +ξλ

Ckf0k_{L}^{1}_{(m}_{2}_{)}, ∀λ≥λ0.

(6) Prove that

kS_{t}f_{0}k_{L}^{1} ≤Θ(t)kf_{0}k_{L}^{1}_{(m}_{2}_{)}, ∀t≥0, ∀f_{0} ∈L^{1}(m_{2}), hfi= 0,
for the function Θ given by

Θ(t) :=Cinf

λ>0{e^{−κt/λ}+ξ_{λ}}.

What is the value of Θ when m_{0} = 1, m_{1} =hxi, m_{2} =hxi^{2}?

Problem II - An application to the Fokker-Planck equation with weak confine- ment

In all the problem, we consider the Fokker-Planck equation

∂_{t}f =Lf := ∆_{x}f + div_{x}(f E) in (0,∞)×R^{d} (0.3)
for the confinement potential E :=∇φ, φ :=hxi^{γ}/γ, hxi^{2} := 1 +|x|^{2}, that we complement
with an initial condition

f(0, x) = f0(x) inR^{d}. (0.4)

Question 1

Give a strategy in order to build solutions to (0.3) when f_{0} ∈L^{p}_{k}(R^{d}), p∈[1,∞], k≥0.

We assume from now on that f_{0} ∈ L^{1}_{k}(R^{d}), k > 0, and that we are able to build a unique
weak (and renormalized) solution f ∈C([0,∞);L^{1}_{k}) to equation (0.3)-(0.4).

We also assume that γ ≥2.

Question 2 Prove

hf(t)i=hf_{0}i and f(t, .)≥0 if f_{0} ≥0.

Question 3

Prove that there exist α >0 and K ≥0 such that

L^{∗}hxi^{k} ≤ −αhxi^{k}+K,
and deduce that

sup

t≥0

kf(t,·)k_{L}^{1}

k ≤C_{1}kf_{0}k_{L}^{1}

k.
(Hint. A possible constant is C_{1} := max(1, K/α)).

Question 4 Prove that

sup

t≥0

kf(t,·)k_{L}^{2}

k ≤C_{2}kf_{0}k_{L}^{2}

k, at least fork > 0 large enough.

Question 5 Prove that

sup

t≥0

kf(t,·)k_{H}^{1}

k ≤C3kf0k_{H}^{1}

k, at least fork > 0 large enough.

Question 6 Prove that

kf(t,·)k_{H}^{1} ≤ C_{4}
t^{α}kf_{0}k_{L}^{1}

k,

at least fork > 0 large enough and for some constant α >0 to be specified.

(Hint. Consider the functional F(t) := kf(t)k_{L}^{1}

k+t^{α}k∇_{x}f(t)k^{2}_{L}2).

We assume from now on thatd= 1, so that C^{0,1/2} ⊂H^{1}.
Question 7

We fix f_{0} ∈L^{1} such thatf_{0} ≥0 and suppf_{0} ⊂B_{R}, R >0. Using question 3, prove that
Z

Bρ

f(t)≥ 1 2

Z

B_{R}

f_{0},

for any t ≥ 0 by choosing ρ > 0 large enough. Using question 6, prove that there exist
r, κ >0 and for any t >0 there exists x_{0} ∈B_{R} such that

f(t)≥κ, ∀x∈B(x_{0}, r).

We accept the spreading of the positivity property, namely that for ant r_{0}, r_{1} >0, x_{0} ∈R^{d},
there existt_{1}, κ_{1} >0 such that

f_{0} ≥1_{B(x}_{0}_{,r}_{0}_{)} ⇒ f(t_{1},·)≥κ_{1}1_{B(x}_{0}_{,r}_{1}_{)}.
Deduce that there exist θ >0 and T >0 such that

f(T,·)≥θ1_{B(0,R)}
Z

BR

f_{0}dx.

Question 8

Prove that for any k >0, there exists C, λ >0 such that f_{0} ∈L^{1}_{k} satisfying hf_{0}i = 0, there
holds

∀t >0, kf(t, .)k_{L}^{1}

k ≤Ce^{−λt}kf_{0}k_{L}^{1}

k. Question 9

We assume nowγ ∈(0,2). We define

Bf :=Lf −M χ_{R}f

with χ_{R}(x) :=χ(x/R),χ∈ D(R^{d}), 0 ≤χ≤1, χ(x) = 1 for any |x| ≤1, and with M, R >0
to be fixed.

We denote by fB(t) = SB(t)f_{0} the solution associated to the evolution PDE corresponding
to the operatorB and the initial condition f_{0}.

(1) Why such a solution is well defined (no more than one sentence of explanation)?

(2) Prove that there exists M, R >0 such that for any k ≥0 there holds d

dt Z

R^{d}

fB(t)hxi^{k}dx≤ −c_{k}
Z

R^{d}

fB(t)hxi^{k+γ−2} ≤0,
for some constant c_{k} ≥0,c_{k} >0 if k >0, and

kSB(t)k_{L}^{1}

k→L^{1}_{k} ≤1.

(3) Prove that for any k_{1} < k < k_{2} there exists θ ∈(0,1) such that

∀f ≥0 Mk ≤M_{k}^{θ}_{1}M_{k}^{1−θ}

2 , M` :=

Z

R^{d}

f(x)hxi^{`}dx
and write θ as a function of k1, k and k2.

(4) Prove that if ` > k >0 there exists α >0 such that
kSB(t)k_{L}^{1}

`→L^{1} ≤ kSB(t)k_{L}^{1}

`→L^{1}_{k} ≤C/hti^{α},
and that α >1 if ` is large enough (to be specified).

(6) Prove that

SL=SB+SB ∗(ASL), and deduce that for k large enough (to be specified)

kS_{L}k_{L}^{1}

k→L^{1}_{k} ≤C.

Question 10

Still in the case γ ∈(0,2), what can we say about the decay of
kf(t, .)k_{L}^{1}