4 Problem IV - An application to the Fokker-Planck equation

Universit´e Paris-Dauphine 2018-2019

An introduction to evolution PDEs November 2018

1 Problem I - Local in time estimates

Consider a weak solution 0≤f ∈C([0, T];L¹(R²)) to the Keller-Segel equation, that is

∂_tf := ∆f + div( ¯Kf), with

K¯ :=K ∗f, K:=∇κ, κ:= 1

2πlog|z|, such that

A_T := sup

[0,T]

Z

R²

f(1 +|x|²+ (logf)₊)dx+ Z T

0

I(f)dt <∞.

(1) Establish thatf²,|∇_xK|¯ f ∈L¹((0, T)×R²) and that for any mollifier sequence (ρ_n), the function fⁿ :=f ∗_xρ_n satisfies

∂_tfⁿ−K · ∇¯ _xfⁿ−∆_xfⁿ=rⁿ →f² in L¹(0, T;L¹_loc(R²)).

(2) Deduce that Z

R²

β(f_tⁿ

1)χ dx+ Z t1

t0

Z

R²

β⁰⁰(f_sⁿ)|∇_xf_sⁿ|²χ dxds= Z

R²

β(f_tⁿ

0)χ dx +

Z t1

t0

Z

R²

n

β⁰(f_sⁿ)rⁿχ+β(f_sⁿ) ∆χ−β(f_sⁿ) div_x( ¯Kχ)o dxds,

for all 0 ≤ t₀ < t₁ ≤ T, 0 ≤ χ ∈ C_c²(R²) and β ∈ C¹(R)∩ W_loc^2,∞(R) such that β⁰⁰ is non-negative and vanishes outside of a compact set, and next that

Z

R²

β(f_t1)dx+ Z t1

t0

Z

R²

β⁰⁰(f_s)|∇f_s|²dxds (1.1)

≤ Z

R²

β(f_t0)dx+ Z t1

t0

Z

R²

(β⁰(f_s)f_s²−β(f_s)f_s)₊dxds.

(3) Deduce that the same inequality holds for any renormalizing function β :R→R which is convex, piecewise of classC¹ and such that

|β(u)| ≤C(1 +u(logu)₊), (β⁰(u)u²−β(u)u)₊ ≤C(1 +u²) ∀u∈R.

(4) We define the renormalizing functionβK :R⁺→R⁺,K ≥e², by

βK(u) :=u(logfu)² if u≤K, βK(u) := (2 + logK)ulogu−2KlogK if u≥K, and the function logf_K by

logf_Ku:=1u≤e+ (logu)1e<u≤K+ (logK)1u>K. Deduce from (1.1) that

Z

R²

β_K(f_t1)dx+ Z t1

t0

Z

R²

|∇f|²

f (logf_Kf)1_f≥edxds (1.2)

≤ Z

R²

β_K(f_t0)dx+ 4 Z t1

t0

Z

R²

f²logf_Kf dxds.

Establish that for any A∈(e, K) Z

R²

f²logf_Kf dx ≤ (AlogA)M +H⁺(f) logA

1/2Z

R²

(flogf_Kf)³dx1/2

,

as well as Z

R²

(flogf_Kf)³dx 1/2

≤ C

M+H⁺(f)

1/2Z

R²

|∇f|²

f (logf_Kf)1f≥edx+I(f)

.

By choosing A >0 large enough, conclude that there existsC :=C(A_T) such that

H2(f(t1))≤ H2(f(t0)) +C, (1.3) with

H₂(g) :=

Z

R²

f(logg)f ²dx, logfu:=1_u≤e+ (logu)1_u>e. (5) We define the new renormalizing function β_K :R+→R+,K ≥2, by

β_K(u) := u^p

p if u≤K, β_K(u) := K^p−1

logK(u logu−u)− 1

p⁰K^p+ K^p

logK if u≥K.

Prove that Z

R²

β_K(f_t1)dx+ 4 p⁰p

Z t1

t0

Z

R²

|∇(f^p/2)|²1_f≤Kdxds+ K^p−1 logK

Z t1

t0

Z

R²

|∇f|²

f 1_f≥Kdxds

≤ Z

R²

β_K(f_t0)dx+ 1 p⁰

Z t1

t0

T₁ds+ 2K^p−1 Z t1

t0

T₂ds, with

T₁ :=

Z

R²

f^p+11f≤Kdx and T₂ :=

Z

R²

f²1f≥Kdx.

On the one hand, prove that 1

p⁰ Z t1

t0

T₁ds ≤ 2^p

p⁰ A^pM T + 1 p⁰p

Z t1

t0

Z

R²

|∇_x(f^p/2)|²1_f≤Kdxds, for A=A(p,AT)>1 large enough.

On the other hand, prove that

T₂ ≤8K^p−1Z

R²

|∇f|1f≥K/2dx2

and next that

T₂ ≤8K^p−1n4 p²

Z

R²

|∇(f^p/2)|²(2

K)^p−11f≤K + Z

R²

|∇f|² f 1f≥K

o H₂(f) (log(K/2))². Finally deduce from (1.3) that

Z t1

t0

T₂ds ≤ 1 p⁰p

Z t1

t0

Z

R²

|∇(f^p/2)|²1_f≤Kdxds+ K^p−1 logK

Z t1

t0

Z

R²

|∇f|²

f 1f≥Kdxds, for any K ≥K^∗ =K^∗(p,A_T)>max(A,4) large enough.

(6) Deduce that any exponent p∈(1,∞) and for any time t₀ ∈(0, T),f satisfies f ∈L^∞(t₀, T;L^p(R²)) and ∇_xf ∈L²((t₀, T)×R²).

2 Exercise II - Decay estimates

(1) Prove that a function u∈C¹(R) which satisfies the differential inequality u⁰ ≤ −λ u, λ >0,

also satisfies the integral inequality

∀t ≥0, λ Z ∞

t

u(s)ds ≤u(t). (2.1)

Prove that if u:R+ →R+ is decreasing and satisfies (2.1), there exists C =C(λ, u(0))>0 such that

∀t≥0, u(t)≤C e^−λt. (Hint. Introduce the function v(t) := R∞

t u(s)ds).

(2) Prove that a function u∈C¹(R) which satisfies the differential inequality u⁰ ≤ −K u^1+α, α >0, K >0,

also satisfies the integral inequality

∀t≥0, K Z ∞

t

u^1+α(s)ds≤u(t). (2.2)

Prove that if u:R+ →R+ is decreasing and satisfies (2.2), there exists C =C(α, u(0))>0 such that

∀t ≥0, u(t)≤C 1 t^1/α.

3 Problem III - Subgeometric Harris estimate

In this part, we consider a Markov semigroup S =SL on L¹(R^d) which fulfills

(H1) there exist some weight functions mi : R^d → [1,∞) satisfying m1 ≥ m0, m0(x) → ∞ as x→ ∞and there exists constant b >0 such that

L^∗m₁ ≤ −m₀+b;

(H2) there exists a constant T >0 and for any R ≥R₀ ≥ 0 there exists a positive and not zero measureν such that

S_Tf ≥ν Z

BR

f, ∀f ∈L¹, f ≥0;

(H3) there exists m₂ ≥m₁ such that and for any λ >0 there exists ξ_λ such that m₁ ≤λm₀+ξ_λm₂, ξ_λ →0 as λ → ∞.

(H4) We also assume that sup

t≥0

kS_tfk_L¹_(mi) ≤M_ikfk_L¹_(mi), M_i ≥1, i= 1,2.

(1) Prove

kS_Tf₀k_L¹ ≤ kf₀k_L¹, ∀T >0, ∀f₀ ∈L¹.

In the sequel, we fix f₀ ∈L¹(m₂) such that hf₀i= 0 and we denote f_t:=S_tf₀. (2) Prove that

d

dtkftk_L¹_(m1) ≤ −kftk_L¹_(m0)+bkftk_L¹, and deduce that

kS_Tf₀k_L¹_(m1)+ T

M₀kS_Tf₀k_L¹_(m0)≤ kf₀k_L¹_(m1)+Kkf₀k_L¹. We define

kfk_β :=kfk_L¹ +βkfk_L¹_(m1), β >0.

We fix R ≥ R₀ large enough such that A := m(R)/4 ≥ 3M₀/T, and we observe that the following alternative holds

kf₀k_L¹_(m0) ≤Akf₀k_L¹ (3.1) or

kf₀k_L¹_(m0) > Akf₀k_L¹. (3.2) (3) We assume that condition (3.1) holds. Prove that

kS_Tf₀k_L¹ ≤γ₁kf₀k_L¹,

with γ1 ∈(0,1). Deduce that

kS_Tf₀k_β ≤γ₁kf₀k_L¹ − βT

M₀kS_Tf₀k_L¹_(m0)+βkf₀k_L¹_(m1)+βKkf₀k_L¹ and next

kS_Tf₀k_β+ βT

M₀kS_Tf₀k_L¹_(m0) ≤ kf₀k_β, for β >0 small enough.

(4) We assume that condition (3.2) holds. Prove that kS_Tf₀k_L¹_(m1)+ T

M₀kS_Tf₀k_L¹_(m0) ≤ kf₀k_L¹_(m1)+ T

3M₀kf₀k_L¹_(m0), and deduce

kS_Tf₀k_β+ βT

M₀kS_Tf₀k_L¹_(m0) ≤ kf₀k_β + βT

3M₀kf₀k_L¹_(m0). (5) Observe that in both cases (3.1) and (3.2), there holds

kS_Tf₀k_β+ 3αkS_Tf₀k_L¹_(m0)≤ kf₀k_β +αkf₀k_L¹_(m0), where from now β and α are fixed constants. Deduce that

Z(u₁+αv₁)≤u₀+αv₀+ ξ_λ λαw₁, with

u_n:=kS_nTf₀k_β, v_n:=kS_nTf₀k_L¹_(m0), w_n:=kS_nTf₀k_L¹_(m2) and for λ≥λ₀ ≥1 large enough

Z := 1 + δ

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

u_n≤Z⁻ⁿ(u₀+αv₀) + Z Z −1

ξ_λα λ sup

i≥1

w_i, and next

kS_nTf₀k_β ≤

e^{−nTλ 2Tδ} +ξ_λ

Ckf₀k_L¹_(m2), ∀λ≥λ₀. (6) Prove that

kS_tf₀k_L¹ ≤Θ(t)kf₀k_L¹_(m2), ∀t≥0, ∀f₀ ∈L¹(m₂), 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 m₀ = 1, m₁ =hxi, m₂ =hxi²?

4 Problem IV - An application to the Fokker-Planck equation

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

∂_tf =Lf := ∆_xf + div_x(f E) in (0,∞)×R^d (4.1) for the confinement potential E :=∇φ, φ :=hxi^γ/γ, hxi² := 1 +|x|², that we complement with an initial condition

f(0, x) = f₀(x) inR^d. (4.2)

Question 1

Give a strategy in order to build solutions to (4.1) when f₀ ∈L^p_k(R^d), p∈[1,∞], k≥0.

We assume from now on that f0 ∈ L¹_k(R^d), k > 0, and that we are able to build a unique weak (and renormalized) solution f ∈C([0,∞);L¹_k) to equation (4.1)-(4.2).

We also assume that γ ≥2.

Question 2 Prove

hf(t)i=hf₀i and f(t, .)≥0 if f₀ ≥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_L¹

k ≤C₁kf₀k_L¹

k. (Hint. A possible constant is C₁ := max(1, K/α)).

Question 4 Prove that

sup

t≥0

kf(t,·)k_L²

k ≤C₂kf₀k_L²

k, at least fork > 0 large enough.

Question 5 Prove that

sup

t≥0

kf(t,·)k_H¹

k ≤C₃kf₀k_H¹

k, at least fork > 0 large enough.

Question 6 Prove that

kf(t,·)k_H¹ ≤ C₄ t^αkf₀k_L¹

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_L¹

k+t^αk∇_xf(t)k²_L2).

We assume from now on thatd= 1, so that C^0,1/2 ⊂H¹. Question 7

We fix f₀ ∈L¹ such thatf₀ ≥0 and suppf₀ ⊂B_R, R >0. Using question 3, prove that Z

Bρ

f(t)≥ 1 2

Z

BR

f₀,

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 x₀ ∈B_R such that

f(t)≥κ, ∀x∈B(x₀, r).

We accept the spreading of the positivity property, namely that for ant r0, r1 >0, x0 ∈R^d, there existt1, κ1 >0 such that

f₀ ≥1_B(x0,r0) ⇒ f(t₁,·)≥κ₁1_B(x0,r1). Deduce that there exist θ >0 and T >0 such that

f(T,·)≥θ1_B(0,R) Z

B_R

f₀dx.

Question 8

Prove that for any k >0, there exists C, λ >0 such that f0 ∈L¹_k satisfying hf0i = 0, there holds

∀t >0, kf(t, .)k_L¹

k ≤Ce^−λtkf₀k_L¹

k. Question 9

We assume nowγ ∈(0,2). We define

Bf :=Lf −M χ_Rf

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)f₀ the solution associated to the evolution PDE corresponding to the operatorB and the initial condition f₀.

(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^kdx≤ −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_L¹

k→L¹_k ≤1.

(3) Prove that for any k₁ < k < k₂ there exists θ ∈(0,1) such that

∀f ≥0 M_k ≤M_k^θ₁M_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_L¹

`→L¹ ≤ kSB(t)k_L¹

`→L<sup>1</sup><sub>k</sub> ≤C/hti<sup>α</sup>, 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)

kSLk_L¹

k→L¹_k ≤C.

Question 10

Still in the case γ ∈(0,2), what can we say about the decay of kf(t, .)k_L¹

when f₀ ∈L¹_k, k >0, satisfies hf₀i= 0?