2 Problem II

Universit´e Paris-Dauphine 2016-2017

An introduction to evolution PDEs January 2017

1 Problem I

We consider the evolution PDE

∂_tf = ∆f+ div(Ef), (1.1)

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) = f₀(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 (2.1) 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∈L²(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

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

Z

h²w G≤ 1 α

Z

h²(b1_BR

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

Question 5. Take a nice functionh:R^d→Rsuch thathhGi= 0 and denoteG(Ω) :=hG1_Ωi.

Prove that for any R≥R₀ there exists κ_R∈(0,∞) such that Z

h²1B_RG≤κR

Z

BR

|∇h|²G+ 1 G(B_R)

Z

B_R^c

h G 2

and deduce that Z

h²1_BRG≤ κ_R 1 +λ

Z

|∇h|²W G+ G(B_R^c) G(B_R)

Z

h²w G.

Question 6. Establish finally that there exist some constants λ, K1 ∈(0,∞) such that 2

K₁ Z

h²wG≤ Z

W|∇h|²− 1

2h²L^∗W G=

Z

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

Question 7. Consider a nice solution f to (2.1) associated to an initial datum f₀ such that hf₀i= 0. Establish thatf satisfies

1 2

d dt

Z

h²_tW G=− Z

|∇h|²W G+1 2

Z

h²GL^∗W.

Deduce that there exists K₂ ∈(0,∞) such that f satisfies the decay estimate Z

f_t²W G⁻¹dx≤e^−K2t Z

f₀²W G⁻¹dx, ∀t≥0.

2 Problem II

We consider the Keller-Segel equation

∂_tf = ∆f + div(K_ff), (2.1)

on the unknown f =f(t, x),t≥0, x∈R², with

Kf :=K ∗f, K:=∇κ = 1 2π

z

|z|². We complement the equation with an initial condition

f(0, x) = f₀(x).

We accept that for any initial datumf₀ ≥0 with finite massM > 0, finite moment of order 2 and finite entropy, there exists at least one nonnegative solutionf ∈C([0, T);L¹) for some T >0, and that any weak solution furthermore satisfies

f ∈C¹((0, T);W^1,p(R²)), ∀p∈[1,∞].

Question 1. Establish that any weak solution satisfies d

dtkfk²_L2 + 2k∇_xfk²_L2 =kfk³_L3 on (0, T).

Deduce that there exists some constantA ∈(0,∞) such that d

dtkfk²_L2 +1

2k∇_xfk²_L2 ≤A²M on (0, T).

Deduce next that there exists a constant c_M >0 such that d

dtkfk²_L2 +cMkfk⁴_L2 ≤A²M on (0, T).

and finally that exists a constant K (which only depends on c , A²M and T) such that

Question 3. Introducing the notation logf₊g := 2 + (logg)₊ and using an H¨older inequality, prove that

kgk_L^4/3 ≤C(H(g), M2(g)) Z

f²(logf₊g)⁻² 1/4

.

Question 4. Establish that

t^1/4kf(t, .)k_L^4/3 →0 as t→0.

Briefly explain how to deduce the uniqueness of weak solutions to the Keller-Segel equation.

3 Problem III -

We consider the relaxation equation

∂tf =Lf :=−v · ∇f +ρfM −f in (0,∞)×R^2d, (3.1) 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) = f₀(x, v) in R^2d. (3.2) Question 1. A priori estimates and associated semigroup. We denote by f a nice solution to the relaxation equation (3.1)–(3.2).

(a) Prove that f is mass conserving.

(b) Prove that

|ρ_g| ≤ kgk_L2

v(M^−1/2), ∀g =g(v)∈L²_v(M^−1/2) and deduce that

kf(t,·)k_L2

xv(M^−1/2)≤ kf₀k_L2

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^λtkf₀k_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 (3.1) for initial date inL^p_xv(m)? Is the semigroup positive? mass conservative? a contraction in some spaces?

Question 3. Establish thatA:L¹_xv(m)→L¹_xL^∞_v (m) where kgk_L¹

xL^pv(m) :=

Z

R^d

kg(x,)k_L^p_(m)dx.

Prove that d dt

Z Z

f^pdx1/p

dv= Z Z

(∂_tf)f^p−1dxZ

f^pdx1/p−1

dv.

Deduce thatSB satisfies a growth estimateO(e^−t) in anyL¹_xL^p_v(m) space forp∈(1,∞), and then in L¹_xL^∞_v (m). Finally prove that SB(t)A is appropriately bounded in B(L¹, L¹_xL^∞_v (m)) and that SL is bounded in L¹_xL^∞_v (m).

Question 4. We define u(t) :=ASB(t). Establish that (u(t)f₀)(x, v) = M(v)e^−t

Z

R^d

f₀(x−v_∗t, v_∗)dv_∗. Deduce that

ku(t)f₀k_L^∞_xv(m) ≤Ce^−t

t^d kf₀k_L¹_xL^∞_{v (m)}.

Question 5. Establish that there exists some constantsn ≥1 andC ∈[1,∞) such that ku^(∗n)(t)k_L¹_xv(m)→L^∞_xv(m) ≤C e^−t/2.

Deduce that SL is bounded in L^∞_xv(m).

Question 6. How to prove that SL is bounded in L²_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).