Lagrangian solutions to the Vlasov-Poisson system with L1 density

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Lagrangian solutions to the Vlasov-Poisson system with L1 density

Anna Bohun, François Bouchut, Gianluca Crippa

To cite this version:

Anna Bohun, François Bouchut, Gianluca Crippa. Lagrangian solutions to the Vlasov-Poisson sys- tem with L1 density. Journal of Differential Equations, Elsevier, 2016, 260 (4), pp.3576-3597.

�10.1016/j.jde.2015.10.041�. �hal-01097356v2�

SYSTEM WITH L¹ DENSITY

ANNA BOHUN, FRANC¸ OIS BOUCHUT, AND GIANLUCA CRIPPA

Abstract. The recently developed theory of Lagrangian flows for trans- port equations with low regularity coefficients enables to consider non BV vector fields. We apply this theory to prove existence and stability of global Lagrangian solutions to the repulsive Vlasov-Poisson system with only integrable initial distribution function with finite energy. These so- lutions have a well-defined Lagrangian flow. An a priori estimate on the smallness of the superlevels of the flow in three dimensions is established in order to control the characteristics.

1. Introduction

We consider the Cauchy problem for the classical Vlasov-Poisson system

Btf v∇_xf E∇_vf 0, (1.1)

f^p0, x, v^q f^0px, v^q, (1.2) where f^pt, x, v^q¥0 is the distribution function, t^¥0, x, v^PR^N, and

E^pt, x^q∇_xU^pt, x^q (1.3) is the force field. The potential U satisfies the Poisson equation

∆_xU ω^pρ^pt, x^qρ_b^px^qq, (1.4) with ω 1 for the electrostatic (repulsive) case, ω 1 for the grav- itational (attractive) case, and where the density ρ of particles is defined through

ρ^pt, x^q

»

R^N

f^pt, x, v^qdv, (1.5) and ρ_b ^¥ 0, ρ_b ^P L^1pR^Nq is an autonomous background density. Since we are in the whole space, the relation (1.3) together with the Poisson equation (1.4) yield the equivalent relation

E^pt, x^q ω

|S^N1| x

|x^{|N p}ρ^pt, x^qρ_b^px^qq, (1.6) where the convolution is in the space variable.

The Vlasov-Poisson system has been studied for long. Existence of local in time smooth solutions in dimension N 3 has been obtained in [21]

after the results of [8]. Global smooth solutions have been proved to exist in [9, 32, 24], with improvements on the growth in time in [33, 37, 19, 34, 28, 30].

Related results are [35, 27, 15]. These solutions need a sufficiently smooth

Key words and phrases. Vlasov-Poisson system, Lagrangian flows, non BV vector fields, superlevels, weakly convergent initial data.

1

initial datum f⁰. In particular, the following theorem is a classical result due to Pfaffelmoser [32].

Theorem 1.1. If N 3, let f⁰ be a non-negative C¹ function of compact support defined on R⁶. Then there are a non-negative f ^P C^1pR^7q and U ^P C^2pR^4q, which tends to zero at infinity for every fixed t, satisfying equations (1.1)-(1.5) with ρ_b 0. For each fixed t, the function f^pt, x, v^q has compact support. The solution is determined uniquely by the initial datum f⁰.

In a different spirit, global weak solutions were proved to exist in [6, 16, 18], with only f^{0 P} L^1pR^6q, f⁰log f^{0 P} L¹, ^|v^|2f^{0 P} L¹, E^{0 P} L² (and ρ_b 0,ω 1). Related results with weakinitial data have been obtained in [31, 22, 38, 29].

In this paper we would like mainly to extend the existence result of [16] to initial data inL¹ with finite energy (in the repulsive caseω 1), avoiding the Llog L assumption. Our existence result is Theorem 8.4. It involves a well-defined flow. Even weaker solutions were considered in [39, 25, 26], where the distribution function is a measure. However, these solutions do not have well-defined characteristics.

Our approach uses the theory of Lagrangian flows for transport equations with vector fields having weak regularity, developed in [17, 1, 4, 14, 5], and recently in [13, 2, 10]. It enables to consider force fields that are not in W_loc^1,1, nor in BV_loc. In this context we prove stability results with strongly or weakly convergent initial distribution function. The flow is proved to converge strongly anyway. Our main results were announced in [12]. Related results can be found in [3].

Acknowledgment. This research has been partially supported by the SNSF grants 140232 and 156112.

2. Conservation of mass and energy

We would like here to recall some basic identities related to the VP system.

They hold for smooth solutions, and a priori not for weak solutions, for which only the associated a priori bounds remain valid.

Integrating (1.1) with respect to v and noting that the last term is in v-divergence form we obtain thelocal conservation of mass

Btρ^pt, x^q divx^pJ^pt, x^qq0, (2.1) where the current J is defined by

J^pt, x^q

»

RN

vf^pt, x, v^qdv. (2.2) Integrating again with respect to x, we obtain the global conservation of

mass d

dt

¼

R^NR^N

f^pt, x, v^qdxdv d dt

»

R^N

ρ^pt, x^qdx0. (2.3)

Multiplying (1.1) by ^|v₂^|2, integrating inx andv, we get after integration by parts in v

d dt

¼

R^NR^N

|v^|2

2 f^pt, x, v^qdxdv

¼

R^NR^N

Evf dxdv0. (2.4) Using (1.6) and (2.1), one has

BtE

N

¸

k1

B

Bxk

ω

|S^N1| x

|x^|N J_k

0, (2.5)

or in other words

BtEω∇_xp∆x^q

1divxJ, (2.6)

which means that ^BtE is the the gradient component of ωJ (Helmholtz projection). We deduce that

»

RN

E^B_tE dxω

»

RN

EJ dx. (2.7)

Using (2.2) in (2.4), we obtain the conservation of energy d

dt

¼

RN

RN

|v^|2

2 f^pt, x, v^qdxdv ω 2

»

RN

|E^pt, x^q|2dx

0. (2.8) The total conserved energy is the sum of the kinetic energy and of the potential energy multiplied by the factor ω 1. In particular, in the electrostatic case ω 1 we deduce from (2.8) a uniform bound in time on both the kinetic and the potential energy, assuming that they are finite initially. In the gravitational case ω 1 it is not possible to exclude that the individual terms of the kinetic and potential energy become unbounded in finite time, while the sum remains constant. Indeed it is known that it does not happen in three dimensions as soon as f⁰ is sufficiently integrable, but we cannot exclude this a priori for only L¹ solutions.

Note that the assumption E^{0 P}L² is satisfied in 3 dimensions as soon as ρ⁰ρ_b ^PL^6{5. However, in one or two dimensions, for E⁰ to be in L² it is necessary that

³

pρ⁰ρ_b^qdx 0, as is easily seen in Fourier variable. It is also necessary that ρ⁰ρ_b has enough decay at infinity. Thus in one or two dimensions, in order to have finite energy, ρ_b cannot be zero identically.

3. Regularity of the force field for L¹ densities

3.1. Singular integrals. We recall here the basic properties of singular integral operators, that can be found in [36] for example.

Definition 3.1. A function K is a singular kernel of fundamental type in R^N if the following properties hold:

(1) K^|R^Nzt0^uPC^1pR^Nzt0^uq.

(2) There exists a constantC₀ ^¥0 such that

|K^px^{q|¤ C0}

|x^|N, x^PR^Nzt0^u. (3.1)

(3) There exists a constantC₁ ^¥0 such that

|∇K^px^{q|¤ C1}

|x^{|N 1}, x^PR^Nzt0^u. (3.2) (4) There exists a constantA₂ ^¥0 such that

»

R1 |x^| R2

K^px^qdx

¤A₂, (3.3)

for every 0 R₁ R₂  ⁸.

Theorem 3.2 (Calder´on-Zygmund [36]). A singular kernel of fundamental typeK has an extension as a distribution onR^N (still denoted byK), unique up to a constant times Dirac delta at the origin, such that Kˆ ^P L^8pR^Nq. Define

SuKu, for u^PL^2pR^Nq, (3.4) in the sense of multiplication in the Fourier variable. Then we have the estimates for 1 p ⁸

||Su^||_Lp

pRN

q

¤C_N,p^pC₀ C₁ ^||Kˆ^||L^8q||u^||_Lp

pRN

q

, u^PL^pXL^2pR^Nq. (3.5) IfK is a singular kernel of fundamental type, we call the associated oper- atorS a singular integral operator onR^N. We define then the Fr´echet space RpR^{Nq X}m^PN,1 p ⁸W^m,ppR^Nq and its dual R¹pR^{Nq €} S¹pR^Nq, where S¹pR^Nq is the space of tempered distributions on R^N. Since all singular integral operators are bounded on RpR^Nq, by duality we can define the op- erator S also R¹pR^{Nq Ñ} R¹pR^Nq. In particular it enables to defineSu for u^PL^1pR^Nq or foru a measure. The resultSu is inR¹pR^Nq€S¹pR^Nq. 3.2. The split vector field. Letρ^pt, x^{q P}L^8pp0, T^q;L^1pR^Nqq. We denote by

b^pt, x, v^qpb₁, b₂^qpt, x, v^qpv, E^pt, x^qq

v,ω∇_xp∆_x^q1pρ^pt, x^qρ_b^px^qq (3.6) the associated vector field on ^p0, T^qR^N R^N. Then the Vlasov equation can be written in the form of the transport equation ^Btf b∇_x,vf 0. In the following subsections we establish bounds on the vector field b.

3.3. Local integrability. ForL¹densities, using the weak Hardy-Littlewood- Sobolev inequality (see Lemma 4.5.7 in [20]) we have that

∇p∆^q1pρ^pt, x^qρ_b^px^qq

M

N N1

pR^Nq

¤

1

|S^N1||x^|1N|ρ^pt, x^qρ_b^px^q|

M

N N1

pR^Nq

¤c_N^||ρ^pt, x^qρ_b^px^q||_L¹pRN

q,

(3.7)

where

|||u^|||_Mp

pR^Nq sup

γ^¡0

γL^Nptx^PR^Ns.t.^|u^px^q|¡γ^uq1{p, (3.8) and L^N denotes the Lebesgue measure inR^N. It follows that

|||E^|||

L^8pp0,T^q;M^NN1pR^Nqq

¤c_N^||ρρ_b^||_L⁸pp0,T^q;L^1pRN

qq, (3.9)

and using the inclusion M^NN1pR^{Nq €} L^p_loc^pR^Nq for 1 ^¤ p   _N^N

1 we con- clude that b ^P L^8pp0, T^q;L^p_loc^pR^Nx R^Nv ^qq for any 1 ^¤ p   _N^N

1, since v^PL^p_loc^pR^Nx R^Nv ^q for any p.

3.4. Spatial regularity. Since b₁ v is smooth, the only non-trivial gra- dient is the one of b2 E, indeed the differential matrix of the vector field is given by

Db

Dxb1 Dvb1

D_xb₂ D_vb₂

0 Id

D_xE 0

. (3.10)

We have by (3.6)

pD_xE^q_ij ^B_xjE_i ω^B_x²_ixj^pp∆x^q

1

pρρ_b^qq for 1^¤i, j^¤N. (3.11) It is well-known that the operator ^B2_xixj^p∆x^q1 is a singular integral oper- ator. Its kernel is

K_ij^px^q 1

|S^N1|

B

Bxj

x_i

|x^|N

, (3.12)

it is given outside of the origin by K_ij^px^q 1

|S^N1|

N x_ix_j

|x^{|N 2} δ_ij

|x^|N

, forx^PR^Nzt0^u. (3.13) The kernel satisfies the conditions of Subsection 3.1, and ˆKij^pξ^qξiξj^{|ξ^|2. Thus (each component of)D_xEis a singular integral of anL^8pp0, T^q;L^1pR^Nqq function.

3.5. Time regularity. According to (2.6),^BtE is a singular integral of the current J defined by (2.2). Using the bounds available for solutions with finite mass and energy

||f^pt,^q||_L¹

pR^N

xR^N

v^q,

¼

|v^|2f^pt, x, v^qdxdv^¤C, (3.14) and since ^|v^|¤1 ^|v^|2, we get that J ^PL^8pp0, T^q;L^1pR^Nx ^qq. Hence ^BtE is a singular integral of an L^8pp0, T^q;L^1pR^Nx^qqfunction. In particular,

BtE ^PL^8pp0, T^q;S¹pR^Nqq. (3.15) 4. Lagrangian flows

Suppose that f is a smooth solution to (1.1)-(1.5) with f^p0, x, v^q f⁰. Thenfis constant along the characteristics^pX^ps, t, x, v^q, V^ps, t, x, v^qq, which solve the system of equations

$

'

'

'

&

'

'

'

%

dX

ds ^ps, t, x, v^qV^ps, t, x, v^q, dV

ds^ps, t, x, v^q E^ps, X^ps, t, x, v^q, V^ps, t, x, v^qq,

(4.1)

with initial dataX^pt, t, x, v^qxandV^pt, t, x, v^q v. Thus the solution can be expressed as f^pt, x, v^qf^0pX^p0, t, x, v^q, V^p0, t, x, v^qq.

In order to extend this notion of characteristics to non-smooth solutions, we define regular Lagrangian flows, which are defined in an almost every- where sense.

Definition 4.1. Let b ^P L¹_loc^pr0, T^sR^2N;R^2Nq, and t ^{P r}0, T^q. A map Z :^rt, T^sR^{2N Ñ} R^2N is a regular Lagrangian flow starting at time t for the vector field b if

(1) for a.e. z ^P R^2N the map s ^ÞÑ Z^ps, z^q is an absolutely continuous integral solution ofβ^9ps^qb^ps, β^ps^qqfors^Prt, T^swith β^pt^qz.

(2) There exists a constantL independent ofssuch that L^2NpZ^ps,^q1pA^qq¤LL^2NpA^q

for every Borel setA^€R^2N.

The constant L is called the compressibility constant ofZ. Definition 4.2. Define the sublevel of the flow as the set

G_λ ^tz^PR^2N :^|Z^ps, z^q|¤λfor almost alls^Prt, T^su. (4.2) 5. Stability estimate for Lagrangian flows

We summarize the main result from [13] in the following regularity setting of the vector field in arbitrary dimension. We say that a vector fieldbsatisfies (R1) ifb can be decomposed as

b^pt, z^q

1 ^|z^| ˜b₁^pt, z^q ˜b₂^pt, z^q (5.1) where ˜b₁ ^PL^1pp0, T^q;L^1pR^2Nqq, ˜b₂ ^PL^1pp0, T^q;L^8pR^2Nqq.

We assume also that b satisfies(R2): for everyj 1, . . . ,2N,

Bzjb

m

¸

k1

S_jkg_jk (5.2)

where S_jk are singular integrals of fundamental type on R^2N and g_jk ^P L^1pp0, T^q;L^1pR^2Nqq. Moreover, we assume condition (R3), that is

b^PL^p_loc^pr0, T^sR^2Nq, for somep^¡1. (5.3) We recall the following stability theorem from [13], where we denote by B_r the ball with center 0 and radius r in R^2N.

Theorem 5.1. Let b, ¯b be two vector fields satisfying (R1), b satisfying also (R2), (R3). Fix t ^{P r}0, T^q and let Z and Z¯ be regular Lagrangian flows starting at time tassociated to b and¯b respectively, with compression constants L and L. Then the following holds.¯

For every γ ^¡0, r ^¡0 andη ^¡0 there existλ^¡0and Cγ,r,η ^¡0 such that L^2N Br^Xt|Z^ps,^qZ¯^ps,^q|¡γ^u

¤Cγ,r,η^||b¯b^||_L¹pp0,T^qBλ^q η for all s^Prt, T^s. The constants λand C_γ,r,η also depend on

The equi-integrability inL^1pp0, T^q;L^1pR^2Nqqofg_jkcoming from(R2),

The norms of the singular integral operators S_jk from(R2),

The norm ^||b^||_L^ppp0,T^qBλ^q corresponding to (R3),

The L^1pL^1q and L^1pL^8q norms of the decompositions of b and¯b in (R1),

The compression constants L andL.¯

We would like now to state a variant of this theorem, where (R1) and (R2) are replaced by (R1a)and (R2a).

We consider the following weakened assumption (R1a): for all regular Lagrangian flows Z : ^rt, T^sR^{2N Ñ} R^2N relative to b starting at time t with compression constant L, and for all r, λ^¡0,

L^2NpB_r^zG_λ^q¤g^pr, λ^q, withg^pr, λ^qÑ0 asλ^Ñ8at fixed r, (5.4) where G_λ denotes the sublevel of the flowZ, defined in (4.2).

We next consider a splitting of the variables, R^2Nz R^Nx R^Nv , and we denote Z^pt, x, v^{q p}X, V^qpt, x, v^q. Noticing the special form (3.6), we assume the condition that bsatisfies(R2a):

b^pt, x, v^qpb₁, b₂^qpt, x, v^qpb₁^pv^q, b₂^pt, x^qq, (5.5) with

b1 ^PLip^pR^Nv ^q, (5.6)

and where b₂ involves singular kernels only in the first set of variables x, that is for every j1, ..., N,

Bxjb₂

m

¸

k1

S_jkg_jk, (5.7)

where S_jk are singular integrals of fundamental type on R^N and g_jk ^P L^1pp0, T^q;L^1pR^Nqq.

Theorem 5.2. Let b, ¯b be two vector fields satisfying (R1a), b satisfying also (R2a), (R3). Fix t ^{P r}0, T^q and let Z and Z¯ be regular Lagrangian flows starting at time tassociated to b and¯b respectively, with compression constants L and L, and sublevels¯ G_λ and G¯_λ. Then the following holds.

For every γ ^¡0, r^¡0andη ^¡0, there existλ^¡0 andC_γ,r,η^¡0such that L^2N B_r^Xt|Z^ps,^qZ¯^ps,^q|¡γ^u

¤C_γ,r,η^||b¯b^||_L¹

pp0,T^qB_λ^q η for all s^Prt, T^s. The constants λand Cγ,r,η also depend on

The equi-integrability inL^1pp0, T^q;L^1pR^Nqqofg_jkcoming from(R2a),

The norms of the singular integral operators S_jk from(R2a),

The Lipschitz constant of b₁ from (R2a),

The norm ^||b^||_L^ppp0,T^qB_λ^q corresponding to (R3),

The rate of decay of L^2NpBr^zGλ^q and L^2NpBr^zG¯λ^q from (R1a),

The compression constants L andL.¯

Proof. We summarize the modifications of the proof of Theorem 5.1 from [13]. The main added difficulty is the singular integral operators on R^Nx

instead of R^2N. More general situations with singular integrals of measures instead of L¹ functions are considered in [10]. Let z ^px, v^{q P} R^N R^N. We estimate the quantity

Φδ^ps^q

»

Br^XGλ^XG^sλ

log

1 ^|Z^ps, z^qZ^sps, z^q|

δ

dz . (5.8)

Differentiating with respect to time, we get Φ¹_δ^ps^q¤

»

Br^XGλ^XG^sλ

|b^ps, Z^ps, z^qqsb^ps,Z^sps, z^qq|

δ ^|Z^ps, z^qZ^sps, z^q| dz

¤

Ls

δ^||b^ps,^qsb^ps,^q||_L¹pB_λ^q

»

Br^XGλ^XG^sλ

min

"

|b^ps, Z^ps, z^{qq| |}b^ps,Z^sps, z^qq|

δ ,

|b1^ps, Z^ps, z^qqb1^ps,Z^sps, z^qq|

|Z^ps, z^qZ^sps, z^q|

|b2^ps, Z^ps, z^qqb2^ps,Z^sps, z^qq|

|Z^ps, z^qZ^sps, z^q|

*

dz.

Using the special form of b from(R2a), we obtain Φ¹_δ^ps^q¤

Ls

δ^||b^ps,^qsb^ps,^q||_L¹pB_λ^q

»

Br^XGλ^XG^sλ

min

"

|b^ps, Z^ps, z^{qq| |}b^ps,Z^sps, z^qq|

δ ,

|b1^pV^ps, z^qqb1^pV^sps, z^qq|

|V^ps, z^qV^sps, z^q|

|b2^ps, X^ps, z^qqb2^ps,X^sps, z^qq|

|X^ps, z^qX^sps, z^q|

*

dz

¤

Ls

δ^||b^ps,^qsb^ps,^q||_L¹pBλ^q Lip^pb₁^qL^2NpB_r^q

»

Br^XGλ^XG^sλ

min

"

|b^ps, Z^ps, z^{qq| |}b^ps,Z^sps, z^qq|

δ ,

|b₂^ps, X^ps, z^qqb₂^ps,X^sps, z^qq|

|X^ps, z^qX^sps, z^q|

*

dz.

(5.9) Using assumption (R2a), we can use the estimate of [13] on the difference quotients of b₂,

|b₂^ps, X^ps, z^qqb₂^ps,X^sps, z^qq|

|X^ps, z^qX^sps, z^{q| ¤}Ups, X^ps, z^qq Ups,X^sps, z^qq, (5.10) where Ups, .^qPM^1pR^Nq for fixed sis indeed given by

Ups,^q

N

¸

j1 m

¸

k1

M_j^pS_jkg_jk^ps, .^qq, (5.11) with M_j a smooth maximal operator onR^N. Next, we can define the func- tion

Zps, x, v^qUps, x^q1

px,v^qPB^„_λ, (5.12) and we notice that since the above integrals are over G_λ ^XG^s_λ, we can replace the right-hand side of (5.10) byZps, Z^ps, z^qq Zps,Z^sps, z^qq. Then, for givenε^¡0, we can decomposeg_jkg¹_jk g_jk² , with^}g_jk^{1 }}_L¹pp0,T^qRN

q

¤ε and^}g_jk^{2 }}_L²

pp0,T^qR^Nq

¤C_ε,g²_jkhaving support in a setA_ε of finite measure.

This gives rise to two type of termsZ¹ andZ². Since all theZ terms have

compact support in v, this allows to perform the same estimates as in [13].

We finally get

L^2NpBr^Xt|Z^pτ,^qZ^spτ,^q|¡γ^uq

¤

1 log^p1 γ^{δ^q

»τ t

Φ¹_δ^ps^qds L^2NpB_r^zG_λ^q L^2NpB_r^zG^s_λ^q. (5.13) By choosing λlarge first, then εsmall, and finallyδ small, we conclude the

proof as in [13].

6. Control of superlevels

In order to apply Theorem 5.2, we need to satisfy (R1a). Therefore, we seek an upper bound on the size of B_r^zG_λ.

6.1. The case of low space dimension. We first recall the following lemma from [13].

Lemma 6.1. Let b : ^p0, T^q R^{2N Ñ} R^2N be a vector field satisfying (R1). Then b satisfies (R1a), where the function g^pr, λ^q depends only on L,^||˜b1^||L^1pp0,T^q;L^1pR²N

qq,^||˜b2^||L^1pp0,T^q;L^8pR²N

qq.

This lemma allows us to control the superlevels of bin 1 or 2 dimensions.

Proposition 6.2. Letbbe the vector field in (3.6), withE^PL^8pp0, T^q;L^2pR^Nqq. For N 2 or N 1, b satisfies (R1), hence also (R1a).

Proof. It is clear that

v

1 ^|x^{| |}v^{| P}L⁸_t ^pL⁸_x,v^q, (6.1) and

E^pt, x^q 1 ^|x^{| |}v^|

E^pt, x^q

1 ^|x^{| |}v^{|1|v|¤|Ept,xq|}

E^pt, x^q

1 ^|x^{| |}v^{|1|v|¡|Ept,xq|} E˜₁ E˜₂. (6.2) Clearly ˜E₂ ^PL⁸_t ^pL⁸_x,v^q, and if N 2, ˜E₁ ^PL^8pp0, T^q;L¹_x,v^qsince

¼

R²R²

|E^pt, x^q|

1 ^|x^{| |}v^{|1|v|¤|Ept,xq|}dxdv

¤

»

R²

|E^pt, x^q|

»

|v^|¤|E^pt,x^q|

1

|v^|dv

dx2π

»

R²

|E^pt, x^q|2dx.

(6.3)

In the caseN 1, we have directly thatE^pt, x^q{p1 ^|v^|qPL^8pp0, T^q;L²_x,v^q. 6.2. The case of three space dimensions. The condition (R1) being not satisfied in 3 dimensions (the above computation would require E ^P L¹_t^pL³_x^q), we need an estimate on^|Z^|in order to control the superlevels. For getting this we integrate in space a function growing slower at infinity than log^p1 ^|Z^|q(this corresponding to the case (R1)).

Proposition 6.3. Letb be as in (3.6) withN 3,E^PL^8pp0, T^q;L^2pR^Nqq, satisfying (2.6) with J ^P L^8pp0, T^q;L^1pR^Nqq. Furthermore, assume that ω 1, ρ ^¥ 0, and ρ_b ^P L^1XL^ppR^3q for some p ^¡ 3^{2. Then (R1a) holds, where the function g depends only on L, T, ^||E^||_L⁸

t ^pL²_x^q, ^||J^||_L⁸

t ^pL¹_x^q,

}p∆^q1ρ_b^}L⁸, and one has g^pr, λ^{q Ñ}0 asλ^Ñ8 at fixed r.

Proof. Step 1.) LetZ :^rt, T^sR³R^3ÑR³R³ be a regular Lagrangian flow relative tobstarting at timet, with compression constantLand sublevel G_λ. DenotingZ ^pX, V^q, we have the ODEs

#

X9^ps, x, v^qV^ps, x, v^q,

V9^ps, x, v^qE^ps, X^ps, x, v^qq. (6.4) Recalling that E ∇_xU, one has

Bs

|V^ps, x, v^q|2

2 V^ps, x, v^qB_sV^ps, x, v^qE^ps, X^ps, x, v^qqB_sX^ps, x, v^q

Bs^rU^ps, X^ps, x, v^{qqs B}_tU^ps, X^ps, x, v^qq. (6.5) This computation is indeed related to the form of the Hamiltonian for (1.1), H |v^|2{2 U^pt, x^q.

We are going to bound the superlevels of V^ps, x, v^q. We claim that

¼

Br

sup

s^Prt,T^s

1 log

1 ^|V^ps, x, v^q|2{2

α

dxdv^¤A, (6.6) where 0   α   1^{3, and for some constant A depending on L, T, r, α, and on the norms ^||E^||_L⁸

t^pL²_x^q, ^||J^||_L⁸

t ^pL¹_x^q, ^}p∆^q1ρ_b^}L⁸. Assume for the moment that this holds. From the lower bound

¼

Br

sup

s^Prt,T^s

1 log

1 ^|V^ps, x, v^q|2{2

α

dxdv

¥L⁶pB_r^zG^r_λ^qp1 log^p1 λ^2{2^qqα,

(6.7) with G^r_λ the sublevel of V, we get that

L⁶pB_r^zG^r_λ^q¤ A

p1 log^p1 λ^2{2^qqα. (6.8) Next, we remark that by the first equation in (6.4), whenever ^px, v^{q P} G^r_λ one has ^|X^ps, x, v^q|¤|x^{| |}st^|λ, and ^|Z^ps, x, v^q|¤|x^{| p}1 T^qλ. Thus forλ^¡r, one hasBr^zG_λ ^€Br^zG^rpλr^q{p1 T^q, which enables to conclude the proposition (for λ^¤r we can just boundL⁶pB_r^zG_λ^qby L⁶pB_r^q).

Step 2.) By Step 1, it is enough to prove that we have a decomposition

1 log

1 ^|V^ps,,^q|2 2

α

¤f₁ f₂^PL^1pR³xR³v^q L^8pR³xR³v^q, (6.9) for ^px, v^qPB_r, where f₁,f₂ are independent of s^Prt, T^s. Let

β^py^qp1 log^p1 y^qqα, fory^¥0. (6.10)