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On uniqueness for a rough transport-diffusion equation

Guillaume Lévy

To cite this version:

Guillaume Lévy.

On uniqueness for a rough transport-diffusion equation.

Comptes rendus de

l’Académie des sciences. Série I, Mathématique, Elsevier, 2016, 354 (8), pp.804-807. �hal-01314860�

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On uniqueness for a rough transport-diffusion equation.

Guillaume L´

evy

a

a

Laboratoire Jacques-Louis Lions, Universit´e Pierre et Marie Curie, Office 15-16 301, Paris Received *****; accepted after revision +++++

Presented by

Abstract

In this Note, we study a transport-diffusion equation with rough coefficients and we prove that solutions are unique in a low-regularity class.To cite this article: G. L´evy, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

R´esum´e

Sur l’unicit´e pour une ´equation de transport-diffusion irr´eguli`ere.Dans cette Note, nous ´etudions une ´

equation de transport-diffusion `a coefficients irr´eguliers et nous prouvons l’unicit´e de sa solution dans une classe de fonctions peu r´eguli`eres.Pour citer cet article : G. L´evy, C. R. Acad. Sci. Paris, Ser. I 340 (2005).

1. Introduction

In this note, we address the problem of uniqueness for a transport-diffusion equation with rough coeffi-cients. Our primary interest and motivation is a uniqueness result for an equation obeyed by the vorticity of a Leray-type solution of the Navier-Stokes equation in the full, three dimensional space. The main theorem of this note is the following.

Theorem 1.1 : Let v be a divergence free vector field in L2(R

+, ˙H1(R3)) and a a function in L2(R+×R3).

Assume that a is a distributional solution of the Cauchy problem (C)    ∂ta+ ∇ · (av) − ∆a = 0 a(0) = 0, (1)

where the initial condition is understood in the distributional sense. Then a is identically zero on R+×R3. Email address: levy@ljll.math.upmc.fr(Guillaume L´evy).

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As a preliminary remark, the assumptions on both v and a entail that ∂tabelongs to L1loc(R+, H−2(R3))

and thus, in particular, a is also in C(R+,D′(R3)). In Theorem 1.1, a is to be thought of as a scalar

component of the vorticity of v, which is in the original problem a Leray solution of the Navier-Stokes equation. In particular, we only know that a belongs to L2(R

+× R3) and L∞(R+, ˙H−1(R3)), though

we will not use the second assumption. The reader accustomed to three-dimensional fluid mechanics will notice that, comparing the above equation with the actual vorticity equations in 3D, a term of the type a∂ivis missing. In the original problem, where this Theorem first appeared, we actually rely on a double

application of Theorem 1.1. For some technical reasons, only the second application of Theorem 1.1 takes in account the abovementioned term.

As opposed to the standard DiPerna-Lions theory, we cannot assume that a is in L∞

(R+, Lp(R3)) for

some p ≥ 1. However, our proof does bear a resemblance to the work of DiPerna and Lions; our result may thus be viewed as a generalization of their techniques. Because of the low regularity of both the vector field v and the scalar field a, the use of energy-type estimates seems difficult. This is the main reason why we rely instead on a duality argument, embodied by the following theorem.

Theorem 1.2 : Given v a divergence free vector field in L2(R

+, ˙H1(R3)) and a smooth ϕ0 in D(R3),

there exists a distributional solution of the Cauchy problem (C′ )    ∂tϕ− v · ∇ϕ − ∆ϕ = 0 ϕ(0) = ϕ0 (2) with the bounds

kϕ(t)kL∞ (R3)≤ kϕ0kL∞ (R3) (3) and k∂jϕ(t)k2L2(R3)+ Z t 0 k∇∂jϕ(s)k2L2(R3)ds≤ k∂jϕ0k2L2(R3)+ kϕ0k2L∞ (R3)k∂jvk2L2(R +×R3) (4)

for j = 1, 2, 3 and any positive time t.

By reversing the arrow of time, this amounts to build, for any strictly positive T , a solution on [0, T ] × R3

of the Cauchy problem

(−C′ )    −∂tϕ− v · ∇ϕ − ∆ϕ = 0 ϕ(T ) = ϕT, (5) where we have set ϕT := ϕ0 for the reader’s convenience.

2. Proofs

We begin with the dual existence result.

Proof (of Theorem 1.2.) : Let us choose some mollifying kernel ρ = ρ(t, x) and denote vδ := ρ δ∗ v,

where ρδ(t, x) := δ−4ρ(δt,xδ). Let (Cδ′) be the Cauchy problem (C ′

) where we replaced v by vδ. The

existence of a (smooth) solution ϕδ to (C

δ) is then easily obtained thanks to, for instance, a Friedrichs

method combined with heat kernel estimates. We now turn to estimates uniform in the regularization parameter δ. The first one is a sequence of energy estimates done in Lp with p ≥ 2, which yields the

maximum principle in the limit. Multiplying the equation on ϕδ by ϕδδ|p−2 and integrating in space

and time, we get 1 pkϕ δ(t)kp Lp(R3)+ (p − 1) Z t 0 k∇ϕδ(s)|ϕδ(s)|p−2 2 k2 L2(R3)ds= 1 pkϕ0k p Lp(R3). (6) 2

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Discarding the gradient term, taking p-th root in both sides and letting p go to infinity gives kϕδ(t)k

L∞

(R3)≤ kϕ0kL

(R3). (7)

To obtain the last estimate, let us derive for 1 ≤ j ≤ 3 the equation satisfied by ∂jϕδ. We have

∂t∂jϕδ− vδ· ∇∂jϕδ− ∆∂jϕδ = ∂jvδ· ∇ϕδ. (8)

Multiplying this new equation by ∂jϕδ and integrating in space and time gives

1 2k∂jϕ δ(t)k2 L2(R3)+ Z t 0 k∇∂jϕδ(s)k2L2(R3)ds= 1 2k∂jϕ0k 2 L2(R3) + Z t 0 Z R3 ∂jϕδ(s, x)∂jvδ(s, x) · ∇ϕδ(s, x)dxds. (9)

Since v is divergence free, the gradient term in the left-hand side does not contribute to Equation (9). Denote by I(t) the last integral written above. Integrating by parts and recalling that v is divergence free, we have I(t) = − Z t 0 Z R3 ϕδ(s, x)∂jvδ(s, x) · ∇∂jϕδ(s, x)dxds ≤ kϕ0kL∞ (R3) Z t 0 k∂jvδ(s)kL2 (R3)k∇∂jϕδ(s)kL2 (R3)ds ≤ 1 2 Z t 0 k∇∂jϕδ(s)k2L2(R3)ds+ 1 2kϕ0k 2 L∞ (R3) Z t 0 k∂jvδ(s)k2L2(R3)ds.

And finally, the energy estimate on ∂jϕδ reads

k∂jϕδ(t)k2L2(R3)+ Z t 0 k∇∂jϕδ(s)k2L2(R3)ds≤ k∂jϕ0k2L2(R3)+ kϕ0k2L∞ (R3)k∂jvk2L2(R +,×R3). (10)

Thus, the family (ϕδ)

δ is bounded in L∞(R+, H1(R3)) ∩ L2(R+, ˙H2(R3)) ∩ L∞(R+× R3). Up to some

extraction, we have the weak convergence of (ϕδ)

δ in L2(R+, ˙H2(R3)) and its weak-∗ convergence in

L∞

(R+, H1(R3)) ∩ L∞(R+× R3) to some function ϕ.

By interpolation, we also have ∇ϕδ ∇ϕ weakly in L4(R +, ˙H

1

2(R3)) as δ → 0. As a consequence,

because vδ → v strongly in L2(R

+, ˙H1(R3)) as δ → 0, the following convergences hold :

∆ϕδ ⇀∆ϕ in L2(R+× R3); vδ· ∇ϕδ, ∂ tϕδ ⇀ v· ∇ϕ , ∂tϕin L 4 3(R +, L2(R3)).

In particular, such a ϕ is a distributional solution of (C′

) with the desired regularity.  We now state a Lemma which will be useful in the final proof.

Lemma 2.1 : Let v be a fixed, divergence free vector field in L2(R

+, ˙H1(R3)). Let (ϕδ)δ be a bounded

family in L∞

(R+× R3). Let ρ = ρ(x) be some smooth function supported inside the unit ball of R3and

define ρε:= ε −3

ρ ·

ε. Define the commutator C ε,δ by Cε,δ(s, x) := v(s, x) · (∇ρε∗ ϕδ(s))(x) − (∇ρε∗ (v(s)ϕδ(s)))(x). Then kCε,δk L2(R +×R3)≤ k∇ρkL1(R3)k∇vkL2(R +, ˙H1(R3))kϕ δk L∞ (R+,H1(R3)). (11)

This type of lemma is absolutely not new. Actually, it is strongly reminiscent of Lemma II.1 in [2] and serves the same purpose. We are now in position to prove the main theorem of this note.

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Proof (of Theorem 1.1.) : Let ρ = ρ(x) be a radial mollifying kernel and define ρε(x) := ε−3ρ(xε).

Convolving the equation on a by ρεgives, denoting aε:= ρε∗ a,

(Cε) ∂taε+ ∇ · (aεv) − ∆aε= ∇ · (aεv) − ρε∗ ∇ · (av). (12)

Notice that even without any smoothing in time, aε, ∂taε are in L ∞

(R+,C∞(R3)) and L1(R+,C∞(R3))

respectively, which is enough to make the upcoming computations rigorous. In what follows, we let ϕδ

be a solution of the Cauchy problem (−C′

δ), with (−C ′

δ) being (−C ′

) with v replaced by vδ. Let us now

multiply, for δ, ε > 0 the equation (Cε) by ϕδ and integrate in space and time. After integrating by parts

(which is justified by the high regularity of the terms we have written), we get Z T 0 Z R3 ∂taε(s, x)ϕδ(s, x)dxds = haε(T ), ϕTiD′ (R3),D(R3)− Z T 0 Z R3 aε(s, x)∂tϕδ(s, x)dxds and Z T 0 Z R3 [∇ · (v(s, x)aε(s, x)) − ρε(x) ∗ ∇ · (v(s, x)a(s, x))] ϕδ(s, x)dxds = Z T 0 Z R3 a(s, x)Cε,δ(s, x)dxds,

where the commutator Cε,δ has been defined in the Lemma. From these two identities, it follows that

haε(T ), ϕTiD′ (R3),D(R3)= Z T 0 Z R3 a(s, x)Cε,δ(s, x)dxds − Z T 0 Z R3 aε(s, x) −∂tϕδ(s, x) − v(s, x) · ∇ϕδ(s, x) − ∆ϕδ(s, x) dxds.

From the Lemma, we know that (Cε,δ)

ε,δ is bounded in L2(R+ × R3). Because v · ∇ϕδ → v · ∇ϕ in

L43(R+, L2(R3)) as δ → 0, the only weak limit point in L2(R+× R3) of the family (Cε,δ)

ε,δ as δ → 0 is

Cε,0. Thanks to the smoothness of a

ε for each fixed ε, we can take the limit δ → 0 in the last equation,

which leads to haε(T ), ϕTiD′ (R3),D(R3)= Z T 0 Z R3 a(s, x)Cε,0(s, x)dxds. (13) Again, the family (Cε,0)

εis bounded in L2(R+× R3) and its only limit point as ε → 0 is 0, simply because

v· ∇ϕε− ρε∗ (v · ∇ϕ) → 0 in L

4

3(R+, L2(R3)). Taking the limit ε → 0, we finally obtain

ha(T ), ϕTiD′

(R3),D(R3)= 0. (14)

This being true for any test function ϕT, a(T ) is the zero distribution and finally a ≡ 0. 

References

[1] L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. math. 158, no.2, 227-260 (2004) [2] R.J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. math. 98,

511-547 (1989)

[3] C. Fabre and G. Lebeau, R´egularit´e et unicit´e pour le probl`eme de Stokes, Comm. Part. Diff. Eq. 27, no. 3-4, 437-475 (2002)

[4] C. Le Bris and P.-L. Lions, Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Part. Diff. Eq. 33, no. 7-9, 1272-1317 (2008)

[5] J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Mathematica 63, 193-248 (1934)

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