Accepted Manuscript Remarks on continuity equations with nonlinear diffusion and nonlocal drifts
Guillaume Carlier, Maxime Laborde
PII: DOI: Reference:
S0022-247X(16)30385-7 http://dx.doi.org/10.1016/j.jmaa.2016.07.061 YJMAA 20623
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
20 April 2016
Please cite this article in press as: G. Carlier, M. Laborde, Remarks on continuity equations with nonlinear diffusion and nonlocal drifts, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.07.061
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Remarks on continuity equations with nonlinear diffusion and nonlocal drifts Guillaume Carlier ∗, Maxime Laborde
†
August 2, 2016
Abstract This paper is devoted to existence and uniqueness results for some classes of nonlinear diffusion equations in the presence of a regular drift term. These equations may be viewed as regular perturbations of Wasserstein gradient flows but the drift terms are not necessarily gradients (which makes it difficult to use Wasserstein gradient flows techniques). We obtain existence by a regularization procedure and parabolic energy estimates and address the uniqueness issue by an elementary H −1 contraction argument if the diffusion is nondegenerate. Our arguments directly extend to systems with diagonal nonlinear diffusions which are coupled through regular drifts.
Keywords: nonlinear diffusion equations and systems, nonlocal drifts, interacting species. MS Classification: 35K15, 35K40.
1
Introduction
The continuity equation with a density-dependent drift ∂t ρ = div(ρv), with v = V [ρ] is ubiquitous in modeling and arises in a variety of domains such as biology, particle physics, population dynamics, crowd modelling, opinion formation... ∗ Universit´ e Paris Dauphine, PSL Research University, CNRS, CEREMADE, 75016 Paris, France and INRIA-Paris, ´equipe projet MOKAPLAN,
[email protected] † Universit´ e Paris Dauphine, PSL Research University, CNRS, CEREMADE, 75016 Paris, France and INRIA-Paris, ´equipe projet MOKAPLAN,
[email protected].
1
It should actually come as no surprise since it captures the dynamics of a population of particles following the ODE X˙ = −v(t, X) where v = V [ρ] depends itself on the density in a way (local, nonlocal, attractive, repulsive etc..) depending on which phenomena (aggregation, diffusion...) one aims to capture and the type of applications. Of course, at this level of generality not much can be said on existence and uniqueness. There are however two cases which may be treated in a rather systematic way. The first one, is the regular case where V [ρ] is a smooth vector field whatever the probability measure ρ is, with some uniform bounds on some of its derivatives and ρ → V [ρ] is Lipschitz in the Wasserstein metric. In this regular case, existence and uniqueness can be proved by the method of characteristics and suitable fixed point arguments (see [7] as well as [6], [5] for a different approach with applications to crowd dynamics). This regular case (a typical example being that of a convolution) is however rather restrictive and for instance rules out diffusion. The second case where there is a general theory is theWasserstein gradient flow case. In this case, at least at a formal level, v may be written as v = ∇ δE that is the gradient δρ of the first variation of a functional E defined on measures. In their seminal paper [9], Jordan, Kinderlehrer and Otto discovered that the heat flow is the gradient flow of the entropy functional E(ρ) = ρ log(ρ) which corresponds to the case v = ∇ρ . The theory of Wasserstein gradient flows has been very ρ succesful in addressing a variety of nonlinear evolution equations such as the porous medium equation [12], aggregation equations [3] or granular media equations [4]. This powerful theory is presented in a complete and detailed way in the reference book of Ambrosio, Gigli and Savar´e [1]. The purpose of the present paper is a contribution to the following general question: can one hope for an existence/uniqueness theory in the case where V is the sum of a Wasserstein gradient flow term and a regular term (not necessarily a gradient). Our motivation for this question actually comes from systems. For instance, a simple but natural model, for the evolution of two (say) interacting species is: ∂t ρ1 = ν1 Δρ1 + div(ρ1 (V11 ρ1 + V12 ρ2 )), ∂t ρ2 = ν2 Δρ2 + div(ρ2 (V21 ρ1 + V22 ρ2 )).
(1.1) (1.2)
If the vector-fields Vij have potentials i.e. Vij = ∇Uij and there is no diffusion i.e. when ν1 = ν2 = 0, this is exactly the system studied by Di Francesco and Fagioli [8]. As emphasized in [8], if cross-interactions are symmetric i.e. U12 = U21 (or more generally U12 and U21 are proportional), this system has a (product) Wasserstein gradient flow structure but this is certainly a restrictive and often unrealistic assumption in applications. This is why Di 2
Francesco and Fagioli, still taking advantage of the similarity with Wasserstein gradient flows used a semi-implicit scheme a` la Jordan-KinderlehrerOtto to obtain existence and uniqueness results. In [8], there is no diffusion and we refer to [10] for an extension of the semi-implicit method to the diffusive case. Clearly the structure of the system (1.1)-(1.2) belongs to the mixed case where drifts can be decomposed as the sum of a Wasserstein gradient and a regular term. It is worth noting though that the semi-implicit scheme of [8] cannot cover the case of general (not gradients) vector-fields Vij . For such general cases and possibly with nonlinear diffusion, we develop some simple arguments in the sequel which give an alternative to mass transport arguments and that enable us to address regular drifts without a gradient structure. The paper is organized as follows. In section 2, we give an existence result by a suitable regularization of the diffusion and energy estimates. In section 3, we give an elementary H −1 -contraction argument from which uniqueness follows. Finally, we observe that these arguments easily extend to the case of systems (without cross-diffusion) in section 4.
2
Existence
For the sake of simplicity, we shall work in the periodic in space case so as to avoid boundary issues and thus take the space variable in the flat torus Td := Rd /Zd . We then consider the following nonlinear diffusion equation with a nonlocal drift: ∂t ρ − Δ(F (ρ)) + div(ρV [ρ]) = 0, ρ|t=0 = ρ0 ,
(2.1)
on (0, T ) × Td . Denoting by H −1 (Td ) the dual of H 1 (Td ) and by P(Td ) the set of probability measures on Td , we assume the following regularity on the drift term V [ρ]: ∀ρ ∈ L2 ∩ P(Td ), V [ρ] ∈ L∞ (Td ) and div(V [ρ]) ∈ L∞ (Td ) with sup ρ∈L2 ∩P(Td )
{ V [ρ] L∞ + div(V [ρ]) L∞ } < +∞
(2.2)
and for every R > 0, there exists a modulus ωR such that, for every (ρ, η) ∈ (L2 (Td ) ∩ P(Td ))2 such that ρ H −1 (Td ) ≤ R and η H −1 (Td ) ≤ R, one has V [ρ] − V [η] L2 (Td ) ≤ ωR ( ρ − η H −1 (Td ) ).
3
(2.3)
Examples: Typical examples of velocity fields ρ → V [ρ] that satisfy the above assumptions (2.2)-(2.3) are those of the form V [ρ](x) = Td B(x, y)ρ(y)dy, in which case (2.2) is satisfied as soon as B and divx B are bounded. As for (2.3), it holds (with a linear modulus) as soon as |Dy B(x, y)|2 dxdy < +∞ Td ×Td
since in this case V [ρ] − V [η] L2 (Td ) ≤ ρ − η H −1 (Td )
|B| + |Dy B| 2
Td ×Td
2
12
.
One can also consider a velocity of the form V [ρ](x) = Td ×Td B(x, y, z)ρ(y)dyρ(z)dz, if B ∈ L∞ (Td × Td × Td ), divx B ∈ L∞ (Td × Td × Td ) then (2.2) obviously holds; if, in addition the constant 2 2 CB := (|B|2 +|Dy B(x, y, z)|2 +|Dz B(x, y, z)|2 +|Dyz B(x, y, z)|2 )dxdydz Td ×Td ×Td
is finite then (2.3) follows from V [ρ] − V [η] L2 (Td ) ≤ ρ − η H −1 (Td ) ( ρ H −1 (Td ) + η H −1 (Td ) )CB . Concerning the diffusion term, we make the following assumptions on F (which are satisfied whenever F (ρ) = ρm with m ≥ 1): F ∈ C 2 (R+ , R), F (0) = F (0) = 0, F is convex,
(2.4)
F is nondecreasing, and for every ρ > 0, F (ρ) > 0
(2.5)
and there is a constant C > 0 such that F (ρ) ≤ C(1 + ρ2 + F (ρ)), ∀ρ ∈ R+ .
(2.6)
Finally, for the initial condition ρ0 we assume that it is a probability density such that ρ0 ∈ L2 (Td ), F (ρ0 ) ∈ L1 (Td ). (2.7) A nonnegative weak solution of the PDE ∂t ρ − Δ(F (ρ)) + div(ρV [ρ]) = 0, ρ|t=0 = ρ0 . is by definition a function ρ ∈ L2 ((0, T ) × Td , R+ ) such that F (ρ) ∈ L2 ((0, T ), H 1 (Td )) 4
(2.8)
and
T 0
Td
(−∂t φρ + ∇F (ρ) · ∇φ − ρV [ρ] · ∇φ) dxdt =
Td
φ(0, x)ρ0 (x)dx (2.9)
for every φ ∈ C ([0, T ] × T ) such that φ(T, .) = 0. Before we proceed to the existence proof, we need some preliminary results. Let us first study the continuity of the drift term ρ = ρ(t, x) → V [ρ(t, .)](x). It is easy to see that when (2.2) and (2.3) are satisfied and ρn converges strongly in L2 ((0, T ) × Td ) (hence in L2 ((0, T ), H −1 (Td ))) to some ρ then V [ρn ] converges to V [ρ] in L2 ((0, T ) × Td ), but we wil need a variant in the sequel: 1
d
Lemma 2.1. Assume that (2.2) and (2.3) are satisfied. Let ρn be a sequence in L2 ((0, T ) × Td ) such that ∂t ρn ∈ L2 ((0, T ), H −1 (Td )) with sup ∂t ρn L2 ((0,T ),H −1 (Td )) < +∞,
(2.10)
n
and ρ ∈ L2 ((0, T ) × Td ) such that ρn ρ in L2 ((0, T ) × Td ), then V [ρn ] converges to V [ρ] strongly in L2 ((0, T ) × Td ). Proof. First observe that (2.10) implies that for some constant C one has (2.11) ρn (t, .) − ρn (s, .) H −1 ≤ C |t − s|, ∀n, ∀(t, s) ∈ (0, T )2 . Let t ∈ (0, T ) and for h ∈ (0, t) define 1 t n 1 t n ρt,h (x) := ρ (s, x)ds, ρt,h := ρ(s, x)ds h t−h h t−h thanks to (2.11), we obtain, for every n, t and h: √ √ ρn (t, .) − ρnt,h H −1 ≤ C h, ρ(t, .) − ρt,h H −1 ≤ C h.
(2.12)
For fixed h > 0, ρnt,h ρt,h in L2 (Td ) as n → ∞, and since the imbedding of L2 (Td ) into H −1 (Td ) is compact we also have ρnt,h − ρt,h H −1 (Td ) → 0 as n → ∞. We then get √ ρn (t, .) − ρ(t, .) H −1 ≤ 2C h + ρnt,h − ρt,h H −1 (Td ) from which we deduce that ρn (t, .) − ρ(t, .) H −1 tends to 0. Thanks to (2.3), this implies that V [ρn (t, .)] − V [ρ(t, .)] L2 (Td ) tends to 0. The claimed L2 convergence then follows from (2.2) and Lebesgue’s dominated convergence Theorem.
5
We now introduce a regularized nonlinearity to approximate (2.8) by a uniformly parabolic equation as follows. Let ε ∈ (0, 1), let δε and Mε be respectively the smallest ρ for which F (ρ) ≥ ε and the largest ρ for which F (ρ) ≤ ε−1 . Let then Fε be defined by ⎧ ε 2 ⎪ ⎨F (δε ) + F (δε )(ρ − δε ) + 2 (ρ − δε ) if ρ ∈ [0, δε ]; (2.13) Fε (ρ) := F (ρ) if ρ ∈ [δε , Mε ], ⎪ ⎩ 1 2 F (Mε ) + F (Mε )(ρ − Mε ) + 2ε (ρ − Mε ) if ρ ≥ Mε . Clearly, by construction Fε is convex and C 2 on R+ with ε ≤ Fε ≤
1 on R+ ε
(2.14)
and Fε converges pointwise to F since δε and Mε converge respectively to 0 and +∞. In fact, this approximation also has good Γ-convergence properties: Lemma 2.2. Let θ ∈ L2 ((0, T ) × Td , R+ ), then T T Fε (θ(t, x))dxdt = lim+ ε→0
Td
0
0
Td
F (θ(t, x))dxdt
(2.15)
moreover if θε ∈ L2 ((0, T ) × Td ), R+ ) weakly converges to θ in ∈ L2 ((0, T ) × Td ), then T T lim inf Fε (θε (t, x))dxdt ≥ F (θ(t, x))dxdt (2.16) + ε→0
Td
0
Proof. Fatou’s lemma first yields T F (θ(t, x))dxdt ≥ lim inf ε + ε→0
0
Td
on the other hand T Fε (θ(t, x))dxdt ≤ 0
Td
Td
0
T 0
T 0
Td
F (θ(t, x))dxdt
Td
F (θ(t, x))dxdt+
{θ≤δε }
(Fε (θ)−F (θ))dxdt
since the second term in the right hand side converges to 0, we easily deduce (2.15). Let us now assume that θε ∈ L2 ((0, T ) × Td , R+ ) weakly converges to θ in ∈ L2 ((0, T ) × Td ). Let γ > 0 (fixed for the moment) and denote by F γ the function defined by F (ρ) if ρ ∈ [0, γ], γ F (ρ) = F (γ) + F (γ)(ρ − γ) if ρ ≥ γ 6
by construction F γ is convex and below F . For ε > 0 small enough so that γ ∈ [δε , Mε ], we similarly define Fε (ρ) if ρ ∈ [0, γ], Fεγ (ρ) = F (γ) + F (γ)(ρ − γ) if ρ ≥ γ so that Fεγ is convex and coincides with F γ on [δε , +∞). We then have T T Fε (θε (t, x))dxdt ≥ lim inf Fεγ (θε (t, x))dxdt lim inf ε→0+ ε→0+ Td Td 0 0 T γ ≥ lim inf F (θε (t, x))dxdt + lim inf (Fε (θε ) − F (θε )) + + ε→0
ε→0
Td
0
{θε ≤δε }
the second term converges to 0 whereas by weak lower semi-continuity (thanks to the convexity of F γ ) we have T T γ lim inf F (θε (t, x))dxdt ≥ F γ (θ(t, x))dxdt, + ε→0
hence
lim inf + ε→0
Td
0
T 0
Td
0
Td
T
Fε (θε (t, x))dxdt ≥ sup γ>0
0
Td
F γ (θ(t, x))dxdt
and then (2.16) easily follows from the previous inequality, the fact that F γ converges monotonically to F and Beppo-Levi’s monotone convergence Theorem. Theorem 2.3. Assume (2.2)-(2.3)-(2.4)-(2.5)-(2.6)-(2.7), then (2.8) admits at least one weak nonnegative solution. Proof. The proof proceeds in three steps. Step 1: Regularized equation. We first prove existence of a weak solution to the regularized equation: ∂t ρε − Δ(Fε (ρε )) + div(ρε V [ρε ]) = 0, ρε |t=0 = ρ0 . Let
(2.17)
X := {η ∈ L2 ((0, T ) × Td , R+ ) :
Td
η(t, x)dx = 1 for a.e. t ∈ (0, T )}
for fixed ε > 0 and η ∈ X, consider the linear parabolic equation in divergence form: ∂t u − div(Fε (η)∇u) + div(uV [η]) = 0, ut=0 = ρ0 7
(2.18)
which can be rewritten in nondivergence form as ∂t u − div(aε (η)∇u) + b[η] · ∇u + c[η]u = 0
(2.19)
where the coefficients aε (η) := Fε (η), b[η] := V [η] and c[η] := div(V [η]) all belong to L∞ ((0, T ) × Td ) with aε (η) ≥ ε by (2.14). It follows from standard linear parabolic theory (see e.g. [11]) that (2.19) admits a unique weak solution which we denote u := T ε (η) ∈ L2 ((0, T ), H 1 (Td )) ∩ C([0, T ], L2 (Td )) with ∂t u ∈ L2 ((0, T ), H −1 (Td )). Obviously, u(t, .) is a probability density for every t ∈ [0, T ]: it remains nonnegative by the maximum principle and its integral over Td is constant in time, in other words T ε (X) ⊂ X. Moreover, multiplying (2.19) by u thanks to (2.2) and (2.14) there is a constant Cε (independent of η) such that u := T ε (η) satisfies T T (|∇u|2 + u2 )dxdt + ∂t u 2H −1 ≤ Cε . (2.20) 0
Td
0
The bound (2.20) and the Aubin-Lions lemma (see [2], [13]) thus imply that T ε (X) is relatively compact in L2 ((0, T )×Td ). Thanks to (2.3), the continuity of Fε and Lebesgue’s dominated convergence theorem, it is easy to check that T ε is continuous with respect to the L2 ((0, T ) × Td ) norm. Schauder’s fixedpoint Theorem then ensures that T ε admits at least one fixed point i.e. a solution of (2.17) which we from now denote ρε . Step 2: A priori estimates. We aim now to derive estimates independent of ε on ρε . Let δ > 0 such that δ ∈ (δε , Mε ), we then take (ρε − δ)+ as test-function in (2.17) (which is actually licit since this test-function belongs to L2 ((0, T ), H 1 (Td ))) integrating between 0 and t ∈ [0, T ] this yields t t t ε ε ε ε 2 ∂t ρ , (ρ −δ)+ H −1 ,H 1 ds+ Fε (ρ )|∇ρ | = ρε V [ρε ]·∇ρε 0
0
{ρε ≥δ}
0
{ρε ≥δ}
hence, using Young’s inequality, for every μ > 0 t 1 1 ε 2 2 (ρ (t, .) − δ)+ L2 − (ρ0 − δ)+ L2 + Fε (ρε )|∇ρε |2 2 2 {ρε ≥δ} 0 t μ t 1 ε 2 ≤C |∇ρ | + (ρε )2 2 0 {ρε ≥δ} 2μ 0 {ρε ≥δ} t μ t 1 ≤C |∇ρε |2 + [(ρε − δ)2+ + δ 2 ] 2 0 {ρε ≥δ} μ 0 {ρε ≥δ} since Fε (δ) = F (δ) > 0 and F nondecreasing, we can choose μ small enough so that the first term in the right hand side is absorbed by the left 8
hand side of the inequality. Gronwall’s lemma then gives sup ρε (t, .) L2 ≤ C
(2.21)
t∈(0,T )
for a constant C that does not depend on ε. Next we take Fε (ρε ) as testfunction which similarly gives: t t ε ε 2 Fε (ρ (t, .)) − Fε (ρ0 ) + |∇Fε (ρ )| = ρε V [ρε ] · ∇Fε (ρε ) d d d d T T T T 0 0 t μ t 1 ε 2 ≤C |∇Fε (ρ )| + (ρε )2 2 0 Td 2μ 0 Td using (2.21) and chosing μ small enough we thus get
sup t∈[0,T ]
Td
T
Fε (ρε (t, .)) + 0
Td
|∇Fε (ρε )|2 ≤ C
(2.22)
for a constant C not depending on ε. Next we use (2.6) and (2.21)-(2.22) to deduce that Fε (ρε ) ≤ C (2.23) sup t∈[0,T ]
Td
together with Poincar´e-Wirtinger inequality, using again (2.22), this gives Fε (ρε ) L2 ((0,T ),H 1 (Td )) ≤ C.
(2.24)
Step 3: Passing to the limit. Let us set uε := Fε (ρε ), σ ε := ∇uε − ρε V [ρε ]
(2.25)
so that (2.17) can be rewritten as ∂t ρε = Δuε − div(ρε V [ρε ]) = div(σ ε ), ρε |t=0 = ρ0 .
(2.26)
We know from the previous step that ρε L∞ ((0,T ),L2 (Td )) + σ ε L2 ((0,T ),L2 (Td )) + uε L2 ((0,T ),H 1 (Td )) ≤ C
(2.27)
as well as ∂t ρε L2 ((0,T ),H −1 (Td )) ≤ C.
(2.28)
Passing to subsequences if necessary, we may therefore assume that ρε ρ in L2 ((0, T ) × Td ), uε u in L2 ((0, T ), H 1 (Td )) 9
(2.29)
and thanks to Lemma 2.1, (2.2) and (2.28), we have σ ε σ := ∇u − ρV [ρ] in L2 ((0, T ) × Td ).
(2.30)
Obviously one then has: ∂t ρ = div(σ) = Δu − div(ρV [ρ]), ρ|t=0 = ρ0 .
(2.31)
So to establish that ρ is a weak nonnegative solution of (2.8), it is enough to prove that u = F (ρ). Thanks to the convexity of F this amounts to prove that T T T F (θ(t, x))dxdt ≥ F (ρ(t, x))dxdt + u(θ − ρ)dxdt Td
0
Td
0
0
Td
(2.32)
for every θ ∈ L ((0, T ) × T , R+ ). By definition of u we know that T T T ε Fε (θ(t, x))dxdt ≥ Fε (ρ (t, x))dxdt + uε (θ − ρε )dxdt. 2
0
d
Td
ε
Td
0
Let us prove that
T
0
T
ε ε
lim ε
0
Td
uρ = 0
Td
(2.33)
Td
uρ.
For that purpose, let ψ ε be the potential defined by −Δψ ε = ρε , ψ ε = 0, ψ ε ∈ H 1 (Td ).
(2.34)
(2.35)
Td
Thanks to (2.21), we have ψ ε ∈ L∞ ((0, T ), H 2 (Td )) with a bound independendent of ε: ∇ψ ε L∞ ((0,T ),H 1 (Td )) ≤ C. (2.36) As for the time derivative of ∇ψ ε we observe that −Δ(∂t ψ ε ) = ∂t ρε = div(σ ε ) so that, thanks to (2.27), we have ∂t ∇ψ ε ∈ L2 ((0, T )×Td ) and more precisely ∂t ∇ψ ε L2 ((0,T )×Td ) ≤ σ ε L2 ((0,T )×Td ) ≤ C this proves that ∇ψ ε is bounded in H 1 ((0, T ) × Td ), hence converges in L2 ((0, T ) × Td ), up to an extraction if necessary, to ψ given by −Δψ = ρ, ψ = 0, ψ ∈ H 1 (Td ). (2.37) Td
10
Weak convergence of ∇uε and strong convergence of ∇ψ ε in L2 then give T T ε ε lim u ρ = lim ∇uε ∇ψ ε ε ε d d T T 0 0 T T = ∇u∇ψ = uρ 0
Td
0
Td
which establishes (2.34). Next, we use Lemma 2.2, letting ε tend to 0+ , using (2.34) we obtain inequality (2.32) which proves that u = F (ρ) and so ρ is a weak solution of (2.8), concluding the proof.
3
H −1 contraction and uniqueness
We still assume that the diffusion F and the drift V satisfy (2.2)-(2.3)-(2.4)(2.5)-(2.6). Given ρ0 and η0 in L2 (Td ) ∩ P(Td ) such that F (ρ0 ) ∈ L1 (Td ) and F (η0 ) ∈ L1 (Td ), we have found in the previous section weak solutions ρ and η of the Cauchy problems:
and
∂t ρ − Δ(F (ρ)) + div(ρV [ρ]) = 0, ρ|t=0 = ρ0 ,
(3.1)
∂t η − Δ(F (η)) + div(ηV [η]) = 0, η|t=0 = η0
(3.2)
such that (recalling (2.21) in the proof of Theorem 2.3) for some R0 > 0 ρ L∞ ((0,T ),L2 (Td )) ≤ R0 , η L∞ ((0,T ),L2 (Td )) ≤ R0 ,
(3.3)
∂t ρ L2 ((0,T ),H −1 (Td )) ≤ R0 , ∂t η L2 ((0,T ),H −1 (Td )) ≤ R0 .
(3.4)
as well as
We then set u := ρ − η (so that u(t, .) has zero mean for a.e. t ∈ (0, T ) and ∂t u ∈ L2 ((0, T ), H −1 (Td )) and our aim is to prove an H −1 contraction estimate, that is some decay estimate on
2 sup u(t, .)ϕ, |∇ϕ| ≤ 1, ϕ=0 Td
Td
Td
slightly abusing notations, we shall denote the previous quantity u(t, .) H −1 (actually since u(t, .) has zero mean, this is indeed an equivalent norm) and thus u(t, .) 2H −1 = ϕ(t, x)u(t, x)dx Td
11
where ϕ(t, .) is the potential of u(t, .) i.e. −Δϕ(t, .) = u(t, .),
Td
ϕ(t, .) = 0.
(3.5)
For this contraction estimate, we need two additional assumptions. The first one is a strong ellipiticty condition, namely that there exists α > 0 such that (F (s) − F (t))(s − t) ≥ α(s − t)2 , ∀(s, t) ∈ R+ × R+ .
(3.6)
The second assumption is on the drift V and requires that for every R > 0, there is some CR > 0 such that for all (ρ, η) ∈ (L2 (Td ) ∩ P(Td ))2 such that ρ L2 (Td ) ≤ R and η L2 (Td ) ≤ R, one has V [ρ] − V [η] L∞ (Td ) ≤ CR ρ − η L2 (Td ) . (3.7) Note that (3.7) is satisfied B ∈ for V [ρ](x) = Td B(x, y)ρ(y)dy with 2 ∞ 2 (L ) or when V [ρ](x) = B(x, y, z)ρ(y)dyρ(z)dz with B ∈ L (L L∞ x y x,y z ). Td Under these extra assumptions, we have the following contraction result: Theorem 3.1. Under the assumptions above there exists a constant λ = λ(α, R0 , CR0 , V [ρ] L∞ ) such that ρ(t, .) − η(t, .) H −1 ≤ eλt ρ0 − η0 H −1 , ∀t ∈ (0, T ),
(3.8)
so that in particular there is uniqueness for (3.1). Proof. Again we define u(t, .) = ρ(t, .) − η(t, .) and ϕ(t, .) by (3.5), setting G(t, .) := ρ(t, .)V [ρ(t, .)] − η(t, .)V [η(t, .)], and defining the potential H(t, .) by: ΔH(t, .) = div(G(t, .)),
Td
H(t, .) = 0,
(3.9)
we then have d u(t, .) 2H −1 = 2∂t u(t, .), ϕ(t, .)H −1 ,H 1 dt = 2Δ(F (ρ(t, .)) − F (η(t, .))) − div(G(t, .)), ϕ(t, .)H −1 ,H 1 = −2 (F (ρ(t, .)) − F (η(t, .)))u(t, .) + 2 H(t, .)u(t, .) Td
Td
so that, thanks to (3.6), we have d u(t, .) 2H −1 ≤ −2α u(t, .) 2L2 + 2 u(t, .) H −1 H(t, .) H 1 . dt 12
(3.10)
But since H(t, .) H 1 ≤ G(t, .) L2 ≤ ρ(t, .) − η(t, .) L2 V [ρ(t, .)] L∞ + η(t, .) L2 V [ρ(t, .)]−V [η(t, .)] L∞ , thanks to (2.2), (3.7) and (3.3), we arrive at H(t, .) H 1 ≤ C u(t, .) L2 , together with (3.10) and Young’s inequality, this gives C2 d u(t, .) 2H −1 ≤ −2α u(t, .) 2L2 + 2C u(t, .) H −1 u(t, .) L2 ≤ u(t, .) 2H −1 dt 2α from which (3.8) directly follows with λ =
C2 . 4α
So far, we have only considered probability distributions as initial conditions, and since the evolution equation conserves total mass, one may think that the previous argument enables one to compare two solutions with the same total mass only. In fact, one can also obtain stability, in a similar ways as above but between solutions ρ and η corresponding to nonnegative initial conditions ρ0 and η0 not necessarily with the same total mass. Indeed, in this case, set again u(t, .) := ρ(t, .) − η(t, .) and define by u its integral (note that this is constant in time) as well as the zero-mean function v = u − u and its potential ϕ by −Δϕ = v, then computing, as before, the 2 time derivative H −1 = ∇ϕ L2 , we just have an extra term to take care of v of, namely u Td (F (ρ(t, .)−F (η(t, .)) but this term typically can be bounded by a constant times |u|, thanks to the uniform in time bounds on ρ(t, .) L2 , η(t, .) L2 , the energies Td F (ρ(t, .)) and Td F (η(t, .)) and inequality (2.6). Doing so, one arrives easily at d v(t, .) 2H −1 ≤ C v(t, .) 2H −1 + C|u| dt which, together with Grownwall’s Lemma, gives the H −1 stability estimate u(t, .) − u 2H −1 ≤ u(0, .) − u 2H −1 eCt + |u|(eCt − 1).
4
Extension to systems
The previous arguments clearly adapt to systems. More precisely, let us consider the system for the evolution of l densities ρ := (ρ1 , . . . , ρl ): ∂t ρi − Δ(Fi (ρi )) + div(ρi Vi [ρ]) = 0, ρi |t=0 = ρi,0
13
(4.1)
on (0, +∞) × Td . Assuming that each function Fi satisfies (2.4)-(2.5)-(2.6), that the initial conditions are probability densities which satisfy ρi,0 ∈ L2 (Td ), Fi (ρi,0 ) ∈ L1 (Td ), ∀i = 1, . . . , l,
(4.2)
and for every i = 1, . . . , l, the map Vi satisfies ∀ρ ∈ L2 (Td )l ∩ P(Td )l , Vi [ρ] ∈ L∞ (Td ) and div(Vi [ρ]) ∈ L∞ (Td ) with sup ρ∈L2 (Td )l ∩P(Td )l
{ Vi [ρ] L∞ + div(Vi [ρ]) L∞ } < +∞
(4.3)
and for every R > 0, there exists a modulus ωR such that, for every (ρ, η) ∈ L2 (Td )l × L2 (Td )l such that ρ H −1 (Td )l ≤ R and η H −1 (Td )l ≤ R, one has Vi [ρ] − Vi [η] L2 (Td ) ≤ ωR
l
ρj − ηj H −1 (Td ) .
(4.4)
j=1
A direct adaptation of the proof of Theorem 2.3 gives Theorem 4.1. Assume that each function Fi satisfies (2.4)-(2.5)-(2.6), and that (4.2)-(4.3)-(4.4) are satisfied for i = 1, . . . , l, then (4.1) admits at least one weak solution (ρ1 , . . . , ρl ) with each ρi nonnegative. As for uniqueness, the H −1 contraction argument of section 3 also easily adapts to systems of the form (4.1). Provided that there is an α > 0 such that (Fi (s) − Fi (t))(s − t) ≥ α(s − t)2 , ∀(s, t) ∈ R+ × R+ , ∀i = 1, . . . , l, (4.5) and, for every R > 0, there is some CR > 0 such that for all (ρ, η) ∈ (L2 (Td )l ∩ P(Td )l )2 such that ρ L2 (Td )l ≤ R and η L2 (Td )l ≤ R, one has Vi [ρ] − Vi [η] L∞ (Td ) ≤ CR ρ − η L2 (Td )l , ∀i = 1, . . . , l,
(4.6)
if ρ = (ρ1 , . . . , ρl ) and η = (η1 , . . . ηl ) both solve the system (4.1) then exactly as in the proof of Theorem 3.1 there is some λ such that ρ(t, .) − η(t, .) H −1 ≤ eλt ρ(0, .) − η(0, .) H −1 , ∀t ∈ (0, T ). In particular, our results apply show well-posedness for systems like (1.1)(1.2) presented in the introduction. Acknowledgements: G.C. gratefully acknowledges the hospitality of the Mathematics and Statistics Department at UVIC (Victoria, Canada), and the support from the CNRS, from the ANR, through the project ISOTACE (ANR-12- MONU-0013). 14
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