On the local uniqueness of steady states for the Vlasov-Poisson system

Motivated by recent results of Lemou-M\'ehats-R\"aphael and Lemou concerning the quatitative stability of some suitable steady states for the Vlasov-Poisson system, we investigate the local uniqueness of steady states near these one. This research is inspired by analogous results of Couffrut and \v{S}ver\'ak in the context of the 2D Euler equations.


Introduction
The gravitational Vlasov-Poisson equation modelizes the evolution of many particles subject to their own gravity, assuming that both the relativistic effects and the collisions between particles can be neglected.We consider the Vlasov-Poisson system in three dimensions: where the Newtonian potential φ f is given in terms of the density ρ f : At the beginning of the last century, astrophysicist Sir J. Jeans used this system to model stellar clusters and galaxies [14] to study their stability properties.In this context, it appears in many textbooks on astrophysics, such as [4,10].In the repulsive case, this system was introduced by A. A. Vlasov around 1937 [22,23].Because of the considerable importance in plasma physics and astrophysics, there is a vast literature on the Vlasov-Poisson system.
The global existence and uniqueness of classical solutions of the Cauchy problem for the Vlasov-Poisson system was obtained by Iordanskii [13] in dimension 1, Ukai-Okabe [21] in the 2-dimensional case, and independently by Lions-Perthame [17] and Pfaffelmoser [19] in the 3-dimensional case (see also [20]).There are currently no results about the existence and uniqueness of classical solutions in dimensions greater than 3.
It is important to mention that, parallel to the existence of classical solutions, there have been a considerable amount of work on the existence of weak solutions, particularly under very low assumptions on the initial data.
We mention in particular the classical result by Arsen'ev [3], who proved the global existence of weak solutions under the hypothesis that f 0 is bounded and has finite kinetic energy, and the result of Horst and Hunze [12], where the authors relax the integrability assumption on f 0 .If one wishes to relax even more the integrability assumptions on the initial data, then one enters into the framework of the so-called renormalized solutions introduced by Di Perna and Lions [7,8,9].The interested reader is referred to the papers [1,5] and the survey [11] for more details and references.
One of the main features of the nonlinear transport flow (1.1) is the conservation of the total energy and of the Casimir functions: for all G ∈ C 1 ([0, ∞], R + ) such that G(0) = 0, G(f 0 (x, v)) dx dv.

Main result
This work aims to prove a local uniqueness result for steady states of (1.1).In the paper [15] (see also [16,18]), the author proves quantitative stability inequalities for the gravitational Vlasov-Poisson system that will be crucial in the following.More precisely, the author considers a class of steady states f to the Vlasov Poisson system, which are decreasing functions of their microscopic energy, and obtains an explicit control of the L 1 distance between f and any function f in terms of the energy H(f ) − H( f ) and the L 1 distance between the rearrangements f * and f * of f and f , respectively.
In the following, we give some definitions, and we state the local functional inequality in [15,Theorem 2].We first recall the notion of equimeasurability and rearrangement.Definition 1.1.Given two integrable nonnegative functions f, g : R n → R, we say that f and g are equimeasurable if for a.e.s > 0.
Then, the (radially decreasing) rearrangement f * of f is defined as the unique radially decreasing function equimeasurable to f .In other words, the level sets of f * are given by The following important result is proved in [15,Theorem 2(ii)].
Theorem 1.2.Consider f a compactly supported steady state solution of (1.1) of the form where F is a continuous function from R to R + that satisfies the following monotonicity property: there exists e 0 < 0 such that F (e) = 0 for e ≥ e 0 and F is a C 1 function on ) has finite kinetic energy and is sufficiently close to a translation of f in the following sense: for some suitable constant R 0 > 0. Then there exists a constant K 0 > 0, depending only on f , such that inf where we denote f ( An immediate consequence is the following estimate, which will be the starting point of our investigation.
Corollary 1.3.Let f, f be as in Theorem 1.2.Assume in addition that f is equimeasurable to f .Then inf Let f be the stationary solution as above.Our goal is to understand if, nearby f , other stationary solutions of (1.1) exist.Because stationary solutions of (1.1) correspond to critical points of H with respect to variations of f generated by Hamiltonian flows (see Lemma 2.3 below), it makes sense to consider a "neighborhood" of f generated by flows of smooth Hamiltonians.Noticing that f is supported in a ball B ρ ⊂ R 3 × R 3 for some ρ > 0 and we shall use the flow of the functions H to move f , it makes sense to consider Hamiltonians H that are all supported inside B 2ρ .Hence, one should consider these functions H as the "tangent space" at f that will generate the admissible variations.
Let us introduce the following notation: where s → Φ H s is the Hamiltonian flow of H, namely In other words, s → f H s is the variation generated by H, and as H varies, this generates a "symplectic" neighborhood of f .Note that, since f H s = f sH 1 , to parameterize a neighborhood of f it is enough to consider the image of the map 1 We now introduce some definitions to clarify the hypotheses that are needed on the Hamiltonian H. Let us start by defining the set Inv f that represents the set of all the Hamiltonians who act trivially on f .Definition 1.4.
This definition is motivated by the following simple result: 1 This resembles the exponential map in Riemannian geometry, where a neighborhood of a point x ∈ M is obtained as the image of a neighborhood of 0 in TxM via the map v → γv (1), where s → γv(s) is the geodesic starting at x with velocity v.
Since J∇H ∈ C 1 and also f solves this equation (because div(J∇H f ) = {H, f } = 0), we obtain f H s ≡ f by uniqueness of solutions to transport equations with C 1 vector-fields.The lemma above shows that if H belongs to Inv f , then Φ H s is not moving f .Since our goal is to use Hamiltonians H to parameterize a neighborhood of f , there is no reason to consider H that belong to Inv( f ), and it makes sense to exclude them.Actually, for some technical reasons that will be clearer later, we must impose a quantitative version of the condition H ∈ Inv f .To do that, we introduce the family of sets Remark 1.6.We note that Because of this observation, we see that Motivated by this fact, in the sequel, we shall fix k and consider only Hamiltonians that belong to A k .Of course, this is more restrictive than assuming only H ∈ Inv f , but at the moment it is not clear to us how to remove such an assumption.
Going further in our preliminary analysis, we observe that all translations of f are trivially stationary solutions.However, the kinetic energy automatically controls translations in v, and indeed, they do not appear in (1.5).To "kill" the space of translations in x, we will assume that Bar x ( f H 1 ) = Bar x ( f ), where denotes the "barycenter in x" of f .We want to emphasize that this is not a restrictive assumption on H since one could remove it by adding to H a Hamiltonian corresponding to translations in the x variable to recenter the barycenter of f .Since this would not add major technical difficulties to the proof but may distract the reader from the essential points, we prefer to impose a barycenter condition on f H 1 .As a final consideration, since our goal is to prove that there are no steady states of (1.1) in a neighborhood of f generated via the map (1.7), we shall need to assume that our Hamiltonians H are small in some suitable topology.
Our main theorem asserts that, for Hamiltonians small enough in a sufficiently strong Sobolev norm that are also quantitatively away from Inv f , there cannot be a stationary point of the form f H 1 .
Theorem 1.7.Let f be as in (1.3), where F is a continuous function from R to R + that satisfies the following monotonicity property: there exists e 0 < 0 such that F (e) = 0 for e ≥ e 0 and F is a C 1 function on (−∞, e 0 ) with F ′ < 0 on (−∞, e 0 ).Let ρ > 0 be such that supp( f ) ⊂ B ρ .Also, assume that f ∈ W 2,q (R 6 ) for some q > 3. Then the following local uniqueness result for steady states holds: Let r ≥ 22, and given ρ, ε, k > 0 consider the space of functions

Comments
Starting from the seminal paper of Arnold about the geometric interpretation of the Euler equations as L 2 -geodesics in the space of measure preserving diffeomorphisms [2], Choffrut and Šverák obtained a related result for the 2D Euler equation [6].The basic idea there is that, under the evolution given by the 2D incompressible Euler equations, the vorticity is transported by an incompressible vector field, hence the measure of all its super-level sets is constant.This means that, given an initial vorticity ω 0 , its evolution ω(t) is in the same equimeasurability class of ω 0 .This allows one to foliate the space of vorticities into a family of leaves O ω 0 (the equimeasurability class of ω 0 ), and the Euler equations preserve these leaves.In addition, thanks to the Hamiltonian structure of the Euler equations, one can characterize stationary solutions as critical points of the Hamiltonian energy E restricted to the orbits.
In other words, one has the following situation: the space of vorticities is foliated by the orbits O ω , and the equilibria are the critical points of E restricted to the orbits.In finite dimension, the implicit function theorem would give the following: if O ω is smooth near a point ω ∈ O ω , and if ω is a non-degenerate critical point of E in O ω , then near ω, the set of equilibria form a smooth manifold transversal to the foliation.In addition, the dimension of this manifold is equal to the co-dimension of the orbits.In particular, in a non-degenerate situation, the equilibria are locally in one-to-one correspondence with the orbits.In [6], the authors obtain an analog of this correspondence in the infinitedimensional context of Euler equations.There, the authors use an infinite dimensional version of the implicit function theorem in the space of C ∞ function via a Nash-Moser's interation.
With respect to their result, here we have different assumptions and results.These are motivated by the following: • Since the Vlasov-Poisson system (1.1) is Hamiltonian, given an initial condition f 0 , its evolution f t under the Vlasov-Poisson system will also be in the same equimesurability class.However, while Hamiltonian maps and measure preserving maps coincide in 2-dimension, they are very different in higher dimensions (for instance, Hamiltonian maps preserve the symplectic structure).Because solutions to the 3D Vlasov-Poisson systems describe a Hamiltonian evolution of particles in the phase-space R 3 × R 3 , there is no natural reason in this context why there should be only one stationary state in the same equimeasurability class.In particular, as already observed before, stationary solutions of (1.1) correspond to critical points of H with respect to variations of f generated by Hamiltonian flows, and not with respect to arbitrary measure preserving variations.This is why we need to look at functions f that can be connected to f via a Hamiltonian flow, namely f = f H 1 for some H.
• The smallness assumption on ∇ r H L 2 is natural, and actually weaker than the one in [6], since smallness there is measured in the C ∞ topology.We note that we did not try to optimize the value of r, which we chose to be equal to 22 for our proof to work, but that can can probably be optimized further.However, improving this number is not the emphasis here.In addition, as explained before, the assumption of compact support of H in the definition of N k ε is not restrictive, since f H 1 does not depend on the behaviour of H outside a large ball.
• As mentioned before, the assumption on Bar x ( f H 1 ) is not fundamental: one could remove it by replacing it with H − H 0 , where H 0 corresponds to a translation in the phase space (multiplied by a suitable cut-off function, to make it compactly supported).What is more essential is our assumption H ∈ A k , that one would like to replace with H ∈ Inv f .Unfortunately, it is unclear to us how to remove it.
The goal of the next section is to prove our main theorem.
2 Proof of the Theorem 1.7

Strategy of the proof
The idea of the proof is the following: first, by exploiting the results in [15], in Lemma 2.1 we prove that if f H 1 has the same barycenter as f , then Secondly, in Proposition 2.2 we show that if f H 1 is stationary, then for some suitable norm • X of ∇H.Combining these two estimates, we get As a next step, in Lemma 2.4 we prove that that combined with (2.1) and our quantitative assumption on the fact that H does not belong to Inv f (namely, H ∈ A k ) allows us to prove that Finally, exploiting the smallness ∇ r H L 2 ≤ ε and interpolation estimates, we relate the two norms above and conclude that for some δ > 0. This yields a contradiction when ∇H L 2 is small enough, concluding the proof.
In the following sections, we provide all the details of the argument outlined above.

Lower bound
Here and in the sequel, Bar x denotes the spacial barycenter as defined in (1.8).
Lemma 2.1.Let f be as in (1.3), where F is a continuous function from R to R + that satisfies the following monotonicity property: there exists e 0 < 0 such that F (e) = 0 for e ≥ e 0 and F is a C 1 function on (−∞, e 0 ) with Assume that η is small enough so that (1.4) holds.Then where C depends on the diameter of the support of f and f H 1 .
Proof.Let (x 0 , 0) be the point where the minimum is achieved in (1.5).By definition, hence, thanks to (2.2), where we used that f and f H 1 are compactly supported, so |(x, v)| is bounded on the support of f and f H 1 (• − (x 0 , 0)).Thus, using (1.5) we obtain which concludes the proof.

Upper bound
The aim of this section is to provide an estimate of the difference between the energy of f and of f H 1 in terms H, under the additional assumption that f H 1 is a stationary solution for (1.1).More precisely, we prove the following: Proposition 2.2.Let f be a compactly supported steady state such that f ∈ L ∞ (R 6 ), and that f ∈ W 2,q (R 6 ) for some q > 3. Let H ∈ C 2 (R 6 ).Also, assume that f H 1 = (Φ H 1 ) # f is a stationary solution for (1.1).Then the following estimate holds: where C is a constant depending only on f .
As a first step towards the proof of the above result, we aim to give a characterization of the stationary solutions of (1.1) in terms of the energy of the system H. Lemma 2.3.Let f : R 6 → R be a compactly supported function.Then f is a steady state for (1.1) if and only if , ), and consider its flow Φ H s .To simplify the notation we set Φ H s = Φ s .Also, it will be convenient to write Φ s = (Φ x s , Φ v s ) : R 6 → R 3 × R 3 .Given a compactly supported function f , we compute the first variation of the Hamiltonian H around f along f H s : recalling that K(x) = − 1 4π|x| denotes the fundamental solution of the Laplacian, we have

Recalling that
we have , so we can rewrite the above expression as Also, using that Φ s preserves the Lebesgue measure and that (Φ s ) −1 = Φ −s , we can rewrite the first variation in the following way: In particular, since Φ s = Id for s = 0, we see that On the other hand, for f to be a stationary solution for the system (1.1) means that div (2.5) proves that Since C 2 functions are dense in C 1 for the C 1 topology, this proves the result.

Second variation for H
As a second step, we compute the second variation for H, in line with the computation of the first variation (2.4).Here we consider as initial condition f and, given a Hamiltonian As before, to simplify the notation, we set Φ s = Φ H s .Also, we define and we observe that Using (2.4) and (2.8), the second variation is given by the following formula: Thus, we obtain: ))g(Φ −s (y, w)) dx dv dy dw .

Proof of Proposition 2.2
As in the previous section, we set f H s := f •Φ H −s and Φ s := Φ H s .Recall that, by assumption, f H 1 is a stationary solution of (1.1).
We now study the Taylor expansion of the Hamiltonian of the gravitational Vlasov Poisson system both at f and at f H 1 .Since f and f H 1 are two stationary solutions, it follows by Lemma 2.3 applied both to f and to f H 1 that and Therefore, we can add a constant term in the integral, and we get Thanks to the latter computation, in order to estimate the left hand side of (2.10), we can estimate in terms of the regularity of H and f .Recalling that Φ 0 = Id and the definition of g in (2.7), we have the following: Thus, where We begin by controlling T 1 .
By the Fundamental Theorem of Calculus, ) and that Φ s preserves the volumes, we get Also, by the definition of g in (2.7), where C depends on ∇ f L 1 and ∇ 2 f L 1 .In conclusion, the first term T 1 can be estimate as follows: We now estimate the second term: Adding and subtracting f (x, v)g(Φ −s (y, w)), we can bound Using as before the Fundamental Theorem of Calculus and the fact that Φ s is measure preserving, we have Using again (2.7), we obtain that combined with Hölder inequality and the fact that Φ −τ s is measure preserving, yields , where p and q are conjugate exponents.In order to have integrability of the gradient of the kernel K, we need p < 3 2 .Therefore, in the previous estimates we need to assume that ∇g L q is finite for some q > 3. Thus where C depends on f L ∞ .As in the estimate of the term T 1 we use (2.7) and (2.12) to get where C depends on ∇ f L q and ∇ 2 f L q for some q > 3. Since by assumption f ∈ W 2,q (R 6 ) for some q > 3, we have prove that where C depends only on f L ∞ , ∇ f L q , and ∇ 2 f L q , for some q > 3.
where C depends only on f .
Then, by the definition of the flow Φ s (see (1.6)), we deduce Thus, Using again (2.15), we get Adding and subtracting ∇ f • Φ H −s , this gives We now estimate the terms I and II.By the Fundamental Theorem of Calculus, By Fubini, we can rewrite the last integral above as and because Φ H −τ is measure preserving we deduce that the term above is equal to Hence, in conclusion, For II, we want to estimate the term Differentiating the equation in (1.6), we deduce that (2.16) Thus, and by Gronwall's inequality Therefore, thanks to (2.17) and (2.18), Combining the bounds on I and II, we conclude that where C is a constant depending only on ∇ f L 1 and ∇ 2 f L 1 .

Proof of Theorem 1.7
In this section we combine the upper and lower bounds obtained in Sections 2.2 and 2.3 with interpolation estimates to obtain a contradiction to the existence of a stationary solution f H 1 , with H as in the statement of Theorem 1.7.We begin by recalling that, by the Sobolev's embedding, given R > 0 and u : In particular, since in our case n = 6, if H is as in the statement of the theorem then ∇H L ∞ + ∇ 2 H L ∞ is as small a desired provided we choose ε small enough.This allows us to apply Lemma 2.1, that combined with Lemma 2.4 yields following bound on g: Then, using Proposition 2.2, We now use the assumption H ∈ A k to get Note that if the norms in the left hand side and in the right hand side were comparable, we would have an inequality of the form X , which is impossible when H is small enough.Thus, the next step is to use interpolation estimates to compare the different norms of ∇H appearing in (2.20).More precisely, we want to use the following elementary interpolation estimates.

Lemma 2.5. For any smooth compactly supported function
Proof.The proof is simple using Fourier analysis: using Hölder inequality with the conjugate exponents m/ℓ and (m − ℓ)/ℓ, we have .
for all k ≥ 0, the result follows.
Since (2.20) involves L 1 and L ∞ norms, to apply Lemma 2.5 we use shall use other interpolation inequalities.More precisely, we recall the classical Nash inequality: (2.21) We now set n = 6, and we let s be a number larger than n/2 = 3 to be fixed later.(2.22)To conclude we recall that, by assumption, H W r,2 ≤ ε, where r ≥ 22, and we want to obtain a contradiction when ε is sufficiently small.To this aim, we first note that, for s ≤ r − 2, This implies that the quadratic terms in (2.22) are much smaller than the term with the power 3/2, therefore (2.22) yields Then we apply Lemma 2.5 with u = ∂ i H, ℓ = s + 1, and m = r − 1 to get (2.24)  which is false for ε small enough.This shows the desired contradiction and completes the proof. ------

Lemma 1.5. If
H ∈ Inv f then (Φ H t ) # f = f , i.e.,the Hamiltonian flow of H does not move f .