On the local uniqueness of steady states for the Vlasov-Poisson system
aa r X i v : . [ m a t h . A P ] J a n On the local uniqueness of steady states for theVlasov-Poisson system
Mikaela Iacobelli ∗ November 5, 2018
Abstract
Motivated by recent results of Lemou, Méhats, and Räphael [15] and Lemou [14]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 Šverák in the context ofthe 2D Euler equations [6].
The gravitational Vlasov-Poisson equation modelizes the evolution of a large number ofparticles subject to their own gravity, under the assumption that both the relativisticeffects and the collisions between particles can be neglected. We consider the Vlasov-Poisson system in three dimension: ( ∂ t f + v · ∇ x f − ∇ φ f · ∇ v f = 0 , ( t, x, v ) ∈ R + × R × R f (0 , x, v ) = f ( x, v ) ≥ , R f dx dv = 1 . (1.1)where the Newtonian potential φ f is given in terms of the density ρ f : ρ f ( x ) = Z R f ( x, v ) dv, and φ f ( x ) = − π | x | ∗ ρ f = K ∗ ρ f At the beginning of the last century the astrophysicist Sir J. Jeans used this systemto model stellar clusters and galaxies [13] and to study their stability properties. In thiscontext it appears in many textbooks on astrophysics such as [4, 10]. In the repulsivecase, this system was introduced by A. A. Vlasov around 1937 [21, 22]. Because of theconsiderable importance in plasma physics and in astrophysics, there is a huge literatureon the Vlasov-Poisson system.The global existence and uniqueness of classical solutions of the Cauchy problem forthe Vlasov-Poisson system was obtained by Iordanskii [12] in dimension 1, Ukai-Okabe[20] in the 2-dimensional case, and independently by Lions-Perthame [16] and Pfaffelmoser[18] in the 3-dimensional case (see also [19]). To our knowledge, there are currently noresults about existence and uniqueness of classical solutions in dimension greater than 3.It is important to mention that, parallel to the existence of classical solutions, therehave been a considerable amount of work on the existence of weak solutions, in particular ∗ University of Cambridge, DPMMS Centre for Mathematical Sciences, Wilberforce road, CambridgeCB3 0WB, UK. Email: [email protected] f is bounded and has finite kinetic energy, and the result of Horst and Hunze [11],where the authors relax the integrability assumption on f . If one wishes to relax evenmore the integrability assumptions on the initial data then one enters into the frameworkof the so called renormalized solutions introduced by Di Perna and Lions [7, 8, 9]. Theinterested reader is referred to the recent papers [1, 5] for more details and references.One of the main features of the nonlinear transport flow (1.1) is the conservation ofthe total energy H ( f ( t )) = 12 Z R | v | f ( t, x, v ) dx dv − Z R |∇ φ f ( t, x ) | dx = H ( f (0)) (1.2)as well as the Casimir functions: for all G ∈ C ([0 , ∞ ] , R + ) such that G (0) = 0, Z R G ( f ( t, x, v )) dx dv = Z R G ( f ( x, v )) dx dv. The goal of this work is to prove a local uniqueness result for steady states of (1.1). In therecent paper [14] (see also [15, 17]), the author proves quantitative stability inequalitiesfor the gravitational Vlasov-Poisson system that will be crucial in the following. Moreprecisely, 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 controlof the L distance between ¯ f and any function f in terms of the energy H ( f ) − H ( ¯ f ) andthe L distance between the rearrangements ¯ f ∗ and f ∗ of ¯ f and f , respectively.In the following we give some definition and we state the local functional inequality in[14, 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 |{ f > s }| = |{ g > s }| for a.e. s > . Then, the (radially decreasing) rearrangement f ∗ of f is defined as the unique radiallydecreasing function that is equimeasurable to f . In other words, the level sets of f ∗ aregiven by { f ∗ > s } = B r ( s ) , with r ( s ) > s.t. | B r ( s ) | = |{ f > s }| for a.e. s > . The following important result is proved in [14, Theorem 2(ii)].
Theorem 1.2.
Consider ¯ f a compactly supported steady state solution of (1.1) of theform ¯ f ( x, v ) = F ( e ( x, v )) , with e ( x, v ) = | v | φ ¯ f ( x ) , (1.3) where F is a continuous function from R to R + that satisfies the following monotonicityproperty: there exists e < such that F ( e ) = 0 for e ≥ e and F is a C function on ( −∞ , e ) with F ′ < on ( −∞ , e ) . Assume that f ∈ L ∩ L ∞ ( R ) has finite kinetic energyand is sufficiently close to a translation of ¯ f in the following sense: inf z ∈ R k φ f − φ ¯ f ( · − z ) k L ∞ + k∇ φ f − ∇ φ ¯ f ( · − z ) k L < R , (1.4)2 or some suitable constant R > . Then there exists a constant K > , depending onlyon ¯ f , such that inf x ∈ R k f − ¯ f ( · − ( x , k L ≤ k f ∗ − ¯ f ∗ k L + K (cid:2) H ( f ) − H ( ¯ f ) + k f ∗ − ¯ f ∗ k L (cid:3) / . where we denote ¯ f ( · − ( x , f ( x − x , v ) . An immediate consequence is the following estimate, that will be the starting point ofour investigation.
Corollary 1.3.
Let f, ¯ f be as in Theorem 1.2. Assume in addition that f is equimeasurableto ¯ f . Then inf x ∈ R k f − ¯ f ( · − ( x , k L ≤ K [ H ( f ) − H ( ¯ f )] , (1.5)Let ¯ f be the stationary solution as above. Our goal is to understand if, nearby ¯ f , thereexist other stationary solutions of (1.1). Because stationary solutions of (1.1) correspondto 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 byflows of smooth Hamiltonians. Noticing that ¯ f is supported in a ball B ρ ⊂ R × R forsome ρ > H to move ¯ f , it makes sense toconsider Hamiltonians H that are all supported inside B ρ . Hence, one should think ofthese functions H as the “tangent space” at ¯ f that will generate the admissible variations.Let us introduce the following notation:¯ f Hs := (Φ Hs ) ¯ f = ¯ f ◦ Φ H − s ∀ s ∈ R , where s Φ Hs is the Hamiltonian flow of H , namely ∂ s Φ Hs = J ∇ H (Φ Hs ) , J ∈ R × , J = Id − Id ! Φ = Id. (1.6)In other words, s ¯ f Hs is the variation generated by H , and as H vary this generates a“symplectic” neighborhood of ¯ f . Note that, since ¯ f Hs = ¯ f sH , to parameterize a neighbor-hood of ¯ f it is enough to consider the image of the map H ¯ f H . (1.7)We now give some definitions in order to clarify the hypothesis that are needed on theHamiltonian H. Let us start with the definition of the set Inv ¯ f that represents the set ofall the Hamiltonians who acts trivially on ¯ f . Definition 1.4.
Inv ¯ f := { H ∈ C , ( R ) : { H, ¯ f } = 0 } This definition is motivated by the following simple result: This resembles to 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 T x M via the map v γ v (1) , where s γ v ( s ) is the geodesic starting at x with velocity v . emma 1.5. If H ∈ Inv ¯ f then (Φ Ht ) ¯ f = ¯ f , i.e., the Hamiltonian flow of H does notmove ¯ f .Proof. The function ¯ f Hs := (Φ Hs ) ¯ f solves the transport equation ∂ s ¯ f Hs + div( J ∇ H ¯ f Hs ) = 0 , ¯ f Hs | s =0 = ¯ f . Since also ¯ f is a solution to this equation (because div( J ∇ H ¯ f ) = { H, ¯ f } = 0) and thevector field J ∇ H is Lipschitz, f s ≡ ¯ f by uniqueness for the above transport equation.It follows by the lemma above that if H belongs to Inv ¯ f then Φ Hs is not moving ¯ f . Sinceour goal is to use Hamiltonians H to parameterize a neighborhood of ¯ f , there is no reasonto consider H that belong to Inv ( ¯ f ), and it make sense to exclude them. Actually, forsome technical reasons that will be more clear later, we shall need to impose a quantitativeversion of the condition H Inv ¯ f . To do that, we introduce the family of sets A k := { H k∇ H k L ≤ k k{ H, ¯ f }k L } , k ≥ . Remark 1.6.
We note that
Inv ¯ f = T k ∈ N A ck . Indeed, if H ∈ T k ∈ N A ck then k∇ H k L /k ≥k{ H, ¯ f }k L for all k ∈ N . Thus { H, ¯ f } ≡ , which implies that H ∈ Inv ¯f . Viceversa, if H ∈ Inv ¯f then clearly H ∈ T k ∈ N A ck . Because of this observation, we see that H Inv ¯ f ⇐⇒ ∃ k such that H ∈ A k . Motivated by this fact, in the sequel we shall fix k and consider only Hamiltonians thatbelong to A k . Of course this is more restrictive than assuming only H Inv ¯ f but at themoment 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 aretrivially stationary solutions. However, translations in v are automatically controlled bythe kinetic energy and indeed they do not appear in (1.5). To “kill” the space of translationsin x , we will assume that Bar x ( ¯ f H ) = Bar x ( ¯ f ) , where Bar x ( f ) := Z R x f ( x, v ) dx dv ∈ R denotes the “barycenter (in x )” of f . We want to emphasize that this is not a restrictiveassumption on H , since one could remove it by adding to H a Hamiltonian correspondingto translations in the x variable in order to recenter the barycenter of f . Since this wouldnot add major technical difficulties to the proof but may distract the reader from theessential points, we decided to impose this barycenter condition on ¯ f H .As a final consideration, since our goal is prove that there are no steady states to (1.1)in a neighborhood of ¯ f generated via the map (1.7), we shall need to assume that ourHamiltonians H are small in some suitable topology.Our main theorem asserts that, for Hamiltonians small enough in a sufficiently strongSobolev norm that are quantitatively away from Inv ¯ f , there cannot be a stationary pointof the form ¯ f H . 4 heorem 1.7. Let ¯ f be as in (1.3) , where F is a continuous function from R to R + thatsatisfies the following monotonicity property: there exists e < such that F ( e ) = 0 for e ≥ e and F is a C function on ( −∞ , e ) with F ′ < on ( −∞ , e ) . Let ρ > be suchthat supp( ¯ f ) ⊂ B ρ . Also, assume that ¯ f ∈ W ,q ( R ) for some q > . Then the followinglocal uniqueness result for steady states holds:Let r ≥ , and given ρ, ε, k > consider the space of functions N kε := { ¯ f H : supp( H ) ⊂ B ρ , Bar x ( ¯ f H ) = Bar x ( ¯ f ) , H ∈ A k , k H k W r, ≤ ε } . Then, fixed k ∈ N , there is no stationary state for (1.1) in N kε for ε small enough. Starting from the seminal paper of Arnold about the geometric interpretation of the Eulerequations as L -geodesics in the space of measure preserving diffeomorphisms [2], Choffrutand Šverák recently obtained a related result for the 2D Euler equation [6]. The basic ideathere is that, under the evolution given by the 2D incompressible Euler equations, thevorticity is transported by an incompressible vector field, hence the measure of all itssuper-level sets is constant. This means that, given an initial vorticity ω , its evolution ω ( t ) is in the same equimeasurability class of ω . This allows one to foliate the spaceof vorticities into a family of leaves O ω (the equimeasurability class of ω ), and theEuler equations preserve these leaves. In addition, thanks to the Hamiltonian structureof the Euler equations, one can characterize stationary solutions as critical points of theHamiltonian energy E restricted to the orbits.In other words, one has the following situation: the space of vorticities is foliated bythe orbits O ω , and the equilibria are the critical points of E restricted to the orbits. Infinite dimension, the implicit function theorem would give the following: if O ω is smoothnear 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, thedimension of this manifold is equation to the co-dimension of the orbits. In particular,in a non-degenerate situation, the equilibria are locally in one-to-one correspondence withthe orbits. In [6] the authors obtain an analogue of this correspondence in the infinitedimensional context of Euler equations. There, the authors use an infinite dimensionalversion of the implicit function theorem in the space of C ∞ function, via a Nash-Moser’sinteration.With respect to their result, here we have different assumptions and results. These aremotivated by the following: • Since the Vlasov-Poisson system (1.1) is Hamiltonian, given an initial condition f itsevolution f t under the Vlasov-Poisson system will also be in the same equimesurabil-ity class. However, while Hamiltonian maps and measure preserving maps coincide in2-dimension, they are very different in higher dimension (for instance, Hamiltonianmaps preserve the symplectic structure). Because solutions to the 3D Vlasov-Poissonsystems describe a Hamiltonian evolution of particles in the phase-space R × R ,there is no natural reason in this context why there should be only one stationarystate in the same equimeasurability class. In particular, as already observed be-fore, stationary solutions of (1.1) correspond to critical points of H with respect tovariations of ¯ f generated by Hamiltonian flows, and not with respect to arbitrarymeasure preserving variations. This is why we need to look at functions f that canbe connected to ¯ f via a Hamiltonian flow, namely f = ¯ f H for some H .5 The smallness assumption on k∇ r H k L is natural, and actually weaker than the onein [6], since smallness there is measured in the C ∞ topology. • As already mentioned before, the assumption on
Bar x ( ¯ f H ) is not fundamental: onecould easily remove it by replacing it with H − H , where H corresponds to atranslation in the phase space (multiplied by a suitable cut-off function, to make itcompactly supported). What is more essential is our assumption H ∈ A k , and it isunclear at the moment how to remove it. It is our plan to address this issue in afuture work.The goal of the next section is to prove our main theorem. The idea of the proof is the following: first, by exploiting the results in [14], we prove thatif ¯ f H has the same barycenter as ¯ f , then k ¯ f H − ¯ f k L ≤ K [ H ( ¯ f H ) − H ( ¯ f )] . Secondly we show that if ¯ f H is stationary, by a Taylor expansion of s
7→ H ( ¯ f Hs ) we canprove that H ( ¯ f H ) − H ( ¯ f ) ≤ C k∇ H k X for some suitable norm k · k X of ∇ H .Combining these two estimates, we get k ¯ f H − ¯ f k L ≤ C k∇ H k / X . We then relate the two quantities appearing in the above expression: more precisely, wefirst show that k ¯ f H − ¯ f k L ≈ k{ H, ¯ f }k L = k∇ H · J ∇ ¯ f k L , and then we use our quantitative assumption on the fact that H does not belong to Inv ¯ f (namely, H ∈ A k ) to say that k{ H, ¯ f }k L ≈ k∇ H k L . In this way we get k∇ H k L ≤ C k∇ H k / X . Finally, exploiting the smallness k∇ r H k L ≤ ε and interpolation estimates, we are able torelate the two norms above and conclude that k∇ H k L ≤ C k∇ H k δL for some δ >
0, which yields a contradiction when k∇ H k L is small enough.6 .2 Lower bound We recall the definition of the baricenter of f : Bar x ( f ) = Z R x f ( x, v ) dx dv. Lemma 2.1.
Let ¯ f be as in (1.3) , where F is a continuous function from R to R + thatsatisfies the following monotonicity property: there exists e < such that F ( e ) = 0 for e ≥ e and F is a C function on ( −∞ , e ) with F ′ < on ( −∞ , e ) . Let H ∈ C ( R ) andconsider the function ¯ f H := (Φ H ) ¯ f for some H ∈ C , with k∇ H k L ∞ + k∇ H k L ∞ ≤ η .Also, assume that Bar x ( ¯ f ) = Bar x ( ¯ f H ) . (2.1) Then, if η is small enough, k ¯ f H − ¯ f k L ≤ ˆ K [ H ( ¯ f H ) − H ( ¯ f )] , (2.2) where ˆ K depends on the diameter of the support of ¯ f and ¯ f H .Proof. Note that, if η is small enough, the function f = ¯ f H satisfies (1.4). Let ( x ,
0) bethe point where the minimum is achieved in (1.5). By definition,
Bar x ( ¯ f H ( · − ( x , Z R x ¯ f H ( x − x , v ) dx dv = Z R ( x − x ) ¯ f H ( x − x , v ) dx dv + x , = Bar x ( ¯ f H ) + x . hence | x | = | Bar x ( ¯ f H ) − Bar x ( ¯ f H ( · − ( x , | (2.1) = | Bar x ( ¯ f ) − Bar x ( ¯ f H ( · − ( x , |≤ Z R | x | | ¯ f − ¯ f H ( · − ( x , v )) | dx dz ≤ C k ¯ f − ¯ f H ( · − ( x , v )) k L ≤ C [ H ( ¯ f H ) − H ( ¯ f )] / , where we used that ¯ f and ¯ f H are compactly supported, so | ( x, v ) | is bounded on thesupport of ¯ f and ¯ f H ( · − ( x , v )). Thus, k ¯ f − f H k L ≤ k ¯ f − ¯ f ( · + ( x , v ))) k L + k ¯ f ( · + ( x , v ))) − ¯ f H k L ≤ | x | k∇ ¯ f k L + [ H ( ¯ f H ) − H ( ¯ f )] / ≤ C [ H ( ¯ f H ) − H ( ¯ f )] / , which concludes the proof. The aim of this section is to provide an estimate of the difference between the energy of ¯ f and of ¯ f H in terms H , under the additional assumption that ¯ f H is a stationary solutionfor (1.1). More precisely, we prove the following: Proposition 2.2.
Let ¯ f be a compactly supported steady state such that ¯ f ∈ L ∞ ( R ) , andthat ¯ f ∈ W ,q ( R ) for some q > . Let H ∈ C ( R ) . Also, assume that ¯ f H = (Φ H ) ¯ f isa stationary solution for (1.1) . Then the following estimate holds: |H ( f H ) − H ( ¯ f ) | ≤ C k∇ H k L ∞ (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) , where C is a constant depending only on ¯ f .
7s a first step towards the proof of the above result, we aim to give a characterizationof the stationary solutions of (1.1) in terms of the energy of the system H . Lemma 2.3.
Let f : R → R be a compactly supported function. Then f is a steady statefor (1.1) if and only if dds H ( f Hs ) | s =0 = 0 for all H ∈ C ( R ) , where f Hs := (Φ Hs ) f .Proof. Fix H ∈ C ( R ), and consider its flow Φ Hs . To simplify the notation we set Φ Hs =Φ s . Also, it will be convenient to write Φ s = (Φ xs , Φ vs ) : R → R × R .Given a compactly supported function f , we compute the first variation of the Hamil-tonian H around f along f Hs : dds H ( f Hs ) = dds (cid:20) Z R | v | f Hs ( x, v ) dx dv − Z R × K ( x − y ) f Hs ( x, v ) f Hs ( y, w ) dx dv dy dw (cid:21) = dds (cid:20) Z R | Φ vs ( x, v ) | f ( x, v ) dx dv − Z R × K (Φ xs ( x, v ) − Φ xs ( y, v )) f ( x, v ) f ( y, w ) dx dv dy dw (cid:21) = Z R Φ vs ( x, v ) ∂ s Φ vs ( x, v ) f ( x, v ) dx dv − (cid:20)Z R × ∇ x K (Φ xs ( x, v ) − Φ xs ( y, v )) ·· ∂ s (Φ xs ( x, v ) − Φ xs ( y, v )) f ( x, v ) f ( y, w ) dx dv dy dw (cid:21) . Recalling that ∂ s Φ s = ( ∇ v H (Φ s ) , −∇ x H (Φ s )) , we have dds H ( f Hs ) = − Z R Φ vs ( x, v ) · ∇ x H (Φ s ( x, v )) f ( x, v ) dx dv − (cid:20)Z R × ∇ x K (Φ xs ( x, v ) − Φ xs ( y, v )) ·· ( ∇ v H (Φ s ( x, v )) − ∇ v H (Φ s ( y, v ))) f ( x, v ) f ( y, w ) dx dv dy dw (cid:21) . Also, since K ( x − y ) = K ( y − x ) , we see that ∇ x K ( x − y ) = −∇ x K ( y − x ) , so we canrewrite the above expression as dds H ( f Hs ) = − Z R Φ vs ( x, v ) · ∇ x H (Φ s ( x, v )) f ( x, v ) dx dv − Z R × ∇ x K (Φ xs ( x, v ) − Φ xs ( y, v )) · ∇ v H (Φ s ( x, v )) f ( x, v ) f ( y, w ) dx dv dy dw. s preserves the Lebesgue measure and that Φ − s = Φ − s , we can rewritethe first variation in the following way: dds H ( f Hs ) = − Z R v · ∇ x H ( x, v ) f (Φ − s ( x, v )) dx dv − Z R × ∇ x K ( x − y ) · ∇ v H ( x, v ) f (Φ − s ( x, v )) f (Φ − s ( y, w )) dx dv dy dw. (2.3)In particular, since Φ s = Id for s = 0, we see that dds H ( f s ) | s =0 = − Z R v · ∇ x H ( x, v ) f ( x, v ) dx dv − Z R × ∇ x K ( x − y ) · ∇ v H ( x, y )) f ( x, v ) f ( y, w ) dx dv dy dw. (2.4)On the other hand, for f to be a stationary solution for the system (1.1) means thatdiv x ( vf ( x, v )) − div v ( ∇ φ f ( x ) f ( x, v )) = 0 , or equivalently, that for all ψ ∈ C , − Z R v · ∇ x ψ ( x, v ) f ( x, v ) dx dv + Z R ∇ x φ f ( x ) · ∇ v ψ ( x, v ) f ( x, v ) dx dv = 0 . (2.5)Since Z ∇ x K ( x − y ) f ( y, w ) dydw = −∇ φ f , (2.4) proves that dds H ( f Hs ) | s =0 = 0 ⇐⇒ (2.5) holds with ψ = H .Since C functions are dense in C for the C topology, this proves the result. H As a second step, we compute the second variation for H , in line with the computation ofthe first variation (2.3) . Here we consider as initial condition ¯ f and, given a Hamiltonian H ∈ C , we consider ¯ f Hs := ¯ f ◦ Φ H − s . As before, to simplify the notation, we set Φ s = Φ Hs .Also, we define g := ∇ ¯ f · J ∇ H = { H, ¯ f } (2.6)and we observe that dds ¯ f (Φ − s ) = g (Φ − s ) . (2.7)Using the equations (2.3) and (2.7) the second variation is given by the following: d d s H ( ¯ f Hs ) = (cid:20)Z R v · ∇ x H ( x, v ) g (Φ − s ( x, v )) dx dv (cid:21) + (cid:20)Z R × ∇ x K ( x − y ) · ∇ v H ( x, v ) g (Φ − s ( x, v )) ¯ f (Φ − s ( y, w )) dx dv dy dw (cid:21) + (cid:20)Z R × ∇ x K ( x − y ) · ∇ v H ( x, v ) ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) dx dv dy dw (cid:21) . d d s H ( ¯ f Hs ) = (cid:20)Z R v · ∇ x H ( x, v ) g (Φ − s ( x, v )) dx dv (cid:21) (2.8)+ (cid:20)Z R × ∇ x K ( x − y ) · [ ∇ v H ( x, v ) − ∇ w H ( y, w )] ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) dx dv dy dw (cid:21) . As in the previous section, we set ¯ f Hs := ¯ f ◦ Φ H − s and Φ s := Φ Hs . Recall that, by assumption¯ f H , is a stationary solution of (1.1).We now study the Taylor expansion of the Hamiltonian of the gravitational VlasovPoisson system both in ¯ f and in ¯ f H . Since ¯ f and f H are two stationary solutions, itfollows by Lemma 2.3 applied both to ¯ f and to ¯ f H that dds H ( ¯ f Hs ) | s =0 = dds H ( ¯ f ◦ Φ − s ) | s =0 = 0and dds H ( ¯ f Hs ) | s =1 = ddτ H ( ¯ f H ◦ Φ − τ ) | τ =0 = 0 . Hence, by Taylor’s formula, H ( ¯ f H ) = H ( ¯ f ) + Z (1 − s ) d d s H ( ¯ f Hs ) ds and H ( ¯ f ) = H ( ¯ f H ) + Z s d d s H ( ¯ f Hs ) ds. Therefore, H ( ¯ f H ) − H ( ¯ f ) = Z (1 − s ) d d s H ( ¯ f Hs ) ds. Since Z (1 − s ) ds = 0we can add a constant term in the integral, and we get H ( ¯ f H ) − H ( ¯ f ) = 12 Z (1 − s ) d d s H ( ¯ f Hs ) − d d s H ( ¯ f Hs ) | s =0 ! ds. (2.9)Thanks to the latter computation, in order to estimate the left hand side of (2.9), we canestimate d d s H ( ¯ f Hs ) − d d s H ( ¯ f Hs ) | s =0
10n terms of the regularity of the Hamiltonian H, and of ¯ f . Recalling that Φ = Id , wehave the following expression: d d s H ( ¯ f Hs ) − d d s H ( ¯ f Hs ) | s =0 = Z R v · ∇ x H ( x, v ) g (Φ − s ( x, v )) dx dv (2.10) − Z R v · ∇ x H ( x, v ) g ( x, v ) dx dv + Z R × ∇ x K ( x − y ) · [ ∇ v H ( x, v ) − ∇ w H ( y, w )] ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) dx dv dy dw − Z R × ∇ x K ( x − y ) · [ ∇ v H ( x, v ) − ∇ w H ( y, w )] ¯ f ( x, v ) g ( y, w ) dx dv dy dw. ≤ T + T , where T := (cid:12)(cid:12)(cid:12)(cid:12) Z R v · ∇ x H ( x, v ) [ g (Φ − s ( x, v ) − g ( x, v )] dx dv (cid:12)(cid:12)(cid:12)(cid:12) ,T : = (cid:12)(cid:12)(cid:12)(cid:12)Z R × ∇ x K ( x − y ) · [ ∇ v H ( x, v ) − ∇ w H ( y, w )] ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) dx dv dy dw − Z R × ∇ x K ( x − y ) · [ ∇ v H ( x, v ) − ∇ w H ( y, w )] ¯ f ( x, v ) g ( y, w ) dx dv dy dw (cid:12)(cid:12)(cid:12)(cid:12) . We begin by controlling T .By the Fundamental Theorem of Calculus we have that T ≤ C k∇ x H k L ∞ Z R (cid:12)(cid:12)(cid:12)(cid:12) g (Φ − s ( x, v )) − g ( x, v ) (cid:12)(cid:12)(cid:12)(cid:12) dx dv ≤ C k∇ x H k L ∞ Z R (cid:18)Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ g (Φ − τs ( x, v )) · ∂ τ Φ − τs ( x, v ) (cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:19) dx dv Using that ∂ s Φ s = J ∇ H (Φ s ) and that Φ s preserves the volumes, we get T ≤ C k∇ H k ∞ Z R (cid:18)Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ g (Φ − τs ( x, v )) (cid:12)(cid:12)(cid:12)(cid:12) dτ (cid:19) dx dv = C k∇ H k ∞ Z R (cid:18)Z (cid:12)(cid:12)(cid:12)(cid:12) ∇ g ( x, v ) (cid:12)(cid:12)(cid:12)(cid:12) ds (cid:19) dx dv ≤ C k∇ H k ∞ k∇ g k L . Thus, T ≤ C k∇ H k ∞ k∇ g k L . By the definition of g in (2.6), we have that ∇ g = ∇ H · J ∇ ¯ f + ∇ H · J ∇ ¯ f . (2.11)Therefore, k∇ g k L ≤ k∇ H k L ∞ k∇ ¯ f k L + k∇ H k L ∞ k∇ ¯ f k L ≤ C (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) , C depends on k∇ ¯ f k L and k∇ ¯ f k L . In conclusion, the first term T can be estimateas follows: T ≤ C k∇ H k ∞ k (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) . (2.12)We now estimate the second term: T ≤ k∇ H k L ∞ Z R × |∇ x K ( x − y ) | | ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) − ¯ f ( x, v ) g ( y, w ) | dx dv dy dw. Adding and subtracting ¯ f ( x, v ) g (Φ − s ( y, w )), we can bound T ≤ k∇ H k L ∞ Z R × |∇ x K ( x − y ) | | ¯ f (Φ − s ( x, v )) g (Φ − s ( y, w )) − ¯ f ( x, v ) g (Φ − s ( y, w )) | dx dv dy dw + k∇ H k L ∞ Z R × |∇ x K ( x − y ) | | ¯ f ( x, v ) g (Φ − s ( y, w )) − ¯ f ( x, v ) g ( y, w ) | dx dv dy dw ≤ C k∇ H k L ∞ k g k L ∞ Z R × B R |∇ x K ( x − y ) | | ¯ f (Φ − s ( x, v )) − ¯ f ( x, v ) | dx dv dy dw + C k∇ H k L ∞ k ¯ f k L ∞ Z B R × R |∇ x K ( x − y ) | | g (Φ − s ( y, w )) − g ( y, w ) | dx dv dy dw. Using as before the Fundamental Theorem of Calculus and the fact that Φ s is measurepreserving, we have T ≤ C k∇ H k L ∞ k g k L ∞ Z R × B R |∇ x K ( x − y ) |·· (cid:18)Z |∇ ¯ f (Φ − τs ( x, v )) · ∂ s Φ − τs ( x, v ) | dτ (cid:19) dx dv dy dw + C k∇ H k L ∞ k ¯ f k L ∞ Z B R × R |∇ x K ( x − y ) |·· (cid:18)Z |∇ g (Φ − τs ( y, w )) · ∂ s Φ − τs ( y, w ) | dτ (cid:19) dx dv dy dw. By the definition of g , (2.6) we obtain T ≤ C k∇ H k L ∞ k g k L ∞ Z R × B R |∇ x K ( x − y ) | (cid:18)Z | g (Φ − τs ( x, v )) | dτ (cid:19) dx dv dy dw + C k∇ H k L ∞ k ¯ f k L ∞ Z B R × R |∇ x K ( x − y ) |·· (cid:18)Z |∇ g (Φ − τs ( y, w )) · J ∇ H (Φ − τs )( − s ) | dτ (cid:19) dx dv dy dw, and using Hölder inequality we get T ≤ C k∇ H k L ∞ k g k ∞ Z B R × B R |∇ x K ( x − y ) | dx dy + C k∇ H k ∞ k ¯ f k L ∞ k∇ g k L q (cid:18)Z B R × B R |∇ x K ( x − y ) | p dx dy (cid:19) p , p and q are conjugate exponents. In order to have integrability of the gradient ofthe kernel K , we need p < . Therefore, in the previous estimates we need to assume that k∇ g k L q is finite for some q > . Thus T ≤ C (cid:0) k∇ H k L ∞ k g k L ∞ + k∇ H k L ∞ k∇ g k L q (cid:1) , q > , where C depends on k ¯ f k L ∞ . As in the estimate of the term T we use (2.6) and (2.11) toget k g k L ∞ ≤ k∇ H k L ∞ k∇ ¯ f k L ∞ ≤ C k∇ H k L ∞ and k∇ g k L q ≤ k∇ H k L ∞ k∇ ¯ f k L q + k∇ H k L ∞ k∇ ¯ f k L q ≤ C (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) , where C depends on k∇ ¯ f k L q and k∇ ¯ f k L q for some q >
3. Since by assumption ¯ f ∈ W ,q ( R ) for some q >
3, we have prove that T ≤ C (cid:0) k∇ H k L ∞ + k∇ H k L ∞ k∇ H k L ∞ (cid:1) . (2.13)Hence, combining (2.10) , (2.12) , and (2.13) we get |H ( ¯ f H ) − H ( ¯ f ) | ≤ C (cid:0) k∇ H k L ∞ + k∇ H k L ∞ k∇ H k L ∞ (cid:1) , where C depends only on k ¯ f k L ∞ and on k∇ ¯ f k L q and k∇ ¯ f k L q , for some q > k ¯ f − ¯ f H k L and k g k L Lemma 2.4.
Let ¯ f be a compactly supported steady state such that ∇ ¯ f , ∇ ¯ f ∈ L ( R ) .Also, let H ∈ C ( R ) , and define g as in (2.6) . Set ¯ f H := ¯ f ◦ Φ H − . Then (cid:12)(cid:12)(cid:12) k ¯ f − ¯ f H k L − k g k L (cid:12)(cid:12)(cid:12) ≤ C k∇ H k L ∞ (cid:16) k∇ H k L ∞ + k∇ H k L ∞ e k∇ H k L ∞ (cid:17) , where C depends only on ¯ f .Proof. Set ¯ f Hs := ¯ f ◦ Φ H − s . Then, by the definition of the flow Φ s (see (1.6)), we deduce ∂ s ¯ f Hs = − J ∇ H · ∇ ¯ f Hs , ∂ s ¯ f Hs | s =0 = − J ∇ H · ∇ ¯ f , (2.14)therefore ¯ f H − ¯ f = Z ∂ s ¯ f Hs ds = ∂ s ¯ f Hs | s =0 + Z ( ∂ s ¯ f Hs − ∂ s ¯ f Hs | s =0 ) ds (2.14) = − J ∇ H · ∇ ¯ f + Z ( ∂ s ¯ f Hs − ∂ s ¯ f Hs | s =0 ) ds (2.6) = − g + Z ( ∂ s ¯ f Hs − ∂ s ¯ f Hs | s =0 ) ds. Thus, (cid:12)(cid:12)(cid:12)(cid:12) Z R | ¯ f H − ¯ f | dx dv − Z R | g | dx dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z R | ∂ s ¯ f Hs − ∂ s ¯ f Hs | s =0 | dx dv ds. (cid:12)(cid:12)(cid:12)(cid:12) Z R | ¯ f H − ¯ f | dx dv − Z R | g | dx dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z Z R | J ∇ H · ∇ ¯ f Hs − J ∇ H · ∇ ¯ f | dx dv ds ≤ Z ds Z R |∇ H | |∇ ¯ f ◦ Φ H − s · ∇ Φ H − s − ∇ ¯ f | dx dv. Adding and subtracting ∇ ¯ f ◦ Φ H − s , this gives (cid:12)(cid:12)(cid:12)(cid:12) Z R | ¯ f H − ¯ f | dx dv − Z R | g | dx dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ k∇ H k L ∞ (cid:18) Z ds Z R |∇ ¯ f ◦ Φ H − s − ∇ ¯ f | dx dv + Z ds Z R |∇ ¯ f ◦ Φ H − s | |∇ Φ H − s − Id | dx dv (cid:19) =: k∇ H k L ∞ ( A + B ) . We now estimate the terms A and B . By the Fundamental Theorem of Calculus, A = Z ds Z R (cid:12)(cid:12)(cid:12)(cid:12)Z s ddτ ∇ ¯ f ◦ Φ H − τ dτ (cid:12)(cid:12)(cid:12)(cid:12) dx dv = Z ds Z R (cid:12)(cid:12)(cid:12)(cid:12)Z s ∇ ¯ f ◦ Φ H − τ · J ∇ H ◦ Φ H − τ dτ (cid:12)(cid:12)(cid:12)(cid:12) dx dv ≤ k∇ H k L ∞ Z ds Z R Z s |∇ ¯ f | ◦ Φ H − τ dτ dx dv By Fubini, we can rewrite the last integral above as Z ds Z s dτ Z R |∇ ¯ f | ◦ Φ H − τ dx dv and because Φ H − τ is measure preserving we deduce that the term above is equal to Z ds Z s dτ Z R |∇ ¯ f | dx dv = 12 k∇ ¯ f k L . Hence, in conclusion, A ≤ k∇ ¯ f k L k∇ H k L ∞ . For B , we want to estimate the term |∇ ¯ f ◦ Φ H − s | |∇ Φ H − s − Id | . Differentiating the equation in (1.6), we deduce that ( ∂ s ∇ Φ Hs = J ∇ H (Φ Hs ) · ∇ Φ Hs ∇ Φ H ( x, v ) = Id . (2.15)Thus, ( dds |∇ Φ Hs | ≤ k∇ H k L ∞ |∇ Φ Hs ||∇ Φ H | = 1 (2.16)and by Gronwall’s inequality |∇ Φ Hs | ≤ e s k∇ H k L ∞ . (2.17)14herefore, |∇ Φ Hs − Id | = (cid:12)(cid:12)(cid:12)(cid:12)Z s ∂ τ ∇ Φ Hτ dτ (cid:12)(cid:12)(cid:12)(cid:12) (2.16) ≤ k∇ H k L ∞ sup τ ∈ [0 ,s ] |∇ Φ Hτ | (2.17) ≤ k∇ H k L ∞ e s k∇ H k L ∞ , which yields B ≤ k∇ H k L ∞ e k∇ H k L ∞ Z R |∇ ¯ f | dx dv. Combining the latter estimates, we obtain (cid:12)(cid:12)(cid:12)(cid:12) Z R | f − ¯ f | dx dv − Z R | g | dx dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ C k∇ H k L ∞ (cid:16) k∇ H k L ∞ + k∇ H k L ∞ e k∇ H k L ∞ (cid:17) , where C is a constant depending only on k∇ ¯ f k L and k∇ ¯ f k L . In this section we combine the upper and lower bounds obtained in Sections 2.2 and 2.3with interpolation estimates to obtain a contradiction to the existence of a stationarysolution ¯ f H with H as in the statement of Theorem 1.7.We begin by recalling that, by the Sobolev’s embedding, given R > u : B R → R compactly supported, k u k L ∞ ( B R ) ≤ C n,R k∇ s u k L ( B R ) ∀ s > n/ . (2.18)In particular, since n = 6, if H is as in the statement of the theorem then k∇ H k L ∞ + k∇ H k L ∞ is as small a desired provided we choose ε small enough. This allows us toapply Lemma 2.1, that combined with Lemma 2.4 yields following bound on g : k g k L ≤ C (cid:18)q H ( ¯ f H ) − H ( ¯ f ) + k∇ H k L ∞ + k∇ H k L ∞ k∇ H k L ∞ e k∇ H k L ∞ (cid:19) . Then, using Proposition 2.2, k g k L ≤ C (cid:18) k∇ H k L ∞ (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) + k∇ H k L ∞ + k∇ H k L ∞ k∇ H k L ∞ e k∇ H k L ∞ (cid:19) . We now use the assumption H ∈ A k to get k∇ H k L ≤ Ck (cid:18) k∇ H k L ∞ (cid:16) k∇ H k L ∞ + k∇ H k L ∞ (cid:17) + k∇ H k L ∞ + k∇ H k L ∞ k∇ H k L ∞ e k∇ H k L ∞ (cid:19) . (2.19)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 k∇ H k X ≤ C k∇ H k / X , which is impossible when H is small enough. Thus, the next step is to use interpolationestimates to compare the different norms of ∇ H appearing in (2.19). More precisely, wewant to use the following interpolation estimates.15 emma 2.5. For any smooth compactly supported function u : R n → R , k∇ ℓ u k L ≤ k u k − ℓ/mL k∇ m u k ℓ/mL ∀ ≤ ℓ ≤ m. Proof.
The proof is immediate by Fourier: using Hölder inequality with the conjugateexponents m/ℓ and ( m − ℓ ) /ℓ , we get Z | ξ | ℓ | ˆ u | = Z (cid:0) | ξ | ℓ | ˆ u | ℓ/m (cid:1) | ˆ u | m − ℓ ) /m ≤ k| ξ | ℓ | ˆ u | ℓ/m k L m/ℓ k| ˆ u | m − ℓ ) /m k L ( m − ℓ ) /ℓ = (cid:18)Z | ξ | m | ˆ u | (cid:19) ℓ/m (cid:18)Z | ˆ u | (cid:19) − ℓ/m . Since k∇ k u k L ( R n ) = k| ξ | k ˆ u k L ( R n ) for all k ≥
0, the result follows.Since (2.19) involves L and L ∞ norms, to apply Lemma 2.5 we use shall use otherinterpolation inequalities. More precisely, we recall the classical Nash inequality: k u k /nL ( R n ) ≤ C n k u k /nL ( R n ) k∇ u k L ( R n ) . (2.20)We now set n = 6, and we let s be a number larger than n/ ∂ i H : R n → R , i = 1 , . . . , n , we get k∇ H k /nL ≤ C k∇ H k /nL k∇ H k L . Let us recall that, by assumption, H is supported in B ρ . Hence, we can apply (2.18) bothwith u = ∂ i H and u = ∂ ij H to get k∇ H k L ∞ ≤ C k∇ s +1 H k L , k∇ H k L ∞ ≤ C k∇ s +2 H k L . Note also that, by Poincaré inequality in B ρ , k∇ s +1 H k L ≤ C k∇ s +2 H k L . Combiningall these estimates with (2.19), we get k∇ H k /nL ≤ Ck (cid:16) k∇ s +2 H k / L + k∇ s +2 H k L + k∇ s +2 H k L e k∇ s +2 H k L (cid:17) /n k∇ H k L . (2.21)To conclude we recall that, by assumption, k H k W r, ≤ ε , where r ≥
22, and we want toobtain a contradiction when ǫ is sufficiently small. To this aim, we first note that, for s ≤ r − k∇ s +2 H k L ≤ k H k W r, ≤ ε ≪ . This implies that the quadratic terms in (2.21) are much smaller than the term with thepower 3 /
2, therefore (2.21) yields k∇ H k /nL ≤ Ck k∇ s +2 H k /nL k∇ H k L . Then we apply Lemma 2.5 with u = ∂ i H , ℓ = s + 1, and m = r − k∇ s +2 H k L ≤ C k∇ H k r − s − r − L k∇ r H k s +1 r − L ≤ Cε s +1 r − k∇ H k r − s − r − L , therefore k∇ H k /nL ≤ Ck (cid:16) k∇ H k r − s − r − L ε s +1 r − (cid:17) /n k∇ H k L . (2.22)Also, by Lemma 2.5 with u = ∂ i H , ℓ = 1, m = r −
1, we have k∇ H k L ≤ C k∇ H k r − r − L k∇ r H k r − L ≤ Cε r − k∇ H k r − r − L . (2.23)16hus, combining (2.22) and (2.23), we obtain k∇ H k /nL ≤ Ckε s +1) /nr − k∇ H k r − s − n ( r − + r − r − L . We finally choose s . Since s is any exponent larger than n/ r − ≥ s = 4. Then, the inequality above becomes k∇ H k / L ≤ Ckε s +1) /nr − k∇ H k r − r − + r − r − L = Ckε r − k∇ H k r − r − L . Since r ≥
22 by assumption, we see that r − r − ≥ /
3, thus we obtain1 ≤ Ckε r − k∇ H k r − r − − L ≤ Ckε r − + r − r − − = Ckε / , which is false for ǫ small enough. This shows the desired contradiction and completes theproof. —————— Acknowledgments:
The author is grateful to Clément Mouhot for proposing to me thisbeautiful problem and for interesting discussions. Also, I wish to thank Pierre Räphael foruseful comments during the preparation of this manuscript. The author would also liketo acknowledge the L’Oréal Foundation for partially supporting this project by awardingthe L’Oréal-UNESCO
For Women in Science France fellowship . References [1] L. Ambrosio, M. Colombo, and A. Figalli. On the Lagrangian structure of transportequations: the Vlasov-Poisson system. Preprint, 2015.[2] V. I. Arnold. Sur la géométrie différentielle des groupes de Lie de dimension infinie etses applications à l’hydrodynamique des fluides parfaits.
Ann. Inst. Fourier (Grenoble) ,16:319–361, 1966.[3] A. A. Arsenev. Existence in the large of a weak solution of Vlasov’s system of equations.
Ž. Vyčisl. Mat. i Mat. Fiz. , 15:136–147, 276, 1975.[4] J. Binney and S. Tremaine.
Galactic Dynamics.
Princeton: Princeton University Press,1987.[5] A. Bohun, F. Bouchut, and G. Crippa. Lagrangian solutions to the Vlasov-Poissonequation with L density. J. Differential Equations
Local structure of the set of steady-state solutions to the 2Dincompressible Euler equations . Geom. Funct. Anal.
C. R. Acad. Sci. ParisSér. I Math. , 307:655-658, 1988.[8] R. J. DiPerna and P.-L. Lions. Global weak solutions of kinetic equations.
Rend. Sem.Mat. Univ. Politec. Torino , 46:259–288, 1988.[9] R. J. DiPerna & P.-L. Lions. Global weak solutions of Vlasov-Maxwell systems.
Comm.Pure Appl. Math. , 42:729–757, 1989. 1710] A. M. Fridman, V. L. Polyachenko.
Physics of Gravitating Systems I . New York:Springer-Verlag 1984[11] E. Horst and R. Hunze. Weak solutions of the initial value problem for the unmodifiednonlinear Vlasov equation.
Math. Methods Appl. Sci.
TrudyMat. Inst. Steklov. , 60:181–194, 1961.[13] J. Jeans. On the theory of star-streaming and the structure of the universe.
Mon.Not. R. Astron. Soc.
Invent. Math. , 187(1):145–194, 2012.[16] P.-L. Lions and B. Perthame. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system.
Invent. Math. , 105:415–430, 1991.[17] C. Mouhot. Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel(d’après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al.). (French) [Orbital stabilityfor the gravitational Vlasov-Poisson system (after Lemou-Méhats-Raphaël, Guo, Lin,Rein et al.)] Séminaire Bourbaki. Vol. 2011/2012. Exposés 1043-1058.
Astérisque
No.352 (2013), Exp. No. 1044, vii, 35-82.[18] K. Pfaffelmoser. Global classical solutions of the Vlasov-Poisson system in threedimensions for general initial data.
J. Differential Equations , 95(2):281–303, 1992.[19] J. Schaeffer. Global existence of smooth solutions to the Vlasov-Poisson system inthree dimensions.
Comm. Partial Differential Equations , 16(8-9):1313–1335, 1991.[20] S. Ukai and T. Okabe. On classical solutions in the large in time of two-dimensionalVlasov’s equation.
Osaka J. Math. , 15:245–261, 1978.[21] A. A. Vlasov. Zh. Eksper.
Teor. Fiz.