Stability estimates for the Vlasov–Poisson system in p$p$ ‐kinetic Wasserstein distances

We extend Loeper's L2$L^2$ ‐estimate (Theorem 2.9 in J. Math. Pures Appl. (9) 86 (2006), no. 1, 68–79) relating the force fields to the densities for the Vlasov–Poisson system to Lp$L^p$ , with 1


General overview
Monge-Kantorovich distances, also known as Wasserstein distances, are used extensively in kinetic theory, in particularly in the context of stability, convergence to equilibrium and meanfield limits.A first celebrated result for the 1-Monge-Kantorovich distance is due to Dobrushin [2, Theorem 1], who proved the well-posedness for Vlasov equations with  1,1 potentials.An explanation of Dobrushin's stability estimate and its consequences on the mean-field limit for the Vlasov equation can be found in [5, chapter 1] and [11, chapter 3], and we refer to [7, section 3] for a survey on well-posedness for the Vlasov-Poisson system.
Regarding the 2-Wasserstein distance, Loeper proved [13, Theorem 1.2] a uniqueness criterion for solutions with bounded density based on a 2-Wasserstein distance stability estimate using both a link between the Ḣ−1 -seminorm and the 2-Wasserstein distance, and the fact that the Coulomb kernel is generated by a potential solving the Poisson equation.In addition to the Vlasov-Poisson system, this criterion gives a new proof of uniqueness à la Yudovich for 2 Euler.Beyond bounded density, Loeper's uniqueness criterion has been extended for some suitable Orlicz spaces using the 1-Monge-Kantorovich distance by Miot [16,Theorem 1.1] and Miot and Holding [9,Theorem 1.1].
On the Torus, Loeper's criterion was improved by Han-Kwan and Iacobelli [8,Theorem 3.1] for the Vlasov-Poisson system, and more recently for the Vlasov-Poisson system with massless electrons by Griffin-Pickering, Iacobelli [6,Theorem 4.1].
The aim of this work is twofold.The first goal is to generalize Loeper's 2-Wasserstein distance stability estimate to -Wasserstein distances for 1 <  < +∞.The second goal is to extend the recent stability estimate [10,Theorem 3.1] by the first author relying on the newly introduced kinetic Wasserstein distance [10,Theorem 3.1] to kinetic Wasserstein distances of order 1 <  < +∞.

Definitions and main results
We first recall the classical Wasserstein distance (see [21, chapter 6]) on the product space  × ℝ  , with  denoting in the sequel either the -dimensional torus   or the Euclidean space ℝ  : Definition 1.1.Let ,  be two probability measures on  × ℝ  .The Wasserstein distance of order , with  ⩾ 1, between  and  is defined as | − |  + | − |  (, , , ) where Π(, ) is the set of couplings; that is, the set of probability measures with marginals  and .A coupling is said to be optimal if it minimizes the Wasserstein distance.
(1) A new   -estimate for the difference of force fields.Loeper estimates the  2 -norm [13, Theorem 2.9] of the difference of force fields with the Wasserstein distance between the densities.We extend the  2 -estimate to   for 1 <  < +∞ using the Helmholtz-Weyl decomposition of   () =   () ⊕   () into its hydrodynamic space that we recall (see [3,  ), while an orthogonal decomposition in  2 is always possible, whatever the domain is.
The validity of the Helmholtz-Weyl decomposition implies the existence of an Helmholtz-Weyl bounded linear projection operator (see [3,Remark III.1.1]) with range   () and with   () as null space.More precisely, there is a constant    > 0 that only depends on  and  such that for all  ∈   (), it holds ‖  ()‖   () ⩽    ‖‖   () . (1.3) Using optimal transport techniques, Loeper manages to link the strong dual homogeneous Sobolev norm and Wasserstein distances between densities, and we recall those notions: Definition 1.5.Let 1 <  < +∞.The homogeneous Sobolev space is the space where [⋅] ∶= {⋅ + ;  ∈ ℝ} denotes the equivalence class of functions up to a constant, together with the norm This is a Banach space for which the equivalence classes of test functions are dense in it (see [17,Theorem 2.1]).
First, we extend this connection for densities to   .Using the machinery of Helmholtz-Weyl decomposition, we generalize [13, Lemma 2.10] into the following: Lemma 1.7.Let  1 ,  2 ∈  ∞ () be two probability measures, and let   satisfy Δ  =   for  = ℝ  , or Δ  =   − 1 for  =   , with  = 1, 2,  = ±1, in the distributional sense.Let 1 <  < +∞.Then there is a constant  HW > 0 that only depends on  and  such that (1.4) Second, we adapt Loeper's argument of the  2 -estimate [13, Theorem 2.9](see also [20,Proposition 1.1] in bounded convex domains) relating negative homogeneous Sobolev norms to Wasserstein distances with this new link on force fields to get the new   -estimate allowing us to generalize stability estimates; Proposition 1.8.Let  1 ,  2 ∈  ∞ () be two probability measures, and let   satisfy Δ  =   for  = ℝ  , or Δ  =   − 1 for  =   , with  = 1, 2,  = ±1, in the distributional sense.Let 1 <  < +∞.Then there is a constant  HW > 0 that only depends on  and  such that (2) Loeper's stability estimate in   .Loeper noted [13, Lemma 3.6] that both the Wasserstein distance of order two of the solutions and of the associated densities are bounded by a flow quantity  given by and the bounds read as Loeper uses the quantity () together with the  2 -estimate on the force fields to prove the stability estimate [13, Theorem 1.2] leading to the uniqueness of weak solutions.By modifying the quantity () to where   ∈ Π( 1 (),  2 ()) and  0 is an optimal   coupling (see [6, section 4] for a construction of   ), we are able to generalise Loeper's stability estimate [13, Theorem 1.2], and [8, Theorem 3.1] both on the torus   and on the whole space ℝ  , to any Wasserstein distance of order , with 1 <  < +∞; Theorem 1.9.Let  1 ,  2 be two weak solutions to the Vlasov-Poisson system on  with respective densities Let 1 <  < +∞, and set which is assumed to be in  1 ([0, )) for some  > 0. Then there is a constant  L > 0 that only depends on  and  such that if which is assumed to be in  1 ([0, )) for some  > 0. Then there is a universal constant  0 > 0 and a constant  KW > 0 that depends only on  and  such that if then . (1.12) The improvement of this stability estimate (1.12) of Theorem 1.11 via -kinetic Wasserstein distance compared to Loeper's stability estimate in   (1.9) of Theorem 1.9 lies in the order of magnitude of the time interval in which the two solutions are close to each other in Wasserstein distance.Indeed, if    ( 1 (0),  2 (0)) =  ≪ 1, then Loeper First, we consider the torus case  =   : We use the Helmholtz-Weyl decomposition given by Theorem 1.4 to write any ℝ  -valued test function Φ ∈  ∞  (  ) as Φ = ∇ + g, where ∇ ∈   ′ (  ) and g ∈   ′ (  ) with 1∕ + 1∕ ′ = 1.By definition, there is a divergence-free sequence of test functions (g  ) ∈ℕ whose   ′ -limit is g.By continuity of the force fields (see [8,Lemma 3.2]), ∇ 1 − ∇ 2 ∈  ∞ (  ), and in particular ∇ 1 − ∇ 2 ∈   (  ).An integration by parts yields As the projection operator   ′ ∶ Φ ↦ g is bounded from   ′ (  ) to   ′ (  ), we have that where    ′ is the constant from (1. the interpolant measure between  1 and  2 , where  is the optimal transport map of Theorem 2.1.Let  ∈  ∞  () be a test function.By the properties of pushforward of measures, it follows immediately that Lebesgue's dominated convergence theorem yields Now, by using Hölder inequality with respect to the measure  1 , we get () The second term in the product is exactly   ( 1 ,  2 ) by Theorem 2.1.For the first one, thanks to [18,Remark 8], the  ∞ -norm of the interpolant is controlled by the one of the two measures; Therefore, Combining the above estimate with the fact that ∫  2 −  1 = 0 and Fubini's theorem yields .
By restricting to quotient test functions [] such that ‖∇[]‖   ′ () ⩽ 1, we get the strong dual homogeneous norm so that for a vector field   related to the Wasserstein distance through Differentiating both sides of Poisson's equation gives and integrating by parts against     itself as test function yields and the conclusion follows after integrating over  ∈ [1,2].
Even though there is a   version of Benamou-Brenier formula [19, Theorem 5.28], there is no analog test function that allows to mimic this proof for   .

Loeper's estimate revisited
The proof of Loeper's stability estimate in   on the torus   is similar to [8, Theorem
Proof of Theorem 1.11.
The last two terms are estimated using Hölder's inequality with respect to the measure  0 , and we have Recall the separation of the difference of force fields; where First, consider the torus case  =   : We estimate  1 (3.8) using the nondecreasing concave function on [0, +∞) given by