Convergence to Equilibrium in Wasserstein distance for damped Euler equations with interaction forces
aa r X i v : . [ m a t h . A P ] J u l CONVERGENCE TO EQUILIBRIUM IN WASSERSTEIN DISTANCE FORDAMPED EULER EQUATIONS WITH INTERACTION FORCES
JOS´E A. CARRILLO, YOUNG-PIL CHOI, AND OLIVER TSE
Abstract.
We develop tools to construct Lyapunov functionals on the space of probabilitymeasures in order to investigate the convergence to global equilibrium of a damped Euler systemunder the influence of external and interaction potential forces with respect to the 2-Wassersteindistance. We also discuss the overdamped limit to a nonlocal equation used in the modellingof granular media with respect to the 2-Wasserstein distance, and provide rigorous proofs forparticular examples in one spatial dimension. Introduction
In this paper, we develop tools to analyse the large-time behavior of second-order dynamicsthat describe evolutions in the space of probability measures driven by a free energy F . Moreprecisely, we consider the evolution of probability measures µ with Lebesgue densities ̺ and theirvelocities u described by damped Euler systems with damping parameter γ > ∂ t ̺ t + ∇ · ( ̺ t u t ) = 0 , ( t, x ) ∈ R + × R d ,∂ t ( ̺ t u t ) + ∇ · ( ̺ t u t ⊗ u t ) = − ̺ t ∇ ( δ µ F )( µ t ) − γ̺ t u t , (1)subject to initial density and velocity conditions(2) ( ̺ t , u t ) | t =0 = ( ̺ , u ) for x ∈ R d , where δ µ F = δ F /δµ is the variational derivative of a free energy F acting on probability measures µ that are absolutely continuous with respect to the Lebesgue measure on R d with Radon–Nikodymderivative dµ/dx = ̺ , given by F ( µ ) := Z R d U ( ̺ ) dx + Z R d V ( x ) dµ ( x ) + 12 Z Z R d × R d W ( x − y ) dµ ( x ) dµ ( y ) . (3)Here, U denotes an increasing function describing the internal energy of the density dµ/dx = ̺ , V : R d → R and W : R d → R are the confinement and the interaction potentials respectively. Inthis case, the variational derivative of F in (3) is given by [1, 62]( δ µ F )( µ ) = U ′ ( ̺ ) + V ( x ) + W ⋆ µ.
In Section 2.1 we will make sufficient assumptions on U , V and W in order to ensure the existenceof a stationary measure µ ∞ for the damped Euler system (1) that may also be characterized as aminimizer of the free energy F . In the sequel, we will identify the measure µ with its density ̺ assoon as the measure µ has a Lebesgue density.While the well-posedness of (1) remains a challenging open question—even for restricted classesof initial data (2)—and not dealt with in this paper, damped Euler systems without confining andinteraction forces ( V = W ≡
0) have been investigated in multiple contexts. For instance, theglobal existence of BV and L ∞ entropy weak solutions for the one-dimensional case were addressedin [29, 40] and [31, 42] respectively. The asymptotic behavior of solutions were also discussed in[39, 41, 42, 43, 58]. For the multi-dimensional case, global existence and pointwise estimates ofsolutions based on the Green’s function approach together with energy estimates were obtainedin [63], while the global existence of classical solutions and the large-time behavior of solutionswere studied in [8, 37, 60] under the smallness assumptions on the initial data. We also refer to Mathematics Subject Classification.
Key words and phrases.
Convergence to equilibrium, Euler equations, overdamped limit, Wasserstein distance. [35, 51] for the study of global well-posedness and asymptotic behavior of solutions based on theframework of Besov spaces. We refer the reader to [25] for a general survey of the Euler equations.An initial attempt at proving equilibration results with explicit decay rates was conducted in[48] for the case U ( s ) = s log s , V ( x ) = | x | / W ≡
0. There, the authors used entropydissipation methods to heuristically derive functional inequalities that provided the decay ratesto equilibrium under relatively strong global regularity assumptions on ( ̺, u ). The results in [48]indicate a convergence behavior similar to spatially inhomogeneous entropy-dissipating kineticequations where hypocoercivity of the operators involved played an important role in determiningconvergence to equilibrium [30, 32, 33, 47, 46, 61]. There, the exponential decay rate λ = λ ( γ )has the property that λ → γ → γ → + ∞ , i.e., the best equilibration rate for (1) holdsfor some γ ∈ (0 , ∞ ).A related equation is the well-known aggregation-diffusion equation ∂ t ¯ ̺ t = ∇ · (cid:0) ¯ ̺ t ∇ ( δ µ F )(¯ ̺ t ) (cid:1) , ( t, x ) ∈ R + × R d , (4)with ¯ ̺ t a probability density on R d . The long-time asymptotics for (4) are given by the minimizerof the free energy F as t → ∞ , whenever the potentials are uniformly convex as in one of the earliestapplications of these equations in granular media modelling [4, 3, 23, 50]. Both equations, (1) and(4), also find numerous applications in mathematical biology and technology such as swarmingof animal species, cell movement by chemotaxis, self-assembly of particles and dynamical densityfunctional theory (DDFT)—see for instance [21, 36, 38, 49] and the references therein.Explicit equilibration rates for the aggregation-diffusion equation (4) have been derived usingentropy dissipation methods [23, 27] or, more recently, by using contraction estimates in the2-Wasserstein distance [9, 10, 11] under convexity assumptions on the potentials. The entropydissipation method is based on studying the time derivative of an appropriate Lyapunov functionalalong the flow generated by the aggregation-diffusion equation (4), and using functional inequalitiesto bound the dissipation from below in terms of the Lyapunov functional. Actually, the solutionsof (4) formally satisfy the free energy dissipation(5) ddt F (¯ ̺ t ) = − Z R d |∇ ( δ µ F )(¯ ̺ t ) | ¯ ̺ t dx . Heuristically, one may view solutions of (4) as gradient flows of the free energy F on the spaceof probability measures, endowed with the 2-Wasserstein distance [1, 24, 59]. On the other hand,the method of using Wasserstein contraction estimates introduced in [10, 11, 24], is based oncomparing the 2-Wasserstein distance with its dissipation along the evolution. This theory can dealwith displacement convex functionals, as introduced in the seminal paper of McCann [56], whichinclude certain non uniformly convex potentials [19, 20]. However, much less is known in terms ofrates of convergence if uniform convexity of the potentials is not present, see [2, 5, 6, 15, 22] andthe references therein for blow-up time, equilibrium solutions and qualitative covergence results.In fact, equation (4) may be seen as an overdamped limit ( γ → + ∞ ) of the damped Eulerequation (1) and have been studied in [28, 44] for the isothermal pressure law case ( U ( s ) = s log s , V = W ≡ U ( s ) = s m , m > V = W ≡ δ -shocks, and their application to sticky particles [13, 14] orto consensus/contagion in swarming/crowd models [7, 17, 18].The objective of this paper is to develop contraction estimates in the 2-Wasserstein distance inthe presence of uniform convexity of the potentials, in order to (a) prove convergence to equilibriumresults for the damped Euler equations (1) in its full generality, and (b) to prove the overdampedlimit ( γ → ∞ ) of (1) to (4) after suitable scaling. The general idea in handling both problemsstems from viewing (1) as a damped harmonic oscillator for the pair ( ̺, u ) with energy H ( ̺ t , u t ) := F ( ̺ t ) + 12 Z R d | u t | ̺ t dx, ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 3 which plays the role of a mathematical entropy and provides for a Lyapunov functional of (1).Indeed, for smooth solutions ( ̺, u ) of (1), the identity ddt H ( ̺ t , u t ) = − γ Z R d | u t | ̺ t dx ≤ , (6)holds. Although this inequality clearly states the dissipation of H with time t ≥
0, one cannot con-clude the convergence to (global) equilibrium, since the right-hand side vanishes at local equilibria( u t ≡ F ( ̺ t ) + 12 Z R d | u t | ̺ t dx + γ Z t Z R d | u s | ̺ s dx ds ≤ F ( ̺ ) + 12 Z R d | u | ̺ dx, for all t ≥
0. A solution ( ̺, u ) of (1) satisfying this estimate is called an energy decaying solution in the sequel. We will assume that these solutions exist globally in time with certain regularityfor their velocity fields. We emphasize otherwise that our results hold without any smallnessassumption on the initial data or closeness assumption to equilibrium solutions.A good intuition for our strategy comes from the finite-dimensional setting. It is well-knownthat finite-dimensional gradient flows of uniformly convex energy landscapes enjoy exponentialequilibration towards their unique global minimum. More precisely, assume E : R d −→ R tobe a uniformly C convex function achieving its global minimum at zero with D E ≥ λI d , forsome λ >
0. Then, a good quantity to estimate the decay to zero of all solutions is given by theeuclidean distance of a trajectory of the gradient flow ˙ x = −∇ E ( x ) to the origin. Actually, onecan show that ddt | x ( t ) | ≤ − λ | x ( t ) | for all t ≥ . The gradient flow ˙ x = −∇ E ( x ) is the finite-dimensional counterpart of the aggregation-diffusionequation (4). For the damped Euler system (1), the finite dimensional counterpart is the classicaldamped oscillator ˙ x = v , ˙ v = −∇ E ( x ) − γv . Observe that the energy E ( x ) is dissipated bythe gradient flow ˙ x = −∇ E ( x ), that is, ddt E ( x ) = −|∇ E ( x ) | that resembles the gradient flowstructure of (4) and its dissipation (5). In the case of the classical damped oscillator, we have thefollowing dissipation of the total energy ddt (cid:20) | v ( t ) | + E ( x ( t )) (cid:21) = − γ | v ( t ) | for all t ≥ , that resembles (6). Since the quantity | x ( t ) | was a good measure of the equilibration of thegradient flow equation, it seems quite natural to check if it is also the case for the classicaldamped oscillator. In fact, one can show that d dt | x ( t ) | + γ ddt | x ( t ) | + λ | x ( t ) | ≤ | v ( t ) | for all t ≥ . This relation together with the energy identity implies the convergence, without rate, for thesolutions of the classical damped oscillator towards the origin. Its proof will be discussed inSection 4 in the framework of solutions to the Euler equation (1).To analyse the evolution of probability measures, it is classical that the euclidean Wassersteindistance towards the global equilibrium of the free energy F plays the role of the euclidean distancein R d to the origin. Therefore, motivated by the finite dimensional computation above and thework in [48] (cf. [61]), we construct a Lyapunov functional based on the weighted sum of theenergy H , the 2-Wasserstein distance and its temporal derivative. In particular, we will require anestimate for the second-order temporal derivative of the 2-Wasserstein distance, which is providedby Theorem 1 in Section 3. Roughly speaking, it states that for solutions ( ̺, u ) satisfying (1), thesecond-order temporal derivative of the 2-Wasserstein distance between µ t , with density ̺ t , andany probability measure σ with finite second moment, is given by12 d + dt ddt W ( µ t , σ ) + γ ddt W ( µ t , σ ) ≤ Z R d | u t | dµ t − Z R d h T t ( y ) − y, ∇ ( δ µ F )( µ t ) ◦ T t ( y ) i dσ, where T t : R d → R d is an optimal transport map between µ t and σ , satisfying T t σ = µ t , i.e., µ t is the push-forward of σ under the map T t . CARRILLO, CHOI, AND TSE
When V and W satisfy certain λ -convexity assumptions ( (H1) and (H2) below) and σ = µ ∞ is a sufficiently smooth minimizer of F , i.e., µ ∞ , with density ̺ ∞ , satisfies µ ∞ ∇ ( δ µ F )( µ ∞ ) = 0,then the previous estimate reduces to (cf. Corollary 1)12 d + dt ddt W ( µ t , µ ∞ ) + γ ddt W ( µ t , µ ∞ ) ≤ Z R d | u t | dµ t − λ W ( µ t , µ ∞ ) , for some λ >
0. It is this form of the estimate, along with estimate (6), that will be usedto construct a strict Lyapunov function for the evolution, thereby resulting in the equilibrationstatements found in Theorems 3 and 4 (see also Corollaries 2 and 3): For initial data ( ̺ , u ) withbounded energy and W ( µ , µ ∞ ) < ∞ , one obtains W ( µ t , µ ∞ ) −→ t → ∞ . In order to also deduce the convergence k u t k L ( µ t ) → t → ∞ , one requires an additionalassumption (H3) on the relationship between F and W , and on the regularity of the solution.A similar approach is used to prove the overdamped limit ( γ → + ∞ ) in Section 5, where wecompare a rescaled version of the solution ( ̺ γ , u γ ) to the Euler system (1) with the solution ¯ ̺ ofthe granular media equation (4). Since both ̺ γ and ¯ ̺ are time dependent, we extend the second-order estimate above to include measures σ that evolve in time (Theorem 2). This enables us toshow in Theorem 5 that Z T W ( ̺ γt dx, ¯ ̺ t dx ) dt −→ γ → ∞ , for any T > δ -shocks in finite time. The example in (b) clearly illustrates thestrength of the calculus even for the one dimensional case, where standard tools fail due to lackof regularity. In short, we show that all global in time Lagrangian solutions in the sense of [13]converge in W towards a Dirac Delta at the center of mass of the initial density.2. Preliminary results
We begin this section by introducing known results and stating restrictions on the free energy F that will be assumed throughout this paper. The next part of this section describes the generalstrategy applied to a toy example. Definition 1.
Let P ( R d ) denote the set of Borel probability measures on R d with finite secondmoment, i.e., R | x | dµ < ∞ for all µ ∈ P ( R d ). The 2-Wasserstein distance between two measures µ and ν in P ( R d ) is defined as W ( µ, ν ) = inf π ∈ Π( µ,ν ) (cid:18)Z Z R d × R d | x − y | dπ ( x, y ) (cid:19) / , where Π( µ, ν ) denotes the collection of all Borel probability measures on R d × R d with marginals µ and ν on the first and second factors respectively. The set Π( µ, ν ) is also known as the set ofall couplings of µ and ν ∈ P ( R d ). We further denote by Π ( µ, ν ) the set of optimal couplingsbetween µ and ν . The Wasserstein distance defines a distance on P ( R d ) which metricizes thenarrow convergence, up to a condition on the moments. We denote the set of probability measureshaving finite second moment with Lebesgue densities by P ac ( R d ).2.1. Existence of stationary measures.
To emphasize on the presentation of the equilibrationmethod, we do not consider the most general assumptions to ensure the existence of minimizersof the free energy F . Throughout this paper, we assume the conditions below: (H1) U ∈ C ([0 , ∞ )) ∩ C ((0 , ∞ )) with U (0) = 0, and the function r r d U ( r − d ) is convexnonincreasing on (0 , ∞ ), or equivalently,( d − p ( r ) ≤ drp ′ ( r ) on (0 , ∞ ) , p ( r ) = rU ′ ( r ) − U ( r ) , ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 5 or r r − /d p ( r ) is nondecreasing on (0 , ∞ ). (H2) V and W are C ( R d ) potentials on R d with W ( − x ) = W ( x ) for all x ∈ R d , satisfying h x − y, ∇ V ( x ) − ∇ V ( y ) i ≥ c V | x − y | h x − y, ∇ W ( x ) − ∇ W ( y ) i ≥ c W | x − y | ) for all x, y ∈ R d , with either c V > c V + c W > V
0, or c V = 0 and c W > V ≡ Proposition 1.
The free energy F : P ( R d ) → ( −∞ , + ∞ ] defined by (3) for absolutely continuousmeasures (w.r.t. the Lebesgue measure) and by + ∞ otherwise achieves its minimum. A minimizer µ ∞ of F has a non-negative density ̺ ∞ on R d satisfying ∇ p ( ̺ ∞ ) + ̺ ∞ ( ∇ V + ∇ W ⋆ ̺ ∞ ) = 0 a.e.In particular, we have U ′ ( ̺ ∞ ) + V + W ⋆ ̺ ∞ = c µ ∞ -a.e.for some constant c ∈ R . Remark 1.
For isothermal/isentropic flows in fluid dynamics the pressure law p ( ̺ ) is typicallyprescribed by p ( ̺ ) = ̺ m with m ≥ . In this case, the internal energy U is uniquely given by U ( ̺ ) = ( ̺ log ̺ if m = 1 ,̺ m / ( m −
1) if m > , and condition (H1) takes the form m ≥ − /d which is now classical. Consequently, the isentropicpressure satisfies condition (H1) for any m ≥ Remark 2.
In the seminal work [56], McCann showed that assumption (H1) is equivalent to therequirement that R R d U ( ̺ ) dx is (geodesically) displacement convex on ( P ( R d ) , W ).2.2. Equilibration of the center of mass.
Here, we provide a simple construction of a strictLyapunov functional for a toy example, for which we obtain exponential decay rates of the centerof mass of µ , with density ̺ , where ( ̺, u ) solves (1) with V ( x ) = c V | x − ¯ x | / c V > x ∈ R d and W ≡
0. In this case, the free energy is given by F ( µ ) = Z R d U ( ̺ ) dx + c V Z R d | x − ¯ x | dµ, with variational derivative ( δ µ F )( µ ) = U ′ ( ̺ ) + c V ( x − ¯ x ).By examining the evolution of the center of mass of µ t , we find that ddt Z R d x dµ t = Z R d u t dµ t , ddt Z R d u t dµ t = − Z R d ∇ V dµ t − γ Z R d u t dµ t . (7)which clearly resembles a damped harmonic oscillator. Note that the computations above are donecomponent-wise, i.e., ddt Z R d x i dµ t = Z R d u i,t dµ t for all i = 1 , . . . , d. From this observation, one may easily deduce the exponential convergence of the center of massof µ t towards ¯ x ∈ R d at a rate λ that has the properties λ ( γ ) → γ → γ → + ∞ .Indeed, it follows from (7) that d dt Z R d ( x − ¯ x ) dµ t + γ ddt Z R d ( x − ¯ x ) dµ t + c V Z R d ( x − ¯ x ) dµ t = 0 . CARRILLO, CHOI, AND TSE
Thus, from classical theory of differential equations, we may solve the second-order linear equationto obtain explicit decay rates. However, we use an alternative approach to estimate the decay rate,which simultaneously illustrates the basic idea behind our strategy. First of all, notice that12 ddt " c V (cid:12)(cid:12)(cid:12)(cid:12)Z R d ( x − ¯ x ) dµ t (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z R d u t dµ t (cid:12)(cid:12)(cid:12)(cid:12) = − γ (cid:12)(cid:12)(cid:12)(cid:12)Z R d u t dµ t (cid:12)(cid:12)(cid:12)(cid:12) , i.e., we are in the same conditions as in equation (6). Now consider the temporal derivative of J ( t ) := α (cid:12)(cid:12)(cid:12)(cid:12)Z R d ( x − ¯ x ) dµ t (cid:12)(cid:12)(cid:12)(cid:12) + 2 (cid:18)Z R d ( x − ¯ x ) dµ t (cid:19) · (cid:18)Z R d u t dµ t (cid:19) + β (cid:12)(cid:12)(cid:12)(cid:12)Z R d u t dµ t (cid:12)(cid:12)(cid:12)(cid:12) =: α J ( t ) + 2 J ( t ) + β J ( t )where α, β > αβ >
1, we have the equivalence p ( J + J ) ≤ J ≤ q ( J + J ) , for some constant p, q >
0, depending only on α and β . Simple computations yield ddt J ( t ) = − c V J ( t ) + 2( α − βc V − γ ) J ( t ) − βγ − J ( t ) . Choosing β = (1 + c V ) /γ and α = βc V + γ , and using the fact that αβ ≥ c V >
1, we obtain ddt J ( t ) = − c V ( J ( t ) + J ( t )) ≤ − c V /q ) J ( t ) . A simple application of the Grownwall inequality provides the exponential decay J ( t ) ≤ J (0) e − (2 c V /q ) t , q = ( α + β ) + p β − α ) . With the explicit choice of α and β , one easily examines the γ dependent decay rate. Remark 3.
Notice that while the above computations provide exponential convergence of thecenter of mass of µ and the momentum uµ towards (¯ x, ̺, u ) itself.3. Temporal derivatives of the Wasserstein distance
Before we show any convergence results, we extend a basic result regarding the time derivativesof the Wasserstein distance between two evolving measures [1, 10, 11, 62]. We begin by recallinga known result for the first temporal derivative [1], [62, Theorem 23.9].
Proposition 2.
Let µ, ν ∈ C ([0 , ∞ ) , P ac ( R d )) be solutions of the continuity equations ∂ t µ t + ∇ · ( µ t ξ t ) = 0 , ∂ t ν t + ∇ · ( ν t η t ) = 0 , in distribution , for locally Lipschitz vector fields ξ and η satisfying Z ∞ (cid:18)Z R d | ξ t | dµ t + Z R d | η t | dν t (cid:19) dt < ∞ , then µ, ν ∈ AC ([0 , ∞ ) , P ac ( R d )) and for almost every t ∈ (0 , ∞ ) , ddt W ( µ t , ν t ) = Z Z R d × R d h x − y, ξ t ( x ) − η t ( y ) i dπ t (8) = Z R d h x − ∇ ϕ ∗ t ( x ) , ξ t ( x ) i dµ t + Z R d h y − ∇ ϕ t ( y ) , η t ( y ) i dν t , where π t ∈ Π ( µ t , ν t ) and ∇ ϕ t ν t = µ t , ∇ ϕ ∗ t µ t = ν t . Remark 4.
Note that when ξ and η are globally Lipschitz vector fields, then the first temporalderivative (8) holds for all t ∈ (0 , ∞ ). We will implicitly use this fact in Theorem 1 below. ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 7
Heuristical ideas.
For any two given measures µ t , ν t ∈ P ac ( R d ), Brenier’s theorem [12, 55] assertsthe existence of a (proper) convex function ϕ t : R d → ( −∞ , + ∞ ] such that ∇ ϕ t ν t = µ t and ∇ ϕ ∗ t µ t = ν t , where ϕ t ( x ) ∗ = sup R d {h x, y i − ϕ ( y ) } is the Legendre–Fenchel dual of ϕ t satisying( ∇ ϕ ∗ t ◦ ∇ ϕ t )( y ) = y ν t -a.e.In particular, we have the change of variables formula Z R d g ( t, ∇ ϕ t ( y )) dν t = Z R d g ( t, x ) dµ t , for any test function g ∈ C b ( R + × R d ). Taking the temporal derivative gives Z R d h∇ g ( t, ∇ ϕ t ( y )) , ∂ t ∇ ϕ t ( y ) i dν t + Z R d g ( t, ∇ ϕ t ( y )) d ( ∂ t ν t ) = Z R d g ( t, x ) d ( ∂ t µ t ) . (9)By choosing g ( t, x ) = | x | / − ϕ ∗ t ( x ), we obtain from [62] (see also [1])12 ddt W ( µ t , ν t ) = 12 ddt Z R d |∇ ϕ t ( y ) − y | dν t = Z R d h∇ ϕ t ( y ) − y, ∂ t ∇ ϕ t ( y ) i dν t + 12 Z R d |∇ ϕ t ( y ) − y | d ( ∂ t ν t )= Z R d g ( t, x ) d ( ∂ t µ t ) − Z R d g ( t, ∇ ϕ t ( y )) d ( ∂ t ν t ) + 12 Z R d |∇ ϕ t ( y ) − y | d ( ∂ t ν t ) . The last two terms on the right hand side may be expressed as Z R d (cid:18) | y | − h∇ ϕ t ( y ) , y i + ϕ ∗ t ( ∇ ϕ t ( y )) (cid:19) d ( ∂ t ν t ) . Since ϕ t and ϕ ∗ t are duals of each other, we have that ϕ t ( y ) + ϕ ∗ t ( ∇ ϕ t ( y )) = h∇ ϕ t ( y ) , y i . Consequently, we obtain12 ddt W ( µ t , ν t ) = Z R d g ( t, x ) d ( ∂ t µ t ) + Z R d h ( t, y ) d ( ∂ t ν t ) , with h ( t, y ) = | y | / − ϕ t ( y ). Finally, inserting the respective continuity equations and integratingby parts yield the required equality in (8). The absolute continuity of t µ t and t ν t followsdirectly from [62, Theorem 23.9]. In fact, they are shown to be H¨older-1 / ∂ t µ t + ∇ · ( µ t ξ t ) = 0 ,∂ t ( µ t ξ t ) + ∇ · ( µ t ξ t ⊗ ξ t ) = − µ t G µ t , where G µ = G µ ( t, x ) is a sufficiently smooth function. Remark 5.
In our particular case (1), G µ is related to the variational derivative of the free energy F and takes the form G µ ( µ, ξ ) = ∇ ( δ µ F )( µ ) + γξ = ∇ (cid:0) U ′ ( ̺ ) + V + W ⋆ µ (cid:1) + γξ , with ̺ being the density of µ . Heuristical ideas.
To simplify the notations, we set T t ( y ) := ∇ ϕ t ( y ) , T ∗ t ( x ) := ∇ ϕ ∗ t ( x ) . Assuming that µ and ν satisfy the generic Euler equations ∂ t µ t + ∇ · ( µ t ξ t ) = 0 ,∂ t ( µ t ξ t ) + ∇ · ( µ t ξ t ⊗ ξ t ) = − µ t G µ t , ∂ t ν t + ∇ · ( ν t η t ) = 0 ,∂ t ( ν t η t ) + ∇ · ( ν t η t ⊗ η t ) = − ν t G ν t , CARRILLO, CHOI, AND TSE with sufficiently smooth velocity fields ξ t and η t , we deduce from (9) that Z R d h∇ g ( t, T t ( y )) , ∂ t T t ( y ) + ∇ T t ( y ) η t ( y ) − ξ t ( T t ( y )) i dν t = 0 , for all smooth test functions g ∈ C b ( R × R d ). This essentially means ∂ t T t ( y ) + ∇ T t ( y ) η t ( y ) = ξ t ( T t ( y )) ν t -a.e.(10)Let us now consider the second temporal derivative of the Wasserstein distance, i.e., we formallytake the temporal derivative of (8) to obtain12 d dt W ( µ t , ν t ) = − Z R d h ∂ t T ∗ t ( x ) , ξ t ( x ) i dµ t − Z R d h ∂ t T t ( y ) , η t ( y ) i dν t + Z R d h x − T ∗ t ( x ) , ∂ t ( µ t ξ t ) i + Z R d h y − T t ( y ) , ∂ t ( ν t η t ) i . (11)For the first term, we notice the fact that0 = ∂ t ( T ∗ t ◦ T t )( y ) = ∂ t T ∗ t ( T t ( y )) + ∇ T ∗ t ( T t ( y )) ∂ t T t ( y ) . Using the previous equality, (10) and the fact that ∇ T ∗ t ( T t ( y )) ∇ T t ( y ) = I d , we obtain Z R d h ∂ t T ∗ t ( x ) , ξ t ( x ) i dµ t = Z R d h ∂ t T ∗ t ( T t ( y )) , ξ t ( T t ( y )) i dν t = − Z R d h∇ T ∗ t ( T t ( y )) ∂ t T t ( y ) , ξ t ( T t ( y )) i dν t = Z R d h∇ T ∗ t ( T t ( y )) (cid:16) ∇ T t ( y ) η t ( y ) − ξ t ( T t ( y )) (cid:17) , ξ t ( T t ( y )) i dν t = Z R d h η t ( y ) , ξ t ( T t ( y )) i dν t − Z R d h∇ T ∗ t ( x ) ξ t ( x ) , ξ t ( x ) i dµ t . Computing the second term analogously gives Z R d h ∂ t T t ( y ) , η t ( y ) i dν t = − Z R d h∇ T t ( y ) η t ( y ) , η t ( y ) i dν t + Z R d h ξ t ( T t ( y )) , η t ( y ) i dν t . The last two terms of (11) may be handled simultaneously to give Z R d h x − T ∗ t ( x ) , ∂ t ( µ t ξ t ) i + Z R d h y − T t ( y ) , ∂ t ( ν t η t ) i = Z R d h ( I d − ∇ T ∗ t ( x )) ξ t ( x ) , ξ t ( x ) i dµ t − Z R d h x − T ∗ t ( x ) , G µ t i dµ t + Z R d h ( I d − ∇ T t ( y )) η t ( y ) , η t ( y ) i dν t − Z R d h y − T t ( y ) , G ν t i dν t . The previous heuristic ideas can now be turned into the following theorem under the right as-sumptions.
Theorem 1.
Let µ ∈ C ([0 , T ) , P ac ( R d )) satisfy the Euler type equation ∂ t µ t + ∇ · ( µ t ξ t ) = 0 ,µ t (cid:0) ∂ t ξ t + ξ t · ∇ ξ t (cid:1) = − µ t G µ t , ) in distribution , with locally in t > and globally in x ∈ R d Lipschitz vector field x ξ t ( x ) satisfying t
7→ k ξ t k L ( µ t ) , k G µ t ( t, · ) k L ( µ t ) ∈ C ([0 , ∞ )) ∩ L ([0 , ∞ )) . For any σ ∈ P ( R d ) , define K ( µ t , σ ) := ddt W ( µ t , σ ) = 2 Z Z R d × R d h x − y, ξ t ( x ) i dπ t . ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 9
Then for any
T > the following inequality holds: K ( µ T , σ ) ≤ K ( µ , σ ) + 2 Z T (cid:18)Z R d | ξ t | dµ t − Z Z R d × R d h x − y, G µ t ( t, x ) i dπ t (cid:19) dt, for the optimal transference plan π t ∈ Π ( µ t , σ ) . In particular, we obtain d + dt ddt W ( µ t , σ ) = 12 d + dt K ( µ t , σ ) ≤ Z R d | ξ t | dµ t − Z Z R d × R d h x − y, G µ t ( t, x ) i dπ t , where d + /dt denotes the upper derivative in almost every t > .Proof. Step 1: For some fixed t ∈ (0 , ∞ ) let ∂ τ Φ τ ( x ) = ( ξ t + τ ◦ Φ τ )( x ) , Φ = x for µ t -a.e. x, (12)be the well-defined global in τ ∈ ( − t, ∞ ) corresponding Lipschitz flow and set µ t + h = Φ h µ t , µ t − h = Φ − h µ t , for each h ∈ (0 , t ). Furthermore, taking the temporal derivative of (12) and using the momentumequation for µ t ξ t provides the representation ∂ τ Φ τ ( x ) = (cid:0) ∂ t ξ t + τ + ξ t + τ · ∇ x ξ t + τ (cid:1) ◦ Φ τ ( x ) = − G µ t ( t + τ, Φ τ ( x )) for µ t -a.e. x, (13)and a.e. τ ∈ ( − t, ∞ ), which will be used in the following steps.Finally let π t ∈ Π ( µ t , σ ) be the unique optimal plan between µ t and σ ∈ P ( R d ). One clearlysees that π τt := (Φ τ × id ) π t induces a transference plan between µ t + τ and σ ∈ P ( R d ). Step 2:
For some fixed t ∈ (0 , ∞ ) and h ∈ (0 , t ), consider the finite difference∆ h K ( µ t , σ ) := ( D sym h D sym h W )( µ t , σ ) , where D sym τ denotes the symmetric difference operator with step τ >
0, i.e.,( D sym τ W )( µ t , σ ) := 12 τ (cid:0) W ( µ t + τ , σ ) − W ( µ t − τ , σ ) (cid:1) . Thus, we explicitly obtain∆ h K ( µ t , σ ) = 1 h (cid:16) W ( µ t + h , σ ) − W ( µ t , σ ) + W ( µ t − h , σ ) (cid:17) , i.e., ∆ h is a second order symmetric difference operator. Notice that, by passing to the limit h → h K ( µ t , σ ), one obtainslim h → ∆ h K ( µ t , σ ) = d dt W ( µ t , σ ) = ddt K ( µ t , σ ) , whenever the right-hand side is well-defined. Unfortunately, the existence of a second temporal2-Wasserstein derivative may not be easily justified. Instead, we proceed with the finite differencecomputations, while mimicking the formal differential computations.Recall that π τt := (Φ τ × id ) π t ∈ Π( µ t + τ , σ ) for any τ ∈ [ − h, h ]. Hence, W ( µ t + τ , σ ) ≤ Z Z R d × R d | x − y | dπ τt = Z Z R d × R d | Φ τ ( x ) − y | dπ t . Consequently, for any h ∈ (0 , t ), we have∆ h K ( µ t , σ ) = 1 h (cid:16) W ( µ t + h , σ ) − W ( µ t , σ ) + W ( µ t − h , σ ) (cid:17) ≤ h Z Z R d × R d | Φ h ( x ) − y | − | x − y | + | Φ − h ( x ) − y | dπ t = (I) . Using the fundamental theorem of calculus and Jensen’s inequality, (I) may be reformulated as(I) = 1 h Z Z R d × R d | Φ h ( x ) − x | + 2 h Φ h ( x ) − x + Φ − h ( x ) , x − y i + | Φ − h ( x ) − x | dπ t ≤ h Z h − h Z R d | ∂ τ Φ τ ( x ) | dµ t dτ + 1 h Z h Z Z R d × R d ( h − τ ) h x − y, ∂ τ Φ τ ( x ) i dπ t dτ + 1 h Z − h Z Z R d × R d ( h + τ ) h x − y, ∂ τ Φ τ ( x ) i dπ t dτ = Z − Z R d | ξ t + sh ( x ) | dµ t + sh ds − Z (1 − s ) Z Z R d × R d h x − y, G µ t + sh ( t + sh, Φ sh ( x )) i dπ t ds − Z − (1 + s ) Z Z R d × R d h x − y, G µ t + sh ( t + sh, Φ sh ( x )) i dπ t ds, where we inserted the representation (13) in the last equality. Step 3:
For a fixed
T >
0, we choose N ∈ N in such a way that h = T /N . Now consider a family { µ nh } n ∈ I N ⊂ P ac ( R d ) for I N = { , . . . , N } ⊂ N , defined recursively by µ ( n +1) h = Φ nh µ nh for n ∈ I N , where Φ nh satisfies ∂ τ Φ nτ ( x ) = ( ξ nh + τ ◦ Φ nτ )( x ) , Φ n = x for µ nh -a.e. x. and τ ∈ ( − h, h ). Then, for each n ∈ I N , Step 2 provides the inequality∆ h K ( µ nh , σ ) ≤ Z − Z R d | ξ ( n + s ) h ( x ) | dµ ( n + s ) h ds − Z (1 − s ) Z Z R d × R d h x − y, G µ ( n + s ) h (( n + s ) h, Φ nsh ( x )) i dπ nh ds − Z − (1 + s ) Z Z R d × R d h x − y, G µ ( n + s ) h (( n + s ) h, Φ nsh ( x )) i dπ nh ds =: (A) + (B) + (C) . (14)Multiplying the inequality with h and summing over n ∈ I N yields for the left-hand side N − X n =1 h ∆ h K ( µ nh , σ ) = ( D sym h W )( µ ( N − ) h , σ ) − ( D sym h W )( µ h , σ ) . Before proceeding, we first note that the fundamental theorem of calculus and the representationof the first temporal 2-Wasserstein derivative provided in Proposition 2 yields( D sym h W )( µ ( n + ) h , σ ) = 1 h (cid:16) W ( µ ( n +1) h , σ ) − W ( µ nh , σ ) (cid:17) = 1 h Z ( n +1) hnh ddτ W ( µ τ , σ ) dτ = 2 h Z ( n +1) hnh Z Z R d × R d h x − y, ξ τ ( x ) i dπ τ dτ = 2 Z Z Z R d × R d h x − y, ξ ( n + s ) h ( x ) i dπ ( n + s ) h ds for any n ∈ I N . Therefore, passing to the limit N → ∞ with T = hN giveslim N →∞ N − X n =1 h ∆ h K ( µ nh , σ ) = 2 (cid:18)Z Z R d × R d h x − y, ξ T ( x ) i dπ T − Z Z R d × R d h x − y, ξ ( x ) i dπ (cid:19) = K ( µ T , σ ) − K ( µ , σ ) , ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 11 which holds due to Lebesgue’s dominated convergence theorem. On the other hand, the followingconvergences hold for the terms on the right-hand side of (14): N − X n =1 h (A) −→ Z T Z R d | ξ t ( x ) | dµ t dt, N − X n =1 h (B) −→ Z T Z Z R d × R d h x − y, G µ t ( t, x ) i dπ t dt N − X n =1 h (C) −→ Z T Z Z R d × R d h x − y, G µ t ( t, x ) i dπ t dt. These convergences hold simply by definition of Riemann integrable functions and the assumedregularity of ξ and G µ . Indeed, due to the assumed continuity of f ( t ) := k ξ t k L ( µ t ) , we know that f is Riemann integrable on [0 , T ]. Therefore, the corresponding upper Darboux sum satisfies N − X n =0 h sup s ∈ [0 , f (( n + s ) h ) −→ Z T f ( t ) dt as N → ∞ . In particular, we have for any s ∈ [0 , I Nf ( s ) := N − X n =0 hf (( n + s ) h ) −→ Z T f ( t ) dt as N → ∞ . Furthermore, we may reformulate the sum to obtain N − X n =1 h (A) = N − X n =1 h Z − Z R d | ξ ( n + s ) h ( x ) | dµ ( n + s ) h ds = Z − N − X n =1 hf (( n + s ) h ) ds = Z N − X n =1 hf (( n + s ) h ) ds + Z − N − X n =1 hf (( n + s ) h ) ds = Z N − X n =1 hf (( n + s ) h ) ds + Z N − X n =0 hf (( n + s ) h ) ds = 2 Z I Nf ( s ) ds − Z hf ( sh ) ds − Z hf ( T − (1 − s ) h ) ds. It is not hard to see that | I Nf ( s ) | ≤ c with some constant c > N ≫ N → ∞ due to the boundedness of f on [0 , T ]. Therefore,an application of the Lebesgue dominated convergence yieldslim N →∞ N − X n =1 h (A) = 2 Z lim N →∞ I Nf ( s ) ds = 2 Z Z T f ( t ) dt ds = 2 Z T f ( t ) dt, as required. The convergence of the sums for (B) and (C) may be shown in a similar fashion.Collecting all the terms together finally yields the statement. (cid:3) Mimicking the strategy of the proof to Theorem 1, we arrive at the following result.
Theorem 2.
Let µ t and ν t ∈ P ac ( R d ) , t ≥ , satisfy Euler type equations of the form ∂ t µ t + ∇ · ( µ t ξ t ) = 0 ,µ t (cid:0) ∂ t ξ t + ξ t · ∇ ξ t (cid:1) = − µ t G µ , ∂ t ν t + ∇ · ( ν t η t ) = 0 ,ν t (cid:0) ∂ t η t + η t · ∇ η t (cid:1) = − ν t G ν , ) in distribution , with locally in t > and globally in x ∈ R d Lipschitz vector fields x ξ t ( x ) , η t ( x ) satisfying t
7→ k ξ t k L ( µ t ) , k η t k L ( ν t ) , k G µ t ( t, · ) k L ( µ t ) , k G ν t ( t, · ) k L ( ν t ) ∈ C ([0 , ∞ )) ∩ L ([0 , ∞ )) . Then for any
T > the following inequality holds: K ( µ T , ν T ) ≤ K ( µ , ν ) + 2 Z T Z R d × R d | ξ t ( x ) − η t ( y ) | − h x − y, G µ t ( t, x ) − G ν t ( t, y ) i dπ t dt, for the optimal transference plan π t ∈ Γ ( µ t , ν t ) . In particular, we obtain d + dt K ( µ t , ν t ) ≤ Z Z R d | ξ t ( x ) − η t ( y ) | dπ t − Z Z R d × R d h x − y, G µ t ( t, x ) − G ν t ( t, y ) i dπ t . A direct consequence of Theorem 2 is the following result.
Corollary 1.
Let ( ̺, u ) be an energy decaying solution of the Euler equations (1) with µ , withdensity ̺ , and u satisfying additionally the assumptions of Theorem 1. Furthermore, let ν , withdensity ω , satisfy ∇ p ( ω )+ ω ( ∇ V + ∇ W ⋆ν ) = 0 dx -almost everywhere. Suppose that p ′ ( ω ) ∈ L ( ν ) and p ′ ( ̺ t ) ∈ L ( µ t ) for almost every t ∈ (0 , ∞ ) . Then, the following inequality d + dt ddt W ( µ t , ν ) ≤ k ξ t k L ( µ t ) − γ ddt W ( µ t , ν ) − J V ( µ t | ν ) − J W ( µ t | ν ) , holds for almost every t > . Here, the functionals J V and J W are defined by J V ( µ t | ν ) := Z R d h y − T t ( y ) , ∇ V ( y ) − ∇ V ( T t ( y )) i dν ( y ) ,J W ( µ t | ν ) := 12 Z Z R d × R d h ( y − ˆ y ) − T t ( y ) − T t (ˆ y ) , ∇ W ( y − ˆ y ) − ∇ W ( T t ( y ) − T t (ˆ y )) i dν ( y ) dν (ˆ y ) , where T t is the unique transport map satisfying T t ν = µ t .Proof. We make use of Theorem 1 with ( ν t , η t ) = ( ν,
0) and G µ = ∇ (cid:0) U ′ ( ̺ ) + V + W ⋆ µ (cid:1) + γξ, G ν = ∇ (cid:0) U ′ ( ω ) + V + W ⋆ ν (cid:1) . In fact, we have that ω G ν ≡ ω G ν = ω ∇ (cid:0) U ′ ( ω ) + V + W ⋆ ν (cid:1) = ∇ p ( ω ) + ω ∇ (cid:0) V + W ⋆ ν (cid:1) = 0 a.e.We begin by computing the term Z R d h y − T t ( y ) , G ν t ( t, y ) i dν = Z R d h y − T t ( y ) , ∇ p ( ω ) i dy + Z R d h y − T t ( y ) , ∇ (cid:0) V + W ⋆ ν (cid:1) i dν, where we used the fact that ω ∇ U ′ ( ω ) = ∇ p ( ω ). Similarly, we obtain Z R d h y − T t ( y ) , G µ t ( t, T t ( y )) i dν = Z R d h T ∗ t ( x ) − x, ∇ p ( ̺ t ) i dx + γ Z R d h T ∗ t ( x ) − x, ξ t ( x ) i dµ t + Z R d h T ∗ t ( x ) − x, ∇ (cid:0) V + W ⋆ µ t (cid:1) i dµ t . Subtracting the equation above from the previous one, we get Z R d h y − T t ( y ) , G ν t ( t, y ) − G µ t ( t, T t ( y )) i dν = Z R d h y − T t ( y ) , ∇ p ( ω ) i dy − Z R d h T ∗ t ( x ) − x, ∇ p ( ̺ t ) i dx + γ ddt W ( µ t , ν )+ J V ( µ t | ν ) + Z R d h y − T t ( y ) , ( ∇ W ⋆ ν )( y ) − ( ∇ W ⋆ µ t )( T t ( y )) i dν = I + γ ddt W ( µ t , ν ) + J V ( µ t | ν ) + I . In order to deal with I , we proceed by a “weak” integration by parts as in [53, 10, 24]. We givesome details on this. We consider a smooth cut-off function χ R ∈ C ∞ ( R d ) satisfying the properties0 ≤ χ R ≤ , |∇ χ R | ≤ CR , χ R ≡ B R (0) , χ R ≡ R d \ B R (0) . ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 13
Under the assumption p ′ ( ̺ t ) ∈ L ( µ t ), we obtain from (H1) Z R d p ( ̺ t ) dx ≤ dd − Z R d ̺ t p ′ ( ̺ t ) dx ≤ dd − (cid:18)Z R d | p ′ ( ̺ t ) | dµ t (cid:19) / , Z R d | T ∗ t − id | p ( ̺ t ) dx ≤ dd − Z R d | T ∗ t − id | p ′ ( ̺ t ) dµ t ≤ dd − W ( µ t , ν ) (cid:18)Z R d | p ′ ( ̺ t ) | dµ t (cid:19) / , and therefore, p ( ̺ t ) and | T ∗ t − id | p ( ̺ t ) are in L ( R d ) a.e. t ∈ (0 , ∞ ). On the other hand, theassumption on G µ ( t, · ) provides Z R d |∇ p ( ̺ t ) | dx = Z R d |∇ U ′ ( ̺ t ) | dµ t ≤ k∇ U ′ ( ̺ t ) k L ( µ t ) a.e. t ∈ (0 , ∞ ) , and therefore, p ( ̺ t ) ∈ W , ( R d ) a.e. t ∈ (0 , ∞ ).Since χ R ( T ∗ t − id ) ∈ L ∞ ( R d ) ∩ BV ( R d ), we perform integration by parts to obtain − Z R d χ R h T ∗ t − id, ∇ p ( ̺ t ) i dx ≥ Z R d ˜ ∇ · (cid:0) χ R ( T ∗ t − id ) (cid:1) p ( ̺ t ) dx = Z R d χ R ( ˜ ∇ · T ∗ t ) p ( ̺ t ) dx − d Z R d χ R p ( ̺ t ) dx + Z R d ∇ χ R · ( T ∗ t − id ) p ( ̺ t ) dx, (15)where we used the fact that the distributional trace of the jacobian ∇ · T ∗ t ≥ ∇· represents the dx -absolutely continuous part of ∇· , defined in the Alexandrovalmost everywhere sense, see [56] for details.Notice that by construction, the following convergences hold: χ R h T ∗ t − id, ∇ p ( ̺ t ) i → h T ∗ t − id, ∇ p ( ̺ t ) i χ R ( ˜ ∇ · T ∗ t ) p ( ̺ t ) → ( ˜ ∇ · T ∗ t ) p ( ̺ t ) χ R p ( ̺ t ) → p ( ̺ t ) as R → ∞ , for almost every t ∈ (0 , ∞ ). Furthermore, we know that ˜ ∇ · T ∗ t ≥
0, and that k χ R h T ∗ t − id, ∇ p ( ̺ t ) ik L ( R d ) ≤ kh T ∗ t − id, ∇ p ( ̺ t ) ik L ( R d ) k χ R p ( ̺ t ) k L ( R d ) ≤ k p ( ̺ t ) k L ( R d ) ) for all R > . For the first bound, we used that ∇ p ( ̺ t ) = ̺ t ∇ U ′ ( ̺ t ) to get kh T ∗ t − id, ∇ p ( ̺ t ) ik L ( R d ) ≤ W ( µ t , ν ) (cid:18)Z R d |∇ U ′ ( ̺ t ) | dµ t (cid:19) / . In particular, using the Lebesgue dominated convergence, we deduce Z R d χ R h T ∗ t − id, ∇ p ( ̺ t ) i dx → Z R d h T ∗ t − id, ∇ p ( ̺ t ) i dx Z R d χ R p ( ̺ t ) dx → Z R d p ( ̺ t ) dx Z R d ∇ χ R · ( T ∗ t − id ) p ( ̺ t ) dx → as R → ∞ . As for the other term, we obtain from Fatou’s lemma Z R d ( ˜ ∇ · T ∗ t ) p ( ̺ t ) dx ≤ lim inf R →∞ Z R d χ R ( ˜ ∇ · T ∗ t ) p ( ̺ t ) dx. Consequently, we pass to the limit as R → ∞ in (15) to obtain the “weak” integration by partsformula − Z R d h T ∗ t − id, ∇ p ( ̺ t ) i dx ≥ Z R d ( ˜ ∇ · T ∗ t − d ) p ( ̺ t ) dx, for almost every t ∈ (0 , ∞ ). Using the same arguments, we can perform a “weak” integration byparts also for the other term in I , thereby obtaining I ≥ − Z R d (cid:0) d − ( ˜ ∇ · T t )( y ) (cid:1) p ( ω ) dy + Z R d (cid:0) ( ˜ ∇ · T ∗ t )( x ) − d (cid:1) p ( ̺ t ) dx = Z R d (cid:16) ( ˜ ∇ · T t )( y ) + ( ˜ ∇ · T ∗ t )( T t ( y )) − d (cid:17) p ( ω ) dy ≥ , (16)where the last inequality follow from [24, 53, 10], where similar arguments were used. Finally,using the fact that ∇ W ( − x ) = −∇ W ( x ) for x ∈ R d , we can rewrite I as I = 12 Z Z R d × R d h y − T t ( y ) + T t (ˆ y ) − ˆ y, ∇ W ( y − ˆ y ) − ∇ W ( T t ( y ) − T t (ˆ y )) i dν ( y ) dν (ˆ y )= J W ( µ t | ν ) . Putting all the terms together and invoking Theorem 2 concludes the proof. (cid:3)
Remark 6.
In the isothermal case with U ( r ) = r log( r ), p ′ ( r ) = rU ′′ ( r ) = 1, and hence theassumption p ′ ( ̺ t ) ∈ L ( µ t ) is trivially satisfied. As for the isentropic case, additional regularity isrequired.A simple outcome of (H2) is the following result which follows from direct computations usingthe convexity assumptions on the potentials, see [23] for more details. Proposition 3.
Under condition (H2) for the potentials V and W , we have that J V ( µ t | µ ∞ ) and J W ( µ t | µ ∞ ) defined in Corollary 1 are bounded from below, where µ ∞ ∈ P ac ( R d ) is a minimizerof the free energy F provided in Proposition 1. In particular, J V ( µ t | µ ∞ ) ≥ c V W ( µ t , µ ∞ ) ,J W ( µ t | µ ∞ ) ≥ c W W ( µ t , µ ∞ ) − c W (cid:12)(cid:12)(cid:12)(cid:12)Z R d x dµ ∞ − Z R d x dµ t (cid:12)(cid:12)(cid:12)(cid:12) , where c V and c W are given in (H2) . We now have the essential ingredients to construct a Lyapunov functional for establishing theconvergence to equilibrium of energy decaying solutions to the damped Euler equations (1) in thespace P ( R d ) endowed with the 2-Wasserstein distance.4. Equilibration in Wasserstein distance
We begin this section by introducing the functionals involved and discuss their properties. Theidea behind lies in the fact that the second temporal derivative of the Wasserstein distance producesa term on the right-hand side which gives a term that dissipates the Wasserstein distance itself(cf. Proposition 3). For this reason, we will have to include the term dW /dt into the Lyapunovfunctional. In addition to the free energy F , we consider the functionals E ( µ t , µ ∞ ) := W ( µ t , µ ∞ ) + Z R d | u t | dµ t , J ( µ t , µ ∞ ) := αW ( µ t , µ ∞ ) + ddt W ( µ t , µ ∞ ) + β Z R d | u t | dµ t , with constants α, β >
0. Since the dW /dt term can be bounded from above by12 (cid:12)(cid:12)(cid:12)(cid:12) ddt W ( µ t , µ ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z R d |h y − T t ( y ) , u t ( T t ( y )) i| dµ ∞ ≤ W ( µ t , µ ∞ ) (cid:18)Z R d | u t | dµ t (cid:19) / , for all t ≥
0, where T t µ ∞ = µ t , we conclude that E and J are equivalent in the following sense: p E ≤ J ≤ q E , (17)for constants p, q >
0, depending only on α and β whenever αβ > ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 15
Remark 7.
If ( ̺, u ) is an energy decaying solution of the damped Euler equations (1), then weobtain the uniform (in t ) boundedness of k u t k L ( µ t ) from the energy estimate (6). Therefore, if W ( µ t , µ ∞ ) is also uniformly bounded in time, thensup t ≥ (cid:12)(cid:12)(cid:12)(cid:12) ddt W ( µ t , µ ∞ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ M, for some constant M < ∞ , which asserts that t W ( ̺ t , µ ∞ ) is uniformly continuous.In order to provide the equilibration also for the velocity field u , we impose additional assump-tions on the free energy F : (H3) The free energy F satisfies the stability estimate F ( µ t ) − F ( µ ∞ ) ≤ c F ( µ t ) W ( µ t , µ ∞ ) , (18) for some time dependent function c F ( µ t ) > t →∞
11 + t Z t c F ( µ s )1 + s ds = 0 . (19) Remark 8.
A well-known inequality which takes the form (18) is the so-called HWI inequality[23, 24, 62]: F ( µ t ) − F ( µ ∞ ) ≤ k∇ ( δ µ F )( µ t ) k L ( µ t ) W ( µ t , µ ∞ ) − ( λ/ W ( µ t , µ ∞ ) , for some λ ≥
0. A classical example of a free energy satisfying the HWI inequality is given by F ( ̺ ) = Z R d ̺ log ̺ dx + Z R d V ( x ) ̺ dx, where V is a smooth convex potential such that R R d exp( − V ( x )) dx = 1. Then the correspondingstationary state is simply ̺ ∞ = e − V . Furthermore, since F ( ̺ ∞ ) = 0, we have0 ≤ F ( ̺ ) − F ( ̺ ∞ ) = Z R d ̺ log( ̺e V ) dx. Then from the standard HWI inequality [62], we obtain (18) with c F ( ̺ ) = R R d |∇ (log ̺ + V ) | d̺ . Remark 9. (i) Observe that (H3) is also an assumption on the regularity of solutions tothe damped Euler equations (1).(ii) A sufficient condition for (19) includes the case c F ( µ t ) ≤ c ∞ uniformly in time. Indeed,11 + t Z t c ∞ s ds = c ∞ (1 + t ) − ln(1 + t ) −→ t → ∞ . In view of condition (H2) , we will study the equilibration for two separate cases.4.1.
The case with confinement.
Here, we consider the case where the confinement potential V is present and satisfies condition (H2) , as well as the interaction potential W . In this case,Proposition 1 provides a stationary measure µ ∞ , with density ̺ ∞ , which satisfies ∇ p ( ̺ ∞ ) + ̺ ∞ ∇ (cid:0) V + W ⋆ ̺ ∞ (cid:1) = 0 a.e.Hence, Theorem 1, with σ = µ ∞ , and Proposition 3 holds true with J V ( µ t | µ ∞ ) + J W ( µ t | µ ∞ ) ≥ c ℓ W ( µ t , µ ∞ ) , c ℓ = ( c V for c W ≥ c V + c W for c W < . Note that in the case c W ≥
0, we have that J W ( µ t | µ ∞ ) ≥ Theorem 3.
Let ( ̺, u ) be an energy decaying solution to the Euler equations (1) , with µ t ∈P ac ( R d ) the measure whose density is ̺ t for all t ≥ , and u satisfying additionally the assumptionsof Theorem 1, and U , V and W satisfying conditions (H1) - (H2) . Furthermore, assume thatthe initial data satisfies F ( µ ) + W ( µ , µ ∞ ) < ∞ , then lim t →∞ W ( µ t , µ ∞ ) = 0 . Proof.
Consider the functional G ( µ t , µ ∞ ) := 2 β (cid:0) F ( µ t ) − F ( µ ∞ ) (cid:1) + J ( µ t , µ ∞ )= αW ( µ t , µ ∞ ) + ddt W ( µ t , µ ∞ ) + 2 β (cid:0) H ( ̺ t , u t ) − H ( ̺ ∞ , (cid:1) ≥ , with constants α, β > G , while the second allows for the computation of the temporal derivative using(6). Taking the temporal derivative of G along the flow generated by the damped Euler equations(1) and applying Corollary 1 we obtain d + dt G ( µ t , µ ∞ ) ≤ − c ℓ W ( µ t , µ ∞ ) + ( α − γ ) ddt W ( µ t , µ ∞ ) − βγ − Z R d | u t | dµ t . Choosing α = γ and β = (1 + c ℓ ) /γ , we further obtain d + dt G ( µ t , µ ∞ ) ≤ − c ℓ E ( µ t , µ ∞ ) . (20)Since αβ = 1 + c ℓ >
1, we have the equivalence between J and E , which concludes the proof.Indeed, integrating (20) over time interval [0 , t ] gives G ( µ t , µ ∞ ) + 2 c ℓ Z t E ( µ s , µ ∞ ) ds ≤ G ( µ , µ ∞ ) , Observe that pW ( µ t , µ ∞ ) ≤ G ( µ t , µ ∞ ) ≤ G ( µ , µ ∞ ) due to (17), which implies the uniformboundedness in time of W ( µ t , µ ∞ ), see Remark 7. Since G is non-negative, we have that2 c ℓ Z ∞ W ( µ s , µ ∞ ) ds ≤ c ℓ Z ∞ E ( µ s , µ ∞ ) ds ≤ G ( µ , µ ∞ ) . Owing to the uniform continuity of t W ( µ t , µ ∞ ) we obtain the asserted convergence [45]. (cid:3) Note that if t
7→ k u t k L ( µ t ) is further assumed to be uniformly continuous in (0 , ∞ ), thenone also obtains k u t k L ( µ t ) → t → ∞ . On the other hand, one may obtain the mentionedconvergence under a different assumption provided in the following result. Corollary 2.
Let ( ̺, u ) be an energy decaying solution to the Euler equations (1) , with µ t ∈P ac ( R d ) the measure whose density is ̺ t for all t ≥ , and u satisfying additionally the assumptionsof Theorem 1, and U , V and W satisfying conditions (H1) - (H3) . Furthermore, assume thatthe initial data satisfies F ( µ ) + W ( µ , µ ∞ ) < ∞ , then lim t →∞ (cid:16) W ( µ t , µ ∞ ) + k u t k L ( µ t ) (cid:17) = lim t →∞ E ( µ t , µ ∞ ) = 0 . Proof.
Notice that for t ≥
0, we have d + dt (cid:0) (1 + t ) G ( µ t , µ ∞ ) (cid:1) = (1 + t ) d + dt G ( µ t , µ ∞ ) + G ( µ t , µ ∞ ) ≤ − c ℓ /q )(1 + t ) J ( µ t , µ ∞ ) + 2 β (cid:0) F ( µ t ) − F ( µ ∞ ) (cid:1) + J ( µ t , µ ∞ ) ≤ − c ℓ /q )(1 + t ) J ( µ t , µ ∞ ) + 2 βc F ( µ t ) W ( µ t , µ ∞ ) + J ( µ t , µ ∞ ) , where we used (20), the equivalence (17) and (H3) . Integrating the equation for t ≥ t ) G ( µ t , µ ∞ ) + 2 c ℓ q Z t (1 + s ) J ( µ s , µ ∞ ) ds ≤ c + 2 β Z t c F ( µ s ) W ( µ s , µ ∞ ) ds, with the constant c := G ( µ , µ ∞ ) + Z ∞ J ( µ s , µ ∞ ) ds < ∞ . For the second term on the right, we estimate from above using Young’s inequality to obtain Z t c F ( µ t ) W ( µ s , µ ∞ ) ds ≤ ε Z t (1 + s ) W ( µ s , µ ∞ ) ds + 12 ε Z t c F ( µ s )1 + s ds ≤ ε p Z t (1 + s ) J ( µ s , µ ∞ ) ds + 12 ε Z t c F ( µ s )1 + s ds, ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 17 where we used the equivalence (17) again. Choosing ε > βε = pc ℓ /q yields(1 + t ) G ( µ t , µ ∞ ) + c ℓ q Z t (1 + s ) J ( µ s , µ ∞ ) ds ≤ c + c Z t c F ( µ s )1 + s ds, which finally provides the required equilibration for t → ∞ . (cid:3) The case with no confinement.
The case without confinement requires special attentionsince, in this case, the free energy F is translational invariant and consequently the center of massis not a priori fixed. However, since the evolution of the center of mass (7) read ddt Z R d x dµ t = Z R d u t dµ t , ddt Z R d u t dµ t = − γ Z R d u t dµ t , solving for the center of mass of µ t clearly implies Z R d x dµ t = Z R d x dµ + 1 γ (1 − e − γt ) Z R d u dµ −→ Z R d x dµ + 1 γ Z R d u dµ as t → ∞ . Owing to the limit above for the center of mass of µ t , we choose a minimizer µ ∞ of F satisfying Z x dµ ∞ = Z x dµ + 1 γ Z u dµ . For this choice of stationary measure µ ∞ ∈ P ac ( R d ) we have the following statement. Theorem 4.
Let ( ̺, u ) be an energy decaying solution to the Euler equations (1) , with µ t ∈P ac ( R d ) the measure whose density is ̺ t for all t ≥ , and u satisfying additionally the assumptionsof Theorem 1, and U , V ≡ and W satisfying conditions (H1) - (H2) . Furthermore, assume thatthe initial data satisfies F ( µ ) + W ( µ , µ ∞ ) < ∞ , then lim t →∞ W ( µ t , µ ∞ ) = 0 . Proof.
As in Theorem 3, we make use of Corollary 1. In this particular case, we have J V ≡
0. Asfor J W we estimate as in Proposition 3 to obtain J W ( µ t | µ ∞ ) ≥ c W W ( µ t , µ ∞ ) − c W (cid:12)(cid:12)(cid:12)(cid:12)Z R d x dµ ∞ − Z R d x dµ t (cid:12)(cid:12)(cid:12)(cid:12) . On the other hand, we have that Z R d x dµ ∞ − Z R d x dµ t = 1 γ e − γt Z R d u dµ , which subsequently gives J W ( µ t | µ ∞ ) ≥ c W W ( µ t , µ ∞ ) − c W γ e − γt Z R d | u | dµ Taking the temporal derivative of the functional G provided in the proof of Theorem 3 gives d + dt G ( µ t , µ ∞ ) ≤ − c W E ( µ t , µ ∞ ) + 2 c W γ e − γt Z R d | u | dµ , where we chose α = γ and β = (1 + c W ) /γ . Integrating the inequality above in time gives G ( µ t , µ ∞ ) + 2 c W Z t E ( µ s , µ ∞ ) ds ≤ G ( µ , µ ∞ ) + c W γ Z | u | dµ . Since G ( µ t , µ ∞ ) ≥ t ≥
0, we finally obtain2 c W Z ∞ W ( µ s , µ ∞ ) ds ≤ c W Z ∞ E ( µ s , µ ∞ ) ds ≤ G ( µ , µ ∞ ) + c W γ Z | u | dµ < ∞ , and consequently the convergence due to the uniform continuity of t W ( µ t , µ ∞ ). (cid:3) Proceeding as in the proof of Corollary 2, we obtain the following result.
Corollary 3.
Let ( ̺, u ) be an energy decaying solution to the Euler equations (1) , with µ t ∈P ac ( R d ) the measure whose density is ̺ t for all t ≥ , and u satisfying additionally the assumptionsof Theorem 1, and U , V ≡ and W satisfying conditions (H1) - (H3) . Furthermore, assume thatthe initial data satisfies F ( µ ) + W ( µ , µ ∞ ) < ∞ , then lim t →∞ (cid:16) W ( µ t , µ ∞ ) + k u t k L ( µ t ) (cid:17) = lim t →∞ E ( µ t , µ ∞ ) = 0 . Overdamped limit ( γ → ∞ )In this section we consider the overdamped limit of (1) for large damping γ ≫
1. In this case,we rescale the time t = γ ˜ t , density ̺ t = ˜ ̺ ˜ t , and velocity u t = ˜ u ˜ t /γ . Dropping the tilde, we obtainthe rescaled Euler equations ∂ t ̺ γt + ∇ · ( ̺ γt u γt ) = 0 ,∂ t ( ̺ γt u γt ) + ∇ · ( ̺ γt u γt ⊗ u γt ) = − γ h ∇ p ( ̺ γt ) + ̺ γt ( ∇ V + ∇ W ⋆ ̺ γt ) + ̺ γt u γt i , (21)where we introduced the superscript γ to make explicit the dependence of the solutions ( ̺ γ , u γ )on the damping parameter γ . For the limit γ → ∞ , we wish to show that solutions ( ̺ γ , u γ )corresponding to (21) converge to the solution (¯ ̺, ¯ u ) of the first order equation ∂ t ¯ ̺ t + ∇ · (¯ ̺ t ¯ u t ) = 0 , ¯ u t = −∇ ( δ µ F )(¯ ̺ t ) . (22)It is well-known that the first order equation (22) has a gradient flow structure in P ( R d ) for thefunctional F and satisfies the decay estimates ddt F (¯ ̺ t ) = − Z R d | ¯ u t | ¯ ̺ t dx, ddt Z R d | ¯ u t | ¯ ̺ t dx = − D (¯ ̺ t ) , (23)with D (¯ ̺ t ) ≥ t ≥ ̺, ¯ u ) satisfies the momentum equation ∂ t (¯ ̺ t ¯ u t ) + ∇ · (¯ ̺ t ¯ u t ⊗ ¯ u t ) = − ¯ ̺ t G ¯ ̺ t , with G ¯ ̺ t = − ( ∂ t ¯ u t + ¯ u t · ∇ ¯ u t ) which we assume to be in L ((0 , T ) , L (¯ ̺ t dx )). In this case, giventhe measures µ γ and ¯ µ , with densities ̺ γ and ¯ ̺ respectively, we get Z R d h y − T t ( y ) , G ¯ µ t ( t, y ) i d ¯ µ t ≤ W ( µ γt , ¯ µ t ) + 12 Z R d | G ¯ µ t | d ¯ µ t , due to Young’s inequality, where T t µ t = µ γt . On the other hand, we have Z R d h y − T t ( y ) , G µ γt ( t, T t ( y )) i d ¯ µ t ≤ − γ (cid:20) c ℓ W ( µ γt , ¯ µ t ) + 12 ddt W ( µ γt , ¯ µ t ) (cid:21) , where c ℓ = c V + min { c W , } . Here, we explicitly used the fact that ¯ u t = −∇ ( δ µ F )(¯ µ t ). Hence, d + dt ddt W ( µ γt , ¯ µ t ) ≤ Z R d | u γt ( T t ( y )) − ¯ u t ( y ) | d ¯ µ t − (2 c ℓ γ − W ( µ γt , ¯ µ t ) − γ ddt W ( µ γt , ¯ µ t ) + Z R d | G ¯ µ t | d ¯ µ t , (24)holds due to Theorem 2. Now define the following functionals E ( µ t , ¯ µ t ) := W ( µ t , ¯ µ t ) + Z R d | u t | dµ t + Z R d | ¯ u t | d ¯ µ t , J ( µ t , ¯ µ t ) := αW ( µ t , ¯ µ t ) + ddt W ( µ t , ¯ µ t ) + β (cid:20)Z R d | u t | dµ t + Z R d | ¯ u t | d ¯ µ t (cid:21) , where α, β > αβ > / p E ≤ J ≤ q E for some constants p, q >
0, depending only on α and β . ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 19
Theorem 5.
Let ( ̺ γ , u γ ) be energy decaying solutions of (21) for all γ ≫ sufficiently large andlet (¯ ̺, ¯ u ) be a gradient flow solution to (22) satisfying additionally the assumptions of Theorem 2.We further assume the initial conditions to be well-prepared, i.e., ̺ γ = ¯ ̺ and u γ = ¯ u such that F (¯ µ ) + W ( µ γ , ¯ µ ) < ∞ , then lim γ →∞ Z T W ( µ γt , ¯ µ t ) dt = 0 , where µ γt and ¯ µ t are the measures with densities ̺ γt and ¯ ̺ t for all t ≥ respectively.Proof. The proof is based on Corollary 1 and follows the same line of arguments as in the previoustheorem. We begin by defining the functional G ( µ γt , ¯ µ t ) := 2 βγ (cid:0) F ( µ γt ) + F (¯ µ t ) (cid:1) + J ( µ γt , ¯ µ t ) . The temporal derivative of G gives d + dt G ( µ γt , ¯ µ t ) ≤ ( α − γ ) ddt W ( µ γt , ¯ µ t ) − βγ (cid:20)Z R d | u γt | dµ γt + Z R d | ¯ u t | d ¯ µ t (cid:21) − βD (¯ ̺ t )+ 2 Z R d | u γt ( T γt ( y )) − ¯ u t ( y ) | d ¯ µ t − (2 c ℓ γ − W ( µ γt , ¯ µ t ) + Z R d | G ¯ µ t | d ¯ µ t . where we used the estimates (23) and (24). Choosing α = γ and noting that D (¯ ̺ t ) ≥ Z R d | u γt ( T γt ( y )) − ¯ u t ( y ) | d ¯ µ t ≤ (cid:20)Z R d | u γt | dµ γt + Z R d | ¯ u t | d ¯ µ t (cid:21) , we further obtain d + dt G ( µ γt , ¯ µ t ) ≤ − (2 c ℓ γ − W ( µ γt , ¯ µ t ) − ( βγ − Z R d | u γt ( T γt ( y )) − ¯ u t ( y ) | d ¯ µ t + Z R d | G ¯ µ t | d ¯ µ t . We now choose β = 2 /γ and integrate in time t over [0 , T ] to obtain G ( µ γT , ¯ µ T ) + (2 c ℓ γ − Z T W ( µ γt , ¯ µ t ) dt ≤ G ( µ γ , ¯ µ ) + Z T Z R d | G ¯ ̺ | d ¯ µ t dt. Since F ( µ γt ) + F (¯ µ t ) ≥ − c for all t ≥ c ≥ Z T W ( µ γt , ¯ µ t ) dt ≤ M ( γ, T ) , (25)where M ( γ, T ) = 1(2 c ℓ γ − c + 8 F (¯ µ ) + (4 /γ ) Z R d | ¯ u | d ¯ µ + Z T Z R d | G ¯ µ t | d ¯ µ t dt ! , whenever γ > / (2 c ℓ ). Passing to the limit γ → ∞ concludes the proof. (cid:3) Remark 10.
Notice that for γ > / (2 c ℓ ), if sup ≤ T < ∞ M ( γ, T ) < ∞ , the estimate (25) with T → ∞ provides the convergence W ( µ γt , ¯ µ t ) → t → ∞ . Indeed, since R ∞ k G ¯ ̺ k L (¯ µ t ) dt isfinite, M ( γ ) < ∞ . Hence, the claim follows again from the uniform continuity of t W ( µ γt , ¯ µ t ).This means that the error between µ γt and ¯ µ t in W goes to zero as t → ∞ .6. Rigorous examples in the 1D case
In spatial dimension one, we obtain an easy representation of the 2-Wasserstein distance givenby the pseudo-inverse χ t defined as follows. Let F t ( x ) = Z x −∞ dµ t = µ t (( −∞ , x ]) ∈ Ω := [0 , , be the cumulative distribution of the probability measure µ t . Then χ t ( η ) = inf { x ∈ R | F t ( x ) > η } , defines the pseudo-inverse corresponding to µ t . In this case, the Wasserstein distance between twoprobability measures µ t and ν t is equivalently expressed as W ( µ t , ν t ) = Z Ω | χ t ( η ) − ζ t ( η ) | dη, where χ t and ζ t are pseudo-inverses corresponding to µ t and ν t respectively.The free energy corresponding to (3) in terms of the pseudo-inverse χ t is given by F ( χ t ) = 1 m − Z Ω ( ∂ η χ t ( η )) − m dη + Z Ω V ( χ t ( η )) dη + 12 Z Ω × Ω W ( χ t ( η ) − χ t (¯ η )) d ¯ η dη , and the entropy reads H ( χ t , v t ) = F ( χ t ) + 12 Z Ω | v t ( η ) | dη. In this case, the damped isentropic Euler equations can be transformed into ∂ t χ t ( η ) = u t ( χ t ( η )) =: v t ( η ) , (26a) ∂ t v t ( η ) = − ∂ η (cid:0) ( ∂ η χ t ( η )) − m (cid:1) − ( ∂ x V )( χ t ( η )) − Z Ω ( ∂ x W )( χ t ( η ) − χ t (¯ η )) d ¯ η − γv t ( η ) , (26b)for the Lagrangian quantities ( χ t , v t ) on Ω × R + with initial condition ( χ , v ) on Ω.As before, a simple verification of the temporal derivative of H for smooth solutions gives ddt H ( χ t , v t ) = − γ Z Ω | v t ( η ) | dη. Temporal derivatives of the Wasserstein distance.
We begin by computing the secondtemporal derivative of the Wasserstein distance. The first temporal derivative reads12 ddt W ( µ t , ν t ) = 12 ddt Z Ω | χ t ( η ) − ζ t ( η ) | dη = Z Ω h χ t ( η ) − ζ t ( η ) , v t ( η ) − w t ( η ) i dη, where v t and w t are the velocities corresponding to χ t and ζ t respectively. This yields12 d dt W ( µ t , ν t ) = Z Ω | v t ( η ) − w t ( η ) | dη + Z Ω h χ t ( η ) − ζ t ( η ) , ∂ t v t ( η ) − ∂ t w t ( η ) i dη =: I + I . The second term on the right-hand side gives I = Z Ω h ∂ η χ t ( η ) − ∂ η ζ t ( η ) , ( ∂ η χ t ( η )) − m − ( ∂ η ζ t ( η )) − m i dη − γ ddt W ( µ t , ν t ) − Z Ω h χ t ( η ) − ζ t ( η ) , ( ∂ x V )( χ t ( η )) − ( ∂ x V )( ζ t ( η )) i dη − Z Ω × Ω h χ t ( η ) − ζ t ( η ) , ( ∂ x W )( χ t ( η ) − χ t (¯ η )) − ( ∂ x W )( ζ t ( η ) − ζ t (¯ η )) i d ¯ η dη. Since the function z − m , m ≥ V and W appeared in ( H2 ), we obtain I ≤ − c ℓ Z Ω | χ t ( η ) − ζ t ( η ) | dη, for some constant c ℓ > d dt W ( µ t , ν t ) + γ ddt W ( µ t , ν t ) ≤ Z Ω | v t ( η ) − w t ( η ) | dη − c ℓ W ( µ t , ν t ) , which resembles the inequalities seen in Section 4. ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 21
In the following examples, we only consider pressureless dynamics, i.e., the damped Euler system(26) without internal pressure. In this case, the system for ( χ t , v t ) reads ∂ t χ t ( η ) = v t ( η ) , (27a) ∂ t v t ( η ) = − ( ∂ x V )( χ t ( η )) − Z Ω ( ∂ x W )( χ t ( η ) − χ t (¯ η )) d ¯ η − γv t ( η ) . (27b)6.2. Smooth solutions with repulsive Newtonian potential.
We consider the damped pres-sureless Euler for ( χ t , v t ) given by (27) with the explicit potentials V ( x ) = | x | / W ( x ) = −| x | .Notice that W now forms a repulsive interaction potential. In spatial dimension one, this interac-tion potential induces a free energy F on P ( R ) that is known to be 1-convex along generalizedgeodesics [20]. The stationary solution is known to be χ ∞ = 2 η − F [ χ t ] = χ t − χ ∞ =: ζ t . In this case, we consider the dynamics ∂ t χ t ( η ) = v t ( η ) , ∂ t v t ( η ) = − F [ χ t ]( η ) − γv t ( η ) , (28)and only consider smooth solutions of the system, i.e., χ t is strictly monotonically increasing upto sets of zero measure. Furthermore, since12 ddt Z Ω | ζ t | dη = Z Ω ζ t v t dη = Z Ω F [ χ t ] v t dη = ddt ( F ( χ t ) − F ( χ ∞ )) , we deduce that F ( χ t ) − F ( χ ∞ ) is essentially k ζ t k L (Ω) up to a constant shift, which allows us toonly work with k ζ t k L (Ω) . The remaining derivatives may be easily computed to obtain12 ddt Z Ω | v t | dη = − Z Ω v t ζ t dη − γ Z Ω | v t | dη,ddt Z Ω ζ t v t dη = Z Ω | v t | dη − γ Z Ω ζ t v t dη − Z Ω | ζ t | dη. Now define the functional G ( χ t , χ ∞ ) := ( β + γ ) Z Ω | ζ t | dη + 2 Z Ω ζ t v t dη + β Z Ω | v t | dη. Taking its temporal derivative gives ddt G = − Z Ω | ζ t | dη − βγ − Z Ω | v t | dη Choosing β = 2 /γ and using the fact that ( β + γ ) β = 2 + β >
1, we have ddt
G ≤ − (2 /q ) G = ⇒ G ( χ t , χ ∞ ) ≤ e − (2 /q ) t G ( χ , χ ∞ ) , for some positive constant q = q ( γ ), i.e., G decays to zero at an exponential rate. Summarizingthe above discussions we have the following theorem. Theorem 6.
Let ( χ, v ) be a global smooth solution of system (27) with potentials V ( x ) = | x | / and W ( x ) = −| x | . Suppose the initial data satisfies k χ − χ ∞ k L + k v k L < ∞ with χ ∞ = 2 η − ,then lim t →∞ (cid:16) k χ t − χ ∞ k L + k v t k L (cid:17) = 0 , exponentially fast. In particular, lim t →∞ W ( µ t , µ ∞ ) = 0 , exponentially fast with µ t = χ t Ω dη and µ ∞ = χ ∞ Ω dη = (1 / ( − , dx . Remark 11.
The system (28) has been extensively studied in [18] via the characteristics formu-lation with a friction coefficient of γ = 1 and is known to either have smooth solutions or solutionsmay blow up in finite time, depending on the total mass. Generalized Lagrangian solutions with attractive Newtonian potential.
The notionof sticky solutions of pressureless Euler systems have been considered since the 70’s to describe δ -shocks that may form in finite time [64]. Since then, numerous works have gone into the con-struction of such solutions, thereby extending the notion of a solution of (27) past the formationof δ -shocks [13, 14, 16, 26, 34, 57]. Here, we adopt the notion of generalized Lagrangian solutionsfor globally sticky dynamics found in [13].Consider the potentials V ( x ) = | x | / W ( x ) = | x | , which provides global attraction for theEuler system (27) with the free energy F ( χ t ) = 12 Z Ω | χ t | dη + 12 Z Ω × Ω | χ t ( η ) − χ t (¯ η ) | d ¯ η dη. Due to sufficiently strong attraction, one expects the formation of δ -shocks in finite time and theonly stable stationary solution of (27) is the Dirac measure µ ∞ = δ at x = 0. This correspondsto the stable stationary pseudoinverse χ ∞ ≡ K the closed convex cone of right-continuousnondecreasing functions in L (Ω), i.e., K = (cid:8) χ ∈ L (Ω) (cid:12)(cid:12) χ is nondecreasing (cid:9) , and I K : L (Ω) → [0 , + ∞ ] be the indicator function of K which is convex and lower semicontinuous.Hence, its subdifferential ∂I K ( χ ) at χ ∈ K is a maximal monotone operator on L (Ω) and it canbe characterized as ∂I K ( χ ) = (cid:26) ζ ∈ L (Ω) (cid:12)(cid:12)(cid:12) Z Ω ζ ( ¯ χ − χ ) dη ≤ χ ∈ K (cid:27) . Now define the setΩ χ := { η ∈ Ω | χ is constant in an open neighborhood of η } , and the closed subspace H χ = (cid:8) ζ ∈ L (Ω) | ζ is constant on each interval ( a, b ) ⊂ Ω χ (cid:9) , for any χ ∈ K . The projection P χ : L (Ω) → H χ is given by P χ ( ζ ) = ζ a.e. in Ω \ Ω χ and P χ ( ζ ) = 1 b − a Z ba ζ ( η ) dη in any maximal interval ( a, b ) ⊂ Ω χ . It was shown in [13] that the tangent cone T χ K to K at χ ∈ K can be characterized as T χ K = (cid:8) ζ ∈ L (Ω) | ζ is nondecreasing in each interval ( a, b ) ⊂ Ω χ (cid:9) . In particular, P χ ( ζ ) ∈ T χ K for any ζ ∈ L (Ω).Therefore, a global sticky dynamics for (27) can be formulated as ∂ t χ t ( η ) = P t ( v t )( η ) , ∂ t v t ( η ) = − P t ( F [ χ t ])( η ) − γv t ( η ) , (29)where F [ χ t ]( η ) = χ t ( η ) + 2 η − P t := P χ t is the projection onto the closed subspace H χ t defined above. For this particular choiceof potentials, there is a unique sticky Lagrangian solution ( χ t , v t ) of (29) with χ ∈ Lip loc ( R + , K ) , v ∈ C ( R + , L (Ω)) , and initial data ( χ , v ) ∈ K × L (Ω).We now compute the evolution of the energy corresponding to the solution pair ( χ t , v t ): d + dt (cid:20) F ( χ t ) + 12 Z Ω | v t | dη (cid:21) = Z Ω F [ χ t ] P t ( v t ) dη − Z Ω v t P t ( F [ χ t ]) dη − γ Z Ω | v t | dη. Notice that the following equalities hold Z Ω P t ( v t )( P t ( F [ χ t ]) − F [ χ t ])) dη = 0 = Z Ω P t ( F [ χ t ])( P t ( v t ) − v t ) dη. ONVERGENCE FOR DAMPED EULER EQUATIONS WITH INTERACTION FORCES 23
This consequently yields Z Ω F [ χ t ] P t ( v t ) − v t P t ( F [ χ t ]) dη = Z Ω P t ( v t ) P t ( F [ χ t ]) − P t ( F [ χ t ]) P t ( v t ) dη = 0 , and the above equality can be simplified to d + dt (cid:20) F ( χ t ) + 12 Z Ω | v t | dη (cid:21) = − γ Z Ω | v t | dη, which gives the same expression as the usual case and also implies the uniform temporal boundson F ( χ t ) and k v t k L (Ω) . In particular, we have the uniform estimatesup t ≥ (cid:16) F ( χ t ) + 12 k v t k L (Ω) (cid:17) ≤ F ( χ ) + 12 k v k L (Ω) =: c . Consequently, we conclude that F ( χ t ) ≤ Z Ω | χ t | dη + Z Ω | χ t | dη ≤ c F k χ t k L (Ω) with c F = 1 + ( √ c /
2) and this satisfies the required assumption in (H3) .Now taking the temporal derivative of the L -distance between χ t and χ ∞ = 0 gives12 d + dt Z Ω | χ t | dη = Z Ω χ t P t ( v t ) dη = Z Ω χ t ( P t ( v t ) − v t ) dη + Z Ω χ t v t dη = Z Ω χ t v t dη, where we used the fact that χ t ∈ H χ t and P t ( v t ) − v t ∈ H ⊥ χ t . Since χ t is only locally Lipschitzcontinuous in time, we are unable to consider the second temporal derivative. Instead we compute d + dt Z Ω χ t v t dη = Z Ω P t ( v t ) v t dη − γ Z Ω χ t v t dη − Z Ω χ t P t ( F [ χ t ]) dη ≤ Z Ω | v t | dη − γ Z Ω χ t v t dη − Z Ω χ t F [ χ t ] dη, where we used the nonexpansive property of the projection P t for all times t ≥ Z Ω χ t F [ χ t ] dη = Z Ω | χ t | dη + Z Ω χ t (2 η − dη = Z Ω χ t dη − Z Ω ∂ η χ t ( η − η ) dη. Since χ t ∈ K and η ( η − ≤ η ∈ Ω, we have that Z Ω ∂ η χ t ( η − η ) dη ≤ , and therefore d + dt Z Ω χ t v t dη ≤ Z Ω | v t | dη − γ Z Ω χ t v t dη − Z Ω | χ t | dη. From here, we may proceed as in the previous section to conclude convergence to equilibrium forthe global sticky dynamics towards the unique stationary solution δ ∈ P ( R ). We can summarizethe discussion above in the following result. Theorem 7.
Let ( χ, v ) be a global Lagrangian solution of system (27) with potentials V ( x ) = | x | / and W ( x ) = | x | . Suppose the initial data satisfies k χ k L + k v k L < ∞ , then lim t →∞ (cid:16) k χ t k L + k v t k L (cid:17) = 0 . In particular, lim t →∞ W ( µ t , µ ∞ ) = 0 , with µ t = χ t Ω dη and µ ∞ = δ . acknowledgements JAC was partially supported by the EPSRC grant number EP/P031587/1. YPC was supportedby National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(No. 2017R1C1B2012918 and 2017R1A4A1014735) and POSCO Science Fellowship of POSCOTJ Park Foundation. The authors are very grateful to the Mittag-Leffler Institute for providing afruitful working environment during the special semester
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Department of Mathematics, Imperial College London,London SW7 2AZ, United Kingdom
E-mail address : [email protected] (Young-Pil Choi) Department of Mathematics and Institute of Applied Mathematics, Inha University,Incheon 402-751, Republic of Korea
E-mail address : [email protected] (Oliver Tse) Department of Mathematics and Computer Science, Eindhoven University of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands
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