Quantifying the hydrodynamic limit of Vlasov-type equations with alignment and nonlocal forces

In this paper, we quantify the asymptotic limit of collective behavior kinetic equations arising in mathematical biology modeled by Vlasov-type equations with nonlocal interaction forces and alignment. More precisely, we investigate the hydrodynamic limit of a kinetic Cucker--Smale flocking model with confinement, nonlocal interaction, and local alignment forces, linear damping and diffusion in velocity. We first discuss the hydrodynamic limit of our main equation under strong local alignment and diffusion regime, and we rigorously derive the isothermal Euler equations with nonlocal forces. We also analyze the hydrodynamic limit corresponding to strong local alignment without diffusion. In this case, the limiting system is pressureless Euler-type equations. Our analysis includes the Coulombian interaction potential for both cases and explicit estimates on the distance towards the limiting hydrodynamic equations. The relative entropy method is the crucial technology in our main results, however, for the case without diffusion, we combine a modulated macroscopic kinetic energy with the bounded Lipschitz distance to deal with the nonlocality in the interaction forces. For the sake of completeness, the existence of weak and strong solutions to the kinetic and fluid equations are also established.


Introduction
Collective self-organized motions of autonomous individuals, such as flocks of birds, crowd dynamics, and aggregation of bacteria, etc, appear in many applications in the field of engineering, biology, and sociology [2,3,4,25,36,39,40,42,45], we refer to [10,19] and references therein for recent surveys. Mathematical modelling of such behaviors is based on Individual-Based Models (IBMs) which are microscopic descriptions, and it includes three basic effects, a short-range repulsion, a long-range attraction, and an alignment in certain spatial regions. These IBMs lead to continuum description by means of mean-field limit [5,7,8,22,31,32], and in particular a second-order N -particle system converges toward a kinetic equation as the number of particles N goes to infinity. In this paper, we study a class of such kinetic-type models which are typically Vlasov-type equations with nonlocal forces. More precisely, let f = f (x, v, t) be the one-particle distribution function at (x, v) ∈ Ω × R d and at time t > 0, where Ω is either T d or R d with d ≥ 1, then our main equation is given by with (x, v, t) ∈ Ω × R d × R + subject to the initial data: where ρ = ρ(x, t) and u = u(x, t) are the local particle density and velocity given by respectively, V : R d → R and W : R d → R are the confinement and the interaction potentials with a positive coefficient λ, respectively. Here N F P denotes the nonlinear Fokker-Planck operator [46] given by with the local Maxwellian and positive constants β and σ. F represents the velocity alignment force fields, where the local average of relative velocities weighted by the function φ, given by where φ : R d → R + is called a communication weight. The confinement and interaction potentials are assumed to be symmetric in the sense V (x) = V (−x) and W (x) = W (−x) on R d due to the action-reaction principle by Newton's third law. The weight function φ is usually assumed to be radially symmetric, i.e., φ(x) =φ(|x|) for someφ : R + ∪ {0} → R + , andφ is decreasing such that the closer particles have more stronger influence than the further ones. The right hand side of (1.1) consists of the local alignment forces and the diffusion term in velocity. Throughout this paper, we also assume that f is a probability density, i.e., f (·, ·, t) L 1 = 1 for t ≥ 0, since the total mass is preserved in time.
In the current work, we are interested in the asymptotic analysis of (1.1) by considering singular parameters. More specifically, we deal with hydrodynamic limits to isothermal/pressureless Euler equations with nonlocal forces.
1.1. Formal derivation from kinetic to isothermal/pressureless Euler equations. Taking into account the moments on the kinetic equation (1.1), we find that the local density ρ and local velocity u satisfy ∂ t ρ + ∇ x · (ρu) = 0, We notice that the above system is not closed in the sense that it cannot be written only in terms of ρ and u. On the other hand, if we consider the singular parameters β = σ = 1/ε in (1.1), i.e., the local alignment and diffusive forces are very strong and consider the limit ε → 0, then at the formal level, we expect that N F P ≃ 0, and this leads that the particle density behaves like: where f ε denotes the corresponding solution of (1.1) with β = σ = 1/ε. This formal procedure gives the isothermal Euler equations with interaction forces: Let us now take into account the hydrodynamic limit without diffusion, i.e., the equation (1.1) with β = 1/ε and σ = 0. Then, for the similar reason, we find that 4) and this induces the following pressureless Euler equations with interaction forces: ∂ t ρ + ∇ x · (ρu) = 0, (x, t) ∈ Ω × R + , φ(x − y)(u(x) − u(y))ρ(y) dy. (1.5) Some previous works closely related to the above asymptotic analysis can be summarized as follows. The asymptotic analysis for the kinetic Cucker-Smale model with a strong local alignment force and a strong diffusion, i.e., (1.1) with γ = 0, λ, α > 0, V, W ≡ 0, σ = β = 1/ε, is investigated in [35]. In this regime, the isothermal Euler system with the nonlocal velocity alignment forces, (1.3) with γ = λ = 0 and α > 0 is rigorously derived, see also [18] for the global regularity of classical solutions of that system. In this work, the relative entropy method is employed, and the presence of the pressure term in the limiting system plays an important role in their strategy: it gives the convexity of the entropy with respect to the density ρ; see Section 3 for details. For the diffusionless case, in [28], the velocity alignment term F [f ] is taken into account in the hydrodynamic limit, i.e., the equation (1.1) with V, W ≡ 0, γ = σ = 0, α > 0, and β = 1/ε in the periodic spatial domain, and the pressureless Euler equations with the velocity alignment forces, (1.5) with λ = γ = 0 and α > 0, which is also referred to Euler alignment system in [11], are rigorously derived. In that work, the modulated macroscopic energy combined with the second-order Wasserstein distance is used. This strategy is improved in a recent work [6] where the whole space case is considered, see also [17] for the relation between modulated macroscopic kinetic energy and the pth order Wasserstein distance. It is worth noticing that the interaction potential W is not taken into account in [28,35], and it is not clear that the strategies used in that work can be applied to the case with the interaction potential W when W has a rather weak regularity, see [6] for the case with regular interaction potentials W . On the other hand, for the Coulombian interactions W , i.e., −∆ x W ⋆ ρ = ρ, the hydrodynamic limit of Vlasov-Poisson equation with strong local alignment forces, which corresponds to (1.1) with γ = α = σ = 0, V ≡ 0, β = 1/ε, is discussed in [33].
The main purpose of this work is to consider the most general form of kinetic swarming models (1.1) and identify regimes where the Euler-type equations (1.3) or (1.5) are well approximated by the kinetic equation (1.1) in a quantifiable way. We first deal with the equation (1.1) with strong local alignment and diffusive forces, that is, we consider a singular parameter in the nonlinear Fokker-Planck operator N F P . In this case, as mentioned above, the limiting system is expected to be the isothermal Euler-type system (1.3). We estimate the relative entropy functional together with the free energy to have the quantitative error estimate between solutions f ε of (1.1) and (ρ, u) of (1.3). In particular, we make the formal observation (1.2) completely rigorous with a quantitive bound in terms of ε, see Corollary 2.2. Due to the presence of pressure, L ∞ bound assumptions for both the interaction potential W and the communication weight function φ are sufficient to have that estimate of hydrodynamic limit. We are also able to deal with the Coulombian potential for W .
In the case without diffusion, the limiting system is a pressureless Euler system (1.5), thus the corresponding macroscopic kinetic energy is not strictly convex with respect to ρ. In this respect, it is not clear to have the quantitative bound error estimate between solutions by means of the estimate of modulated kinetic energy only. For that reason, we combine the modulated kinetic energy estimate and the bounded Lipschitz distance between local particle density ρ ε and the fluid density ρ. Note that the bounded Lipschitz distance and the first order Wasserstein distance are equivalent in the set of probability measure with a bounded first moment. Thus our result improves the previous works [6,28], where the second-order Wasserstein distance is used as mentioned above. We show that the bounded Lipschitz distance between densities can be bounded from above by the modulated macroscopic kinetic energy, see Lemma 4.1. Compared to the case with pressure, we need rather stronger assumptions for W and φ, bounded and Lipschitz continuity. Combining these observations, we close the modulated kinetic energy estimates and obtain the quantitative error estimates between solutions f ε of (1.1) with β = 1/ε, σ = 0 and (ρ, u) of (1.5). As we expected from the formal derivation (1.4), the particle distribution function f ε converges to the monokinetic ansatz in the sense of distributions also quantified in terms of the bounded Lipschitz distance, see Corollary 2.3 and the proofs in Subsection 2.3. Even in the pressureless case, we are also able to take into account the Coulombian interaction potential W and establish the same convergence estimates with the regular interaction potential case. Our main mathematical tool is based on the weak-strong uniqueness principle [26], and thus for the rigorous asymptotic analysis mentioned above, the existence of weak solutions of the kinetic equation (1.1) and strong solutions to the limiting systems (1.3) and (1.5) should be obtained at least locally in time. We emphasize that it is important to have the global-in-time weak solutions of (1.1) satisfying the free energy estimate.
Here we introduce several notations used throughout the current work. For functions, f (x, v) and g(x), f L p and g L p represent the usual L p (Ω × R d )-and L p (Ω)-norms, respectively. We denote by C a generic positive constant which may differ from line to line. For simplicity, we often drop x-dependence of differential operators, that is, ∇f := ∇ x f and ∆f := ∆ x f . For any nonnegative integer k and p ∈ [1, ∞], W k,p = W k,p (Ω) stands for the k-th order L p Sobolev space. In particular, if p = 2, we denote by H k = H k (Ω) = W k,2 (Ω). C k ([0, T ]; E) is the set of k-times continuously differentiable functions from an interval [0, T ] ⊂ R into a Banach space E, and L p (0, T ; E) is the set of functions from an interval (0, T ) to a Banach space E. ∇ k denotes any partial derivative ∂ α with multi-index α, |α| = k.
The rest of this paper is organized as follows. In the next section, we provide several a priori estimates of free energy inequalities. We also give precise statements of our main results on the asymptotic analysis of (1.1). In Section 3, we consider our main equation (1.1) in the regime of strong local alignment and diffusion, i.e., β = σ = 1/ε. We show that the weak solution to the kinetic equation (1.1) strongly converges to the strong solution to the isothermal Euler equations with nonlocal interaction forces (1.3). Section 4 is devoted to the asymptotic analysis for the diffusionless case, i.e., (1.1) with σ = 0. In this case, we consider the strong local alignment regime, β = 1/ε and provide the rigorous convergence estimates of solutions f ε to the pressureless Euler system with nonlocal interactions forces (1.5). Finally, in Sections 5 and 6 we provide the details on the global-in-time existence of weak solutions for the kinetic equation (1.1) satisfying the free energy estimate and the local-in-time existence and uniqueness of classical solutions to the isothermal/pressureless Euler equations (1.3) and (1.5).

Preliminaries and main results
2.1. Free energy estimates. In this part, we provide free energy estimates. For this, we introduce the free energy F and the associated dissipations D 1 , D 2 , and D 3 as follows: and respectively.
Then we have the following free energy estimate.
Lemma 2.1. Suppose that f is a solution of (1.1) with sufficient integrability. Then we have In particular, we have d dt when σ = 0.

Proof. A straightforward computation gives
where I i , i = 1, 2, 3, can be estimated as follows: We also estimate the kinetic energy as Combining the above estimates yields Lemma 2.1 shows that the linear damping in velocity and nonlocal velocity generate the free energy increase. In the proposition below, we show that they are controlled by the dissipations and the free energy.
Proposition 2.1. Suppose that f is a solution of (1.1) with sufficient integrability. Then we have Furthermore, we obtain where C > 0 depends only T , f 0 and φ L ∞ .
Proof. It follows from [ where C depends only on T , φ L ∞ . On the other hand, a straightforward computation gives where J 2 can estimated as For the estimate of J 1 , we use Hölder inequality to get i.e., Now we combine the above estimates together with Lemma 2.1 to obtain d dt Applying Gronwall's inequality to the above concludes the desired first result. The second inequality just follows from the first result and the above inequality.
We next state definitions of strong solutions to the systems (1.3) and (1.5) below.
(ρ, u) satisfies the following free energy estimate in the sense of distributions: (iii) (ρ, u) satisfies the system (1.5) in the sense of distributions.
Before providing our results on the hydrodynamic limits, we list our main assumptions on the initial data below.
(H1) The initial data related to the entropy are well-prepared: and (H2) The initial data related to the kinetic energy part in the entropy are well-prepared: and (H3) The bounded Lipschitz distance between initial local densities satisfies Remark 2.1. If we choose the initial data f ε 0 as then we obtain Let us define the classical relative entropy between two probability densities ρ 1 and ρ 2 as and analogously for two densities f 1 and f 2 in phase space as Remark 2.2. The first assumptions in (H1) and (H2) imply that (ii) Weakly regular case ∇W ∈ L ∞ (Ω): Here C > 0 is a positive constant independent of ε.

Corollary 2.2.
Suppose that all the assumptions in Theorem 2.1 hold. Moreover, we assume that the confinement potential V satisfies |∇V (x)| 2 ≤ C|V (x)| for some C > 0 and the solution ρ to the limiting system is regular such that ∇W ⋆ ρ ∈ L ∞ (Ω × (0, T * )). Then for t ≤ T * , we have for the weakly regular potential case (ii), and for the Coulombian potential case (i), where C > 0 is independent of ε > 0. In particular, if the right hand side of the above inequality convergences to zero, then we have as ε → 0.
Proof. Since this proof is lengthy and technical, we postpone it to Appendix A.  (i) Coulombian case ∆W = −δ 0 : (ii) Strongly regular case ∇W ∈ W 1,∞ (Ω): Here C > 0 is a positive constant independent of ε.
Remark 2.5. Compared to Theorem 2.1 (ii), the pressureless case requires higher regularity for W , like ∇W ∈ W 1,∞ (Ω) due to the lack of convexity of the entropy with respect to ρ. If for Coulombian interaction case, then the following convergences hold: Here M is the space of nonnegative Radon measures.
Remark 2.6. The convergence of d BL (ρ ε , ρ) directly gives Remark 2.7. Our results on the hydrodynamic limit also hold in a bounded domain with the specular reflection boundary condition. In this case, the limiting system has a kinematic boundary condition. Concerning this, we provide the existence theory in Section ??. For the hydrodynamic limit estimate, we refer to [21] where the hydrodynamic limit of nonlinear Vlasov-Fokker-Planck/Navier-Stokes equations in a bounded domain is discussed.

2.3.
Proofs of Corollaries 2.1 and 2.3. Before proceeding, for the readers' convenience, we provide the details of proofs of convergences in Corollaries 2.1 and 2.3. In fact, we provide quantitative bounds of convergences.
Lemma 2.2. There exists a positive constant C depending only on u W 1,∞ such that the following inequalities hold.
(i) Error estimate between moments: (ii) Error estimate between convections: (iii) Error estimate between particle distribution and mono-kinetic ansatz: Proof. For any ψ ∈ (L ∞ ∩ Lip)(Ω), we get This asserts the inequality (i). For the estimate of (ii), we notice that Using this identity, we obtain This yields where C > 0 depends only on u W 1,∞ . For (iii), we find for any ϕ ∈ (L ∞ ∩ Lip)(Ω × R d ) that This concludes the inequality (iii).
Proof of Corollary 2.1. We first obtain where C > 0 depends only on ρ ε L 1 and ρ L 1 , see (3.4) for details. This together with Lemma 2.2 yields On the other hand, we find from [35,Lemma 4.8] This yields Combining this, (2.5), and Proposition 2.1 with This completes the proof.
Proof of Corollary 2.3. A simple combination of inequalities in Lemma 2.2 together with Theorem 2.2 gives This asserts the first two convergences. Note that This yields On the other hand, it follows from (2.1) with β = 1/ε that This together with (2.7) implies the third assertion. We also use Lemma 2.2 and (2.

Hydrodynamic limit from kinetic to isothermal Euler equations
In this section, we study the rigorous derivation of the isothermal Euler equations (1.3) from the kinetic equation (1.1) with β = σ = 1/ε as ε → 0. As mentioned before, we use the relative entropy argument based on the weak-strong uniqueness principle to have the quantitative error estimates between the kinetic equation and the limiting system.

3.1.
Relative entropy inequality. We rewrite the equations as a conservative form: Here I d×d denotes the d × d identity matrix. The free energy of the above system is given by We now define the relative entropy functional E between two states of the system U andŪ as follows.
where DE(U ) denotes the derivation of E with respect to ρ, m, i.e., This yields where H(ρ|ρ) is the relative entropy between densities given by (2.3). By Taylor's theorem, we readily see and moreover, we get Thus we obtain where C > 0 only depends on ρ L 1 and ρ L 1 .
Remark 3.1. The free energy of the system (3.1) is given bỹ and we can also define its modulated energy, also often called the relative entropy, as A straightforward computation shows and by symmetry of W , we obtain This functionalẼ is used in the study of large friction limit of Euler equations with nonlocal forces [14,37,38], see also [17] for the pressureless case. However, we employ the form (3.2) to use the estimates in [35] providing the relation between E(U ) and the flux A(U ), see the estimate of I 3 in the proof of Lemma 3.1 below.
Lemma 3.1. The relative entropy E defined in (3.3) satisfies the following equality: where A(Ū |U ) is the relative flux functional given by We first use the integration by parts [35,Lemma 4.1] to get For the estimate I 4 , we notice that Then, by direct calculation, we find Thus we obtain Here we follow the same argument as in [35] to get We next estimate I 3 4 + I 4 4 as Combining the above estimates yields This completes the proof.
We now set where f ε is a weak solution to the equation (1.1).
Proof. It follows from Lemma 3.1 that Here J ε i , i = 1, · · · , 8 can be estimated as follows.
Estimate of J ε 1 : By the assumption, we get Thus, by adding and subtracting the functional K(f ε ), we find Estimate of J ε 3 : It follows from [35,Lemma 4.3] that

This yields
This implies On the other hand, we recall (2.6) that By this and Proposition 2.1, we obtain where we used Hölder inequality to find Here C > 0 depends on T , ∇u L ∞ . It is worth emphasizing that our estimate gives that the constant C depends on ∇u L ∞ , while [35,Lemma 4.8] provides that it also depends on ∇ log ρ L ∞ .
Estimate of J ε 5 : We again use (3.5) to get Thus we have Estimate of J ε 6 : A straightforward computation gives where C > 0 depends on u L ∞ , ρ ε L 1 , and ρ L 1 .
Estimate of J ε 7 : Integrating by parts gives Thus we get Estimate of J ε 8 : Note that This yields We now combine the estimates J ε i , i = 2, 5, 7, 8 to get i∈{2,5,7,8} We then use Proposition 2.1 to find i∈{2,5,7,8} We finally combine all the above estimates to conclude the proof.

Singular/weakly regular interactions
. In this part, we consider the Coulombian interactions W satisfying ∆W = −δ 0 . Motivated from [13,37,38], we use a particular structure of the Poisson equation.
Proof. Using the continuity equations of ρ and ρ ε , we find λ 2 We then use the symmetry of W to get We finally combine (3.6) and (3.7) to conclude the proof.
Then we are now ready to provide the details of the proof of Theorem 2.1 for the singular interactions case. Since the strong convergences (2.4) can be obtained from the inequalities in Theorem 2.1 and it is also already discussed in [35], we skip the details of that proof.
Proof of Theorem 2.1 (i). Using the integration by parts, we estimate This together with Lemma 3.2 and Proposition 3.1 yields We finally apply Grönwall's lemma to the above to conclude the desired result. This completes the proof.
Indeed, we can easily find . In this part, we deal with the weakly singular interactions case.
Proof. We use Hölder inequality to get On the other hand, L 1 -norm of ρ − ρ ε can be estimated as Proof of Theorem 2.1 (ii). By combining Lemma 3.3 and Proposition 3.1, we find We complete the proof by using the Gronwall inequality to the above.

Hydrodynamic limit from kinetic to pressureless Euler equations
In this section, we consider the hydrodynamic limit from (1.1) with σ = 0 to the pressureless Euler equations with nonlocal interaction forces (1.5). Similarly as before, we rewrite the limiting system (1.5) as the following conservative form: We then consider the kinetic energy of the above system: Note that the entropy defined above is not strictly convex with respect to ρ. We also define the modulated kinetic energy asÊ Compared to the previous diffusive case, our functionalÊ does not include the relative pressure, and as a consequence we cannot deal with the L 1 -norm of theρ − ρ. Thus we need to estimate the nonlocal interaction forces in a different way. For this, we will use a bounded Lipschitz distance for local densities, and this requires a higher regularity for the communication weight φ.

Modulated kinetic energy inequality.
In the proposition below, we provide the modulated kinetic energy estimate.
Proof. Employing almost the same arguments as in Lemma 3.1 and Proposition 3.1, we find whereK(f ) denotes the kinetic energy for the kinetic equation, i.e., On the other hand, we notice that and this yields For the term with the communication weight function φ, we denoted it by I ε and split into two terms: where I ε 1 can be estimated as Here we used the fact that y → φ(·, y)u(y) is bounded and Lipschitz continuous. Similarly, we can also show that and this yields where C > 0 is independent of ε > 0. This completes the proof.

Singular/strongly regular interactions:
In this subsection, we consider the Coulombian interaction potential W , i.e., W satisfies ∆W = −δ 0 . We first notice from Lemma 3.2, see also proof of Theorem 2.1 (i), that the last term on the right hand side of (4.1) can be bounded from above by We next use the free energy estimate (2.1) to show where C > 0 is independent of ε. Combining those observations with Proposition 4.1 yields the following proposition.
In order to close the modulated kinetic energy inequality, we show that the bounded and Lipschitz distance d BL between local densities can be bounded from above by the modulated kinetic energy, which directly gives the quantitative error estimate between ρ and ρ ε .
Although it has been already studied in [17], see also [6,28], we give the details of proof for the completeness of our work. Let us define forward characteristics X(t) := X(t; 0, x) which solves Then X(t) uniquely exists on the time interval [0, T ] since u is bounded and Lipschitz continuous. Moreover, the solution ρ can be determined as the push-forward of the its initial densities through the flow maps X, i.e., ρ(t) = X(t; 0, ·)#ρ 0 . On the other hand, we cannot consider the characteristic for the continuity equation of ρ ε due to the lack of regularity of u ε . Regarding this problem, we recall the following proposition from  (i) η is concentrated on the set of pairs (γ, x) such that γ is an absolutely continuous curve satisfyinġ Proof of Lemma 4.1. It follows from Lemma 2.1 that thus, the integrability condition (4.3) holds for p = 2, and thus by Proposition 4.3, we obtain a probability measure η ε in Γ T × R concentrated on the set of pairs (γ, x) such that γ is a solution oḟ with γ(0) = x ∈ Ω. Moreover, ρ ε satisfies Proposition 4.3 (ii), i.e., We now consider the push-forward of ρ ε 0 through the flow map X and denote it byρ ε , i.e.,ρ ε = X#ρ ε 0 . We first estimate the error betweenρ ε and ρ ε in bounded Lipschitz distance. By the disintegration theorem of measures, see [1], we can write x } x∈Ω is a family of probability measures on Γ T concentrated on solutions of (4.4). By using this newly introduced measure η ε , we define a measure ν ε on Γ T × Γ T × Ω by We further take into account an evaluation map E t : Γ T ×Γ T ×Ω → Ω×Ω defined as E t (γ, σ, x) = (γ(t), σ(t)). Then we find that measure π ε t := (E t )#ν ε on Ω × Ω has marginals ρ ε (x, t) dx andρ ε (y, t) dy for t ∈ [0, T ], see (4.5). This implies We then apply Grönwall's lemma to the above to yield where C > 0 is independent of ε > 0. Putting this into (4.6) entails where C > 0 is independent of ε > 0, and we used (4.5).
We next estimate the bounded Lipschitz distance betweenρ ε and ρ. For bounded Lipschitz function φ, we find where C > 0 is independent of ε, and we used the bounded Lipschitz continuity of φ(X(t; 0, ·)). More precisely, we have |u(X(s; 0, x)) − |u(X(s; 0, y))| ds and applying Grönwall's lemma to the above yields the Lipschitz continuity of the characteristic flow X(t; 0, x) in x. Furthermore, we have where · Lip denotes the Lipschitz constant given by This together with (4.8) implies for t ∈ [0, T ], where C > 0 is independent of ε > 0. Finally, we combine this with (4.7) to conclude where C > 0 is independent of ε > 0.

Proof of Theorem 2.2 (i). Applying Lemma 4.1 to Proposition 4.2 yields
We then use Grönwall's lemma to the above and the assumptions (H2)-(H3) to conclude the desired result.
where C > 0 is independent of ε.
Proof of Theorem 2.2 (ii). We first claim that Indeed, since ∇W ∈ W 1,∞ (Ω), we get This yields

This together with Proposition 4.4 provides
Hence, by applying Gronwall's lemma to the above, we complete the proof.

Global existence of weak solutions to the kinetic equation (1.1)
In this section, we provide the global-in-time existence of weak solutions to the system (1.1). For notational simplicity, we set γ = λ = α = β = 1. We also only consider the Coulombian case since the weakly singular case ∇ x W ∈ L ∞ (Ω) can be easily obtained similarly as in [34]. Here, the domain of our interest is Ω = T d or R d with d ≥ 3. Since the analysis on T d is almost similar to the R d case, we mostly consider the case Ω = R d . Furthermore, f satisfies the entropy inequality (2.2).

Regularized kinetic equation.
For the existence of weak solutions to (1.1), we first regularize the system with respect to regularization parameters η := (R, ζ, ε) as follows: subject to initial data: where c d is a normalization constant. Moreover, χ ζ is given as and V R is given as Here M (x) ∈ C ∞ c (R d ) is a smooth function given by Now, we partially linearize (5.1) as follows: whereũ is in S := L 2 (Ω × (0, T )). Once we note that ∇W ε is bounded and Lipschitz continuous, the existence of weak solutions to (5.2) comes from almost the same argument in [34, Theorem 6.3]. Moreover, we estimate . This together with Grönwall's lemma gives In particular, we have We next estimate higher-order velocity moments and entropy inequality of solutions to (5.2). Then, we estimate where C = C(d, η, N, T ) is a positive constant and we used Young's inequality. Since m 0 (f η ) is just f η L 1 = f η 0 L 1 , one uses Grönwall's lemma and induction argument to conclude that Moreover, for N ∈ R + \ {0, 2, 4, · · · }, we can find l ∈ N ∪ {0} that satisfies 0 < N − 2l < 2, and this gives This asserts our desired result.
Proof. First, it directly follows from Lemma 5.1 that On the other hand, we get d dt This yields d dt We then combine the previous estimates with the following entropy estimate to conclude the desired result.

5.2.
Existence of the regularized kinetic equation. Now, we provide the existence of weak solutions to (5.1) and their energy estimates. Similarly as in [9,41], we define the mapping T : S → S, where S = L 2 (Ω × (0, T )) byũ First, we prove that the operator T is well-defined.
For a weak solution f η to (5.2), the averaged quantities (ρ η , ρ η u η ) satisfy and as a consequence, T is well-defined.
Proof. Since the proof for the first assertion can be found in [34], we omit its proof. Since ρ η is bounded, it suffices to show the boundedness of u η ε . Obviously, and since ρ η u η is bounded, T is well-defined.
Next, we discuss the compactness of T . Here, we state the velocity averaging lemma from [43].
We then use the previous lemma to obtain the following result, which is very similar to [ Then, for any ϕ(v) satisfying |ϕ(v)| ≤ c|v|, the sequence is relatively compact in L q (R d+1 ) for any q ∈ 1, d+r d+1 .
Thanks to Lemma 5.4, we can prove the compactness of T . Proof. For the convergence of {(u η ε ) m }, we set then it is easy to see G m ∈ L p loc (R 2d+1 ). Let us choose r appeared in (5.3) sufficiently large. Then, we set ϕ(v) = 1 and ϕ(v) = v in Lemma 5.4, respectively, and obtain the following strong convergence up to a subsequence: (ρ η ) m → ρ η in L 2 (Ω × (0, T )) and a.e., and consequently, it gives the convergence of {(u η ε ) m } up to a subsequence.
Proof. From the existence of the fixed point of T and Proposition 5.1, it is obvious that Here, we note that 5.3. Proof of Theorem 5.1. Now, we provide the existence of a weak solution to (1.1) based on the energy inequality, compactness argument and velocity averaging lemma.
• (Step B: Existence of weak solutions and entropy inequality) Now, it remains to show that the limit f ε satisfies the following equation in distributional sense: and also the entropy inequality: For f ε to be a weak solution to (5.5), it suffices to show the following convergence in distribution sense since the others are obvious: Then, we have For K 1 1 , since f η is uniformly bounded in L ∞ (Ω × R d × (0, T )) and Ψ is compactly supported, it is obvious that ρ η Ψ ∈ L p (0, T ; L q (Ω)) for any p, q ∈ [1, ∞], uniformly in η.
Thus, due to the weak convergence of f η , we get K 2 1 → 0 as R → ∞.
For K 2 2 , we use the compact support of Ψ to get for some p ∈ (1, ∞), and we combine this with the weak convergence of f η to yield K 2 2 → 0 as R → ∞.
⋄ (Step B-4: Entropy inequality) For (5.6), we first take the liminf on the left hand side of Corollary 5.2, convexity of the entropy and use f η Here, we use the reverse Fatou's lemma and the pointwise convergences and we claim that the following convergence holds: For this, we present a theorem similar to Vitali convergence theorem whose proof is presented in Appendix B. Note that when Ω = T d , the condition (ii) is unnecessary. (iii) for every ε > 0, there exists δ = δ(ε) > 0 such that whenever m(E) < δ, In our case, since f η ⇀ f ε in L 1 (Ω × R d × (0, T )) and φ ∈ L ∞ (Ω), we obtain for each x ∈ Ω and t ∈ (0, T ). This implies the convergence of φ ⋆ ρ η to φ ⋆ ρ ε almost everywhere in Ω × (0, t).
For the condition (ii) in Theorem 5.2, we let L > 0 and , and we can deduce the condition (ii) from the above. For the third condition (uniform integrability condition), we choose a measurable set E ⊂ Ω with m(E) < δ. Then, we have where p ∈ (1, (d + 2)/d) and C is a constant independent of η, and this implies the uniform integrability condition. This concludes our desired result.

5.3.2.
Convergence ε → 0. Finally, it remains to prove the convergence as ε → 0. We note that the weak convergence of f η to f ε implies the following uniform upper bound estimate: for p ∈ [1, ∞]. Thus, we combine (5.8) with the entropy inequality (5.6) to get the following weak convergence up to a subsequence as before: p ∈ [1, (d + 2)/(d + 1)).
Moreover, applying the velocity averaging lemma, Lemma 5.4, asserts the strong convergence up to a subsequence for p ∈ (1, (d + 2)/(d + 1)): T ) × Ω) and a.e., ρ ε u ε → ρu in L p ((0, T ) × Ω) (5.9) as ε → 0. Now, to show that f is a weak solution to (1.1), it suffices to show the following convergence in distribution sense, since the others are obvious or can be obtained in the same way from the previous argument: Since the estimates for K 2 4 and K 3 4 are similar to those for K 1 1 and K 2 1 , respectively, we only need to show K 1 4 → 0 as ε → 0. Still, thanks to the uniform-in-ε estimate for f ε in L ∞ (Ω × R d × (0, T )) and the compact support of Ψ, we find ρ ε Ψ ∈ L p (0, T ; L q (Ω)) for any p, q ∈ [1, ∞] uniformly in ε. This gives where p ∈ (1, (d + 2)/(d + 1)) and we used the uniform bound for ρ ε , ρ ε Ψ and the dominated convergence theorem for the convergence of the interaction potential term.
⋄ (Convergence of (5.10) (ii)) Although the proof is almost the same as that of [34,Lemma 4.4], we present here for readers' convenience. First, consider a test function Ψ of the form Ψ(x, v, t) := ψ(x, t)ϕ(v). Thus, ϕ ∈ C ∞ c (R d ) and we similarly write ρ ε ϕ : Here, for p ∈ (1, (d + 2)/(d + 1)), we get , which gives the uniform bound for u ε ε ρ ε ϕ in L p (Ω), since p/(2 − p) ∈ (1, (d + 2)/d), and we already have the uniform bound for ρ ε in L q with q ∈ (1, (d + 2)/d) and |v| 2 f ε in L 1 . Thus, we can find m such that, up to a subsequence, u ε ε ρ ε ϕ ⇀ m in L ∞ (0, T ; L p (Ω)), ∀ p ∈ (1, (d + 2)/(d + 1)). It remains to show that m = uρ ϕ . For this, we let h 1 , h 2 > 0 and define For each h 1 and h 2 , we combine the pointwise convergence of ρ ε to ρ with Egorov's theorem to deduce that for every δ > 0, we may choose A δ ⊂ A h2 h1 satisfying |A h2 h1 \ A δ | < δ and ρ ε → ρ as ε → 0 uniformly on A δ . Then, for a sufficiently small ε, we have ρ ε > h 2 /2 on A δ and thus we get Since the choices of h 1 , h 2 and δ were arbitrary, we now obtain m = uρ ϕ on {ρ > 0}, and therefore, we have for all test functions Ψ of the form Ψ(x, v, t) = ψ(x, t)ϕ(v). Thus, we conclude that f is a weak solution to (1.1). It remains to show that the weak solution obtained above satisfies the entropy inequality (2.2). Note that the regularized solutions f ε satisfies the entropy inequality (5.6) and the strong compactness of macroscopic fields ρ ε , ρ ε u ε are obtained in (5.9) via the velocity averaging lemma. Thus we can use a similar argument as in the previous step together with Fatou's lemma to have the following entropy inequality: Then, since we also have ∇W ε is bounded and smooth, global-in-time existence of weak solutions to the equation (5.2) is clear. Moreover, we can also deduce that the entropy inequality (5.4) and the following upper bound estimate hold: where C = C(T ) is a positive constant independent of η. When d = 1, one uses Young's inequality to get and this gives where C = C(T ) is a constant independent of η, which implies the desired uniform upper bound estimate and this can be also used when ε → 0. For d = 2, we note that the following inequality holds: which subsequently gives log(ε + |x − y| 2 ) ≤ log 2(1 + ε) + log(1 + |x| 2 ) + log(1 + |y| 2 ).
We use the above inequality and log(1 + x) ≤ x on x ≥ 0 to get Moreover, the integral of | log(ε + |x|)| p on |x| ≤ 1 can be bounded uniformly in ε for every p ∈ [1, ∞) and thus we have where C is a constant independent of η and 1/q + 2/p = 2 with p ∈ (1, (d + 2)/d). Hence, for d = 2, we can obtain which gives the desired uniform upper bound estimate. For the free energy inequality, we first notice that the following inequality still holds for d = 1, 2: When d ≥ 3, the interaction potential W is positive, thus we used Fatou's Lemma to obtain the desired inequality. Although it is no longer possible to use Fatou's Lemma when d = 1, 2, we use Theorem 5.2 instead to show that lim for each t ∈ [0, T ]. Since the proof for the ε → 0 case is similar, we only consider the case R → ∞. First, we show that W ε ⋆ ρ η converges to W ε ⋆ ρ ε pointwise. Indeed, the pointwise convergence ρ η → ρ ε implies W ε (x − ·)ρ η (·) converges to W ε (x − ·)ρ ε (·) for each x. If d = 1, for each x.
When d = 2, one uses | log x| ≤ max{x, x −1 }, x > 0, and chooses L sufficiently large so that L ≫ |x| to get |y|≥L W ε (x − y)ρ η (y) dy ≤ 1 2π |y|≥L log ε + |x − y| 2 ρ η (y) dy ≤ 1 2π |y|≥L max{ ε + |x − y| 2 , (ε + |x − y| 2 ) −1/2 }ρ η (y) dy which also gives the desired estimate Theorem 5.2 (ii). For the last condition (iii) in Theorem 5.2, we choose a measurable set E. Then for d = 1, we use Hölder's inequality and the uniform bounds for ρ η in L p with p ∈ (1, 2) to get where p ′ = p/(p − 1) is the Hölder conjugate of p. When d = 2, we obtain which guarantees the condition (iii). Thus, we have the pointwise convergence of W ε ⋆ ρ η to W ε ⋆ ρ ε and hence (W ε ⋆ ρ η )ρ η converges to (W ε ⋆ ρ ε )ρ ε almost everywhere. To prove the desired convergence, we notice that where C is a constant independent of η. More precisely, we find for d = 2. Thus, we use the above estimate to validate the conditions in Theorem 5.2. For the condition (ii) in Theorem 5.2, we choose L > 0 to get For the condition (iii) in Theorem 5.2, we have Hence, we can obtain the entropy inequality similarly as in the case d ≥ 3. In this section, we study the local-in-time existence and uniqueness of strong solutions to (1.3) and (1.5) in the periodic domain Ω = T d . Since the proof for the system (1.5) is similar to that for (1.3), we only provide the details of the proof for the system (1.3), see Section 6.4 for the brief idea of the proof for the pressureless case. For the case with smooth interaction potential W , we briefly mention the existence result in Remark 6.1 below.
To be more specific, we are mainly interested in the local-in-time solvability of the following isothermal Euler-Poisson system with nonlocal forces: Here we set γ = λ = α = 1 for the sake of simplicity. We then reformulate the above system by setting g := log ρ and rewrite it as follows: subject to initial data: We now state the result on the well-posedness of the system (6.2)-(6.3).
Theorem 6.1. Let s > d/2 + 1. Suppose that the confinement potential and communication weight satisfy Then for any positive constants ǫ 0 < M 0 , there exists a positive constant T * such that if g 0 H s + u 0 H s < ǫ 0 , then the system (6.2)-(6.3) admits a unique solution (g, u) ∈ C([0, Note that the solution (g, u) obtained above has C 1 -regularity and in particular g is bounded, we can easily show that (ρ, u) with ρ := e g is a strong solution to (6.1). More precisely, we can have the equivalence relation between the classical solutions to the systems (6.1) and (6.2). The well-posedness theory for the equation (6.1) has not been developed so far to the best of our knowledge. On the other hand, if the velocity alignment forces, the last term on the right hand side of the momentum equations in (6.1) and the confinement forces are ignored, the system (6.1) reduces to the damped isothermal Euler-Poisson system. For that system, the global existence of weak/strong solutions is studied in [23,24,29,30,44]. We refer to [16] for a general survey on the Euler equations and related conservation laws. Critical thresholds phenomena leading to a finite-time blow-up or a global regularity of strong solutions for the Euler-Poisson system are also investigated in [12,27]. 6.1. Solvability for the linearized system. In this subsection, we linearize the system (6.2) and discuss the local-in-time estimates of solutions to that system. More precisely, for a given we consider the associated linear system: with the initial data (g 0 , u 0 ) ∈ H s (T d ) × H s (T d ).
Proof. We first easily obtain the existence and uniqueness of solutions to (6.4) by a standard linear theory of PDEs. Thus, it suffices to provide bound estimates for g and u. A straightforward computation gives Then we use Sobolev inequality and Young's inequality to get where C > 0 only depends on s, d, ∇V and φ.
For higher-order estimates, we first recall the Moser-type inequality: For 1 ≤ k ≤ s, we estimate ∇ k g as where C > 0 depends only on d and s. Similarly, we estimate ∇ k u as where C > 0 only depends on d, s, ∇V , ∇W L 1 and φ. To estimate the Poisson interaction term, we let a k := ∇ k (eg) L 2 . As shown previously, we have a 0 ≤ e g L ∞ ≤ Ce CM . Then, we use the Moser-type inequality and Sobolev inequality to obtain Here C > 0 only depends on d and k. Now, we combine the estimates for ∇ k g and ∇ k u to yield We sum the relation (6.7) over 1 ≤ k ≤ s and combine this with (6.6) to get This concludes the desired result.
Proof. For the proof, we use the inductive argument. Since the initial step (n = 0) is obvious, it suffices to consider the induction step. We recall from Lemma 6.1 that In the lemma below, we show that the approximation sequence (g n , u n ) is a Cauchy sequence in C([0, T * ]; L 2 (T d ))× C([0, T * ]; L 2 (T d )).
In this case, we use the following estimate for the Poisson interaction term: Thus, under suitable assumptions on the confinement potential ∇V and the communication weight φ, we can use the above estimate to get H s+1 -estimates for u, i.e., for any M > N , if then there exists T * > 0 such that sup 0≤t≤T * g n (·, t) 2 H s + u n (·, t) H s+1 < M, ∀ n ∈ N.
Then, the similar argument as the above provides the local-in-time existence and uniqueness of strong solutions to the pressureless Euler system (6.9).