Lorentz spaces in action on pressureless systems arising from models of collective behavior

We are concerned with global-in-time and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior (like traffic, crowds or birds). The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role.


Introduction
We are concerned with some models coming from a special type of hydrodynamical systems, that do not include the effects of the internal pressure. The mathematical properties of these models are not quite the same as those that arise from the physical description of common fluids. The simplest example, giving the nature of the phenomenon, is the motion of dust, that is, of free particles evolving in the space. Here one can mention examples in astrophysics [13], or in multi-fluid systems [1,3]. But leaving the world of inanimate matter, we can find models that describe collective behavior, where particles or rather agents exhibit some intelligence, and for which having a force like internal pressure is not so natural. A well known example in this area is given by equations of traffic flow [2,19], where particles are just cars.
In order to specify and understand this class, let us go back to the kinetic description of a collective behaviour. Consider equations of the following form is a distribution function of gas in the phase-space. Classically, if the operator K is given by the Poisson potential, then we obtain the Vlasov system. If taking a less singular K, then one may obtain for example the Cucker-Smale system that models collective behavior like flocking of birds [4].
Assuming a very special form of f , the so-called mono-kinetic ansatz, one can pass formally from the kinetic model to the hydrodynamical system, putting just This amounts to saying that the distribution of the gas under consideration is located on the curve v = u(t, x). Although one cannot expect this simplification to be a correct description of a gas, it may be relevant for modelling collective behavior phenomena. Typically, one can expect a crowd of individuals to have the same speed (or tendency) at one point [26,28]. Now, plugging (1.2) in (1.1) leads to the following general form of hydrodynamical system: (1.3) ρ t + div (ρu) = 0, ρu t + ρu · ∇u = A(ρ, u).
The first equation is the mass continuity law, and the second one is the momentum balance. Examples of A can be found in [30,31] and in the survey [25]. In the case A ≡ 0, one just recovers inviscid compressible transport, namely the pressureless compressible Euler system, and there is no interaction whatsoever between the individuals.
The first case is a viscous regularization of (1.4) that can be viewed as a simplification of the Euler alignment system. It corresponds to the hydrodynamical version of the Cucker-Smale model, namely (1.5) ρ t + div (ρu) = 0, ρu t + ρu · ∇u = R d u(t, y) − u(t, x) |x − y| d+α ρ(t, y)ρ(t, x)dy, α ∈ (0, 2]. The right-hand side of (1.5) involves the fractional Laplacian (−∆) α/2 (see details in [12,15]) and, at least formally, the first case in (1.4) thus meets α = 2. The second case of (1.4) is the Lamé operator that can be obtained from the Vlasov-Boltzmann equation (for more explanation, one may refer to the introduction of [29]). The form of (1.4) does not take into account the effects of internal pressure. From the mathematical viewpoint, the lack of the pressure term P causes serious problems. In particular, all techniques for the compressible viscous systems based on the properties of the so-called effective viscous flux, namely F := div u − P, which has better regularity than div u and P taken separately, are bound to fail. Recall that using F is one of the keys to the theory of weak solutions of the compressible Navier-Stokes equations [20,21,22,16,17], as it allows to exhibit compactness properties of the set of weak solutions. In the theory of regular solutions [24,5,27], the effective viscous flux provides the decay properties for the density.
In the case of pressureless systems, there is no such a possibility, so that we need to resort to more sophisticated techniques to control the density. This may partially explain the reason why the mathematical theory of pressureless models is poorer than the classical one.
The aim of this note is to present a novel technique coming from the maximal regularity theory for analytic semi-groups, in order to prove global-in-time properties of solutions to (1.3), (1.4). It will enable us to show existence and uniqueness results under rough assumptions on the density (only bounded), even though one cannot take advantage of the effective viscous flux. More precisely, by combining interpolation arguments, subtle embeddings, suitable time weighted norms and the magic properties of Lorentz spaces, we succeed in obtaining the L 1 (R + ; L ∞ ) regularity for the gradient of the solution to the linearized momentum equation in (1.3) and, eventually, to produce global-in-time strong solutions. Our main results concern global in time solvability for the two dimensional case for large velocity, and the three dimensional case in the small data regime.

Results
The idea of this note is to present an interesting application of Lorentz spaces for parabolic type systems. Lorentz spaces can be defined on any measure space (X, µ) via real interpolation between the classical Lebesgue spaces, as follows: Lorentz spaces may be endowed with the following (quasi)-norm (see e.g. [18,Prop. 1.4.9]): The reason for the pre-factor p 1 r is to have f Lp,p = f Lp , according to Cavalieri's principle. Let us first describe what we want to prove in the case where A(ρ, u) = µ∆u + µ ′ ∇div u, if the gas domain is the whole plane R 2 . So we consider: Following recent results of ours in [8,10,11] (in different contexts, though), we strive for global results for general initial velocities provided the volume (bulk) viscosity µ ′ is large enough. Owing to our approach based on a perturbative method, we need moreover the density to be close to a constant.
Our key solution space will be the setẆ 2,1 4/3,1 of functions z : . The corresponding trace space on constant times is the subspaceẆ 1/2 4/3,1 ofẆ 1/2 4/3 that is obtained if replacing the Lebesgue space L 4/3 by the Lorentz space L 4/3,1 . That space may be characterized in terms of the heat semi-group (e t∆ ) t>0 as follows: Being a subspace ofẆ More generally, we define for any 1 < p < ∞ and 1 ≤ r ≤ ∞, that coincides with the trace space of the seṫ Our first main result is a global existence and uniqueness statement for (2.3).
Theorem 2.1. Let us fix some M > 0 and consider any vector field u 0 on R 2 with components inẆ 1/2 4/3,1 such that, denoting by P and Q the Helmholtz projectors on divergence-free and potential vector fields, we have There exist two universal constants c and C such if then System (2.3) admits a global finite energy solution (ρ, u) with ρ ∈ C * w (R + ; L ∞ (R 2 )) and u ∈Ẇ 2,1 4/3,1 , satisfying Furthermore ∇u ∈ L 1 (R + ; L ∞ (R 2 )) and the following decay property holds: Finally, the solution (ρ, u) is unique in the above regularity class.
Some comments are in order: -Condition (2.5) means that global existence holds true for large ν, provided the divergence part of the velocity is O(ν −1/4 ). A similar restriction (with other exponents, though), was found in our prior works dedicated to the global existence of strong solutions for the compressible Navier-Stokes equations with increasing pressure law [8,10,11]. -The above statement involves only quantities that are scaling invariant for System (2.3). -Having Lorentz spaces with second index equal to 1 is crucial as it allows us to capture the limit case of the Sobolev embedding -see (A.2) in the Appendix. Other choices than L 4/3,1 and L 4,1 might be possible.
In the three-dimensional case, the energy space L 2 (R 3 ) is super-critical by half a derivative, and there is no chance to prove a general result for large data, assuming only that one of the viscosity coefficients is large. Therefore, we decided for simplicity to focus on the second case of (1.4) and the system we want to consider is thus: To simplify our analysis, we chose a functional framework for the velocity that is well beyond critical regularity. Here is our main result for (2.9): then (2.9) has a global in time unique solution (ρ, u) with finite energy, and such that ρ ∈ C w (R + ; L ∞ (R 3 )) and u ∈Ẇ 2,1 5/2,1 ∩Ẇ 2,1 5/4 . Furthermore, . Remark 2.1. Let us emphasize that the initial velocity is actually is L 2 , a consequence of the following interpolation inequality: .
We also want to stress the fact that the smallness condition (2.10) is scaling invariant. The rest of the paper is structured as follows. In the next section, we prove our twodimensional global result for (2.3). Section 4 is devoted to the proof of Theorem 2.2. In Appendix, we recall some useful results on interpolation and Lorentz spaces.
We shall use standard notations and conventions. In particular, C will always designate harmless constants that do not depend on 'important' quantities, and we shall sometimes This part is dedicated to the proof of Theorem 2.1. The key observation is that the energy balance associated to (2.3) combined with some interpolation argument enables to bound the norm of u in L 4,2 (R + × R 2 ), in terms of u 0 L 2 (R 2 ) . This will enable us to get a priori estimates for u inẆ 2,1 4/3,1 , then for tu inẆ 2,1 4,1 provided the density is close to 1. Interpolating again will give a control on div u in L 1 (R + ; L ∞ (R 2 )) and thus, going to the mass conservation equation, on ρ−1. At this stage, we will use a bootstrap argument so as to justify a posteriori that, indeed, if ν := µ + µ ′ is large enough, then all the above global-in-time estimates hold true. Then, we observe that the very same arguments leading to the control of div u also allow to bound ∇u in L 1 (R + ; L ∞ (R 2 )). From that point, we follow the energy method of [7, Sec. 4] going to Lagrangian coordinates in order to prove uniqueness, and the rigorous proof of existence is obtained by compactness arguments, after constructing a sequence of smoother solutions.
Let us now go to the details of the proof. We assume throughout that µ ′ ≥ 0 and, to simplify the computations, we take µ = 1. That latter assumption is not restrictive, since (ρ, u) satisfies (2.3) with coefficients (µ, µ ′ ) if and only if 3) with coefficients 1 and µ ′ /µ.
Step 1. The energy balance and the control of the norm in L 4,2 (R + ×R 2 ). As already pointed out, the space for u 0 is continuously embedded in L 2 (R 2 ). Hence the initial data have finite energy. Now, the energy balance for (2.3) reads Provided (2.7) is fulfilled with small enough c, we thus have, denoting by P the L 2 orthogonal projector on solenoidal vector-fields, and ν := 1 + µ ′ , We claim that the above inequality implies that To prove our claim, we shall use (3.3) and the fact thatḢ 1 (R 2 ) ֒→ BMO(R 2 ). Hence it suffices to check that Let us omit R + and R 2 in the following lines for better readability. As a start, we observe that, as a consequence of e.g. [32, Section 1.17], Thus, one can write However, in light of the first relation of (3.6) and of obvious embedding, Analogously, Plugging this in (3.7), we get which completes the proof of (3.4).
Step 2. Control of the norm of the solution inẆ 2,1 4/3,1 . Rewrite the velocity equation as: Using the Helmholtz projectors P and Q on solenoidal and potential vector fields, respectively, yields (Pu) t − ∆Pu = Pf and (Qu) t − ν∆Qu = Qf. Hence, thanks to parabolic maximal regularity (see Proposition A.1), we have and To bound the terms Pf and PQ, we observe that P and Q are continuous on L 4/3,1 , so that it is thus enough to estimate f in L 4/3,1 (R + ×R 2 ). We find that Assuming (2.7), it will be eventually possible to absorb the first term in the right-hand side. Furthermore, by Hölder inequality for Lorentz spaces (see the Appendix), and thus, thanks to the first step, . Therefore, one eventually gets, as ν ≥ 1, Now, a real interpolation argument that is carried out at the end of the appendix implies that we also have the following control that will be the key to the next step: Step 3. A time weighted estimate. We now look at the momentum equation in the form By definition, the initial data for tu is zero, and we know from (3.10) that the term ρu is in L 4,1 (R 2 × R + ). This gives us some hint on the regularity of the whole right-hand side. Now, projecting (3.11) by means of P and Q, using the maximal regularity estimates of the appendix, and still assuming (2.7), one gets for all 0 ≤ T ≤ T ′ ≤ ∞, T ′ ]×R 2 ) · Since we haveẆ 1/2 4,1 (R 2 ) ֒→ L ∞ (R 2 ) (see the Appendix), the term with u · ∇u may be bounded as follows: then Inequality (3.12) reduces to · Of course, if one can take T = 0 and T ′ = ∞ in (3.14), then we control the left-hand side of (3.12) on R + , so assume from now on that u L 4,1 (R + ×R 2 ) > c. We claim that there exists a finite sequence 0 = T 0 < T 1 < · · · < T K−1 < T K = ∞ such that (3.14) if fulfilled on [T k , T k+1 ] for each k ∈ {0, · · · , K − 1}. In order to prove our claim, we introduce and recall that (up to some constant) From Lebesgue dominated convergence theorem, we have Hence one can construct inductively a family 0 = T 0 < T 1 < · · · < T k < · · · such that By simple Hölder inequality on series, we have  Hence we find that for all k = 1, ..., K, Consequently, we mus have So, arguing by induction, we eventually get for all m ∈ {0, · · · , K − 1}, Reverting to (3.15), then using (3.20) and (3.10), we conclude that Step 4. Bounding div u. In order to keep the density close to 1, we need to bound div u in L 1 (R + ; L ∞ (R 2 )). To get it, the key observation is that (3.24) div (tu) ∈ L 4,1 (R + ;Ẇ 1 4,1 (R 2 )) and div u ∈ L 4/3,1 (R + ;Ẇ 1 4/3,1 (R 2 )). Now, from Gagliardo-Nirenberg inequality, we see that (3.25) z L∞(R 2 ) ≤ C ∇z . So we have, thanks to Hölder inequality in Lorentz spaces, Hence, thanks to (3.9) and (3.23), Step 5. Bounding the density. The discrepancy of the density to 1 (that is a := ρ − 1) may be controlled by means of the mass equation: ∂ t a + u · ∇a + (1 + a)div u = 0, which gives and thus Hence, provided we have Step 6: Uniqueness. The key to uniqueness is that ∇u is in L 1 (R + ; L ∞ ). To get that property, one may proceed exactly as for bounding div u, writing that and using that the r.h.s. is bounded in terms of u 0 according to (3.9) and (3.23). There is no need of bootstrap argument here.
Because of the hyperbolic nature of the continuity equation, the uniqueness issue is not straightforward in our case, as the regularity of the density is very low. However, having a control on ∇u in L 1 (R + ; L ∞ (R 2 )) enables us to rewrite our system in Lagrangian coordinates. More precisely, for all y ∈ R 2 , consider the following ODE: (3.30) dX dt (t, y) = u(t, X(t, y)), X| t=0 = y.
Having (3.29) at hand guarantees that (3.30) defines a C 1 flow X on R + × R 2 .
Then, the system for (η, v) reads (see details in e.g. [6]): . Since, in our framework the Lagrangian and Eulerian formulations are equivalent (see e.g. [6,27]), it suffices to prove uniqueness at the level of Lagrangian coordinates. Therefore, consider two solutions (η, v) and (η,v) of (3.32) emanating from the data (ρ 0 , u 0 ). Then, the difference of velocities δv :=v − v satisfies : ∇v · Now, taking the L 2 scalar product of (3.33) with δv and integrating by parts delivers Then, all terms like Id −A w , 1 − J w or Id − adj(DX w ) (with w = v,v ) may be computed by using Neumann series expansions, and we end up with pointwise estimates of the following type: From this, we deduce that Because we have, by Cauchy-Schwarz inequality, integrating the above inequality (and using again Cauchy-Schwarz inequality) yields Hence, there exists c > 0 such that if, in addition to (3.34), we have In light of the above arguments, in order to get uniqueness on the whole R + , it suffices to show that our solutions satisfy not only ∇u ∈ L 1 (R + ; L ∞ (R 2 )), but also This is a consequence of (3.25), as it gives Step 7: Proof of existence. The idea is to smooth out the data, and to solve (2.3) supplemented with those data, according to the local-in-time existence result of [7] (that just requires the initial velocity to be smooth enough, and the initial density to be close to 1 in L ∞ ). Then, the previous steps provide uniform bounds that allow to show that those smoother solutions are actually global, and one can eventually pass to the limit. The reader may refer to the end of the next part where more details are given both for Theorems 2.1 and 2.2.

The three dimensional case
Our aim here is to prove a global existence result in the small data regime case for System (2.9). We want to stress the dependency of the smallness condition in terms of the viscosity coefficient, and to get the optimal one, it is convenient to use again the rescaling (3.1). So we assume from now on that µ = 1.

(4.2)
From it, we will deduce a bound of u in the Lorentz space L 10/3,1 (R + × R 3 ) (that will play the same role as L 4,1 (R + × R 2 ) in the previous section), and get a control on tu iṅ W 2,1 10/3,1 (R 3 × R + ), which will be the key to eventually bound ∇u in L 1 (R + ; L ∞ (R 3 )). From that stage, the proof of uniqueness follows the same lines as for (2.3).
Step 1. Control by the energy. Remembering that our assumptions imply that u 0 is in L 2 (R 3 ), we start with the basic energy balance: By Sobolev embedding and provided that this implies the following bound on u: .
That relation will enable us to control higher norms of the solution, globally in time, provided some scaling invariant quantity involving u 0 is small enough.
In order to estimate the nonlinear term, we start from Hölder inequality: Using suitable embedding, one may prove that Hence, owing to the definition of Π, we have In order to bound u in L 5,1 (R + × R 3 ), we start with the observation that and that, by Hölder inequality and (4.5), Putting together with (4.16) and reverting to (4.14), we end up with 0 Π. Therefore, using also (4.12), we get the following inequality for Π: Step 5. Bounding ∇u. It is now easy to get the desired control on ∇u: we start from the following Gagliardo-Nirenberg inequality: Using Hölder inequality (A.1) with respect to time in Lorentz spaces, we find that 1 . As the right-hand side is bounded, owing to (4.8) and (4.17), one may conclude that ∇u is in L 1 (R + ; L ∞ (R 3 )). More importantly, we have the inequality Therefore, arguing on the mass equation exactly as in the 2D case, one can justify (4.4), and thus all the previous steps provided (2.10) is satisfied. Indeed, in light of (2.12), the above smallness condition is stronger than (4.8).
Step 6. Uniqueness. Arguing as in the previous section and knowing (4.21) (so as to put our system in Lagrangian coordinates), it suffices to establish the additional property that t 1/2 ∇u is in L 2 (0, T ; L ∞ (R 3 )). Now, one may write, owing to (4.18), that Because t∇ 2 u is in L 10/3,1 (R + × R 3 ) and ∇ 2 u is in L 5/2 (R + × R 3 ), the right-hand side is indeed bounded. This completes the proof of uniqueness.
Step 7. Existence. Here we sketch the proof of the existence of a global solution under our assumptions on the data. The overall strategy is essentially the same in dimensions d = 2 and d = 3.
As a first, we truncate ρ 0 and smooth out u 0 to meet the conditions of the local-in-time existence theorem of [7]. We get a sequence (ρ n 0 , u n 0 ) n∈N of data that generates a sequence of local solutions (ρ n , u n ) n∈N on [0, T n ], in the classical maximal regularity space , with e.g. p = 2d and r = 7/6.
It is shown in [7] that those solutions satisfy the energy balance and (4.4), and are such that ∇u n ∈ L r 1 (0, T n ; L p (R d )) with 1 Since the computations of the previous step just follow from the properties of the heat flow and of basic functional analysis, each (ρ n , u n ) satisfy the estimates therein. In particular, ∇u n L 1 (0,T n ;L∞) is uniformly bounded like in (4.21), which provides control of (4.4). Now, applying the standard maximal regularity estimates to 2 ∂ t u n − ∆u n − µ ′ ∆u n = (1 − ρ n )∂ t u n + ρ n u n · ∇u n , one gets for all T < T n , u n Ep,r(T ) u n 0 E 0 p,r + u n · ∇u n Lr(0,T ;Lp) .
Therefore, one can eventually bound u n in E p,r (T n ) independently of T n . Then, applying standard continuation arguments allow to prove that (ρ n , u n ) is actually global, and may be bounded in terms of the original data (ρ 0 , u 0 ) in the spaces of our main theorems, independently of n.
From this stage, passing to the limit in the slightly larger spaceẆ 2,1 5/2 (R 3 × R + ) ∩Ẇ 2,1 5/4 (R 3 × R + ) (or inẆ 2,1 4/3 (R 2 ×R + )) for the velocity can be done as in [7] (passing to the limit directly in the nonreflexive spaceẆ 2,1 5/2,1 would require more care). The mass conservation equation may be handled according to Di Perna and Lions' theory [14] (see details in [7]) and the momentum equation does not present any difficulty compared to works on global weak solutions, since a lot of regularity is available on the velocity and there is no pressure term.
Next, once we know that (ρ, u) is a solution, there is no difficulty to recover all the additional regularity, that are just based on 'linear' properties like interpolation or parabolic maximal regularity.
A.2. Maximal regularity. Consider the heat semi-group (e t∆ ) t>0 on R d and denote The fact that ∇ 2 M : L q (R + ; L p (R d )) → L q (R + ; L p (R d )) whenever 1 < p, q < ∞ belongs to the mathematical folklore (see the pioneering work in [23] for the particular case p = q ). Now, observing that, for all 1 ≤ r ≤ ∞ and θ ∈ (0, 1), we have one may conclude that (A.3) ∇ 2 M : L p,r (R + × R d ) → L p,r (R + × R d ).
Next, observe that, for all T ≥ 0, we have, by definition of Mf and of the spaceẆ Hence, combining with a density argument, one can deduce that M maps L p,r (R + × R d ) to C b (R + ;Ẇ 2−2/p p,r (R d )) (only weak continuity if r = ∞).