Eulerian dynamics with a commutator forcing II: flocking

We continue our study of one-dimensional class of Euler equations, introduced in \cite{ST2016}, driven by a forcing with a commutator structure of the form $[\aL_\phi,u](\rho)=\phi*(\rho u)- (\phi*\rho)u$, where $u$ is the velocity field and $\phi$ belongs to a rather general class of \emph{influence} or interaction kernels. In this paper we quantify the large-time behavior of such systems in terms of \emph{fast flocking} for two prototypical sub-classes of kernels: bounded positive $\phi$'s, and singular $\phi(r) = r^{-(1+\a)}$ of order $\alpha\in [1,2)$ associated with the action of the fractional Laplacian $\aL_\phi=-(-\partial_{xx})^{\alpha/2}$. Specifically, we prove fast velocity alignment as the velocity $u(\cdot,t)$ approaches a constant state, $u \to \bar{u}$, with exponentially decaying slope and curvature bounds $|u_x(\cdot,t)|_{\infty}+ |u_{xx}(\cdot,t)|_{\infty}\lesssim e^{-\d t}$. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state $\rho(\cdot,t) - {\rho_{\infty}}(x - \bar{u} t) \rightarrow 0$.

1. Introduction and statement of main results.
1.1. Flocking hydrodynamics. In this paper we continue our study initiated in [ST2016], of Eulerian dynamics driven by forcing with a commutator structure. In the one-dimensional case, the dynamics of a velocity u : Ω × R + → R is governed by the system of form (1.1) ρ t + (ρu) x = 0, where the commutator on the right, [L φ , u](ρ) := L φ (ρu) − L φ (ρ)u, involves a convolution kernel The motivation for (1.1) comes from the hydrodynamic description of a large-crowd dynamics driven by Cucker-Smale agent-based model Here, φ is a positive, bounded influence function which models the binary interactions among agents in Ω. We focus our attention on the periodic or open line setup, Ω = T, R. For large crowds, N ≫ 1, the dynamics can be encoded in terms of the empirical dis- , so that its limiting moments lead to a density, ρ(x, t) = lim N →∞ R f N (x, v, t)dv, and momentum, ρu(x, t) = lim N →∞ R vf N (x, v, t)dv, governed by (1.1), [HT2008, CCP2017].
In particular, there is a fast alignment if the flocking convergence rate is exponential. In the present case of symmetric interactions, the conservation of averaged mass and momentum, M(t) := 1 2π T ρ(x, t)dx ≡ M 0 , P(t) := 1 2π T (ρu)(x, t)dx ≡ P 0 implies that a limiting flocking velocity, provided it exists, is given byū = P 0 /M 0 .
Remark 1.1. In the case when the dynamics of (1.4) takes place over the line Ω = R as in [HT2008, TT2014], the flocking phenomenon assumes a compactly supported initial configuration with finite initial velocity variation, It requires that, in addition to (1.6a), the flow remains compactly supported This reflects the corresponding flocking behavior in the agent-based Cucker-Smale model, max with strictly positive kernel, ι k (t) = inf x,y∈T k(x, y, t) > 0. Then there is a flocking alignment In particular, the case of symmetric interaction (1.4) admits fast alignment, Then the maximal value does not exceed, Similarly, we have the lower bound Subtracting the latter implies that the velocity diameter V (t) = max and the result readily follows.
We demonstrate the generality of lemma 1.2 with the following two examples.
Example 1.3 (an example on non-symmetric kernel). The Mostch-Tadmor model [MT2011] uses an adaptive normalization, where the pre-factor 1/N on the right of (1.3) is replaced by 1/ j φ(|x i − x j |), leading to the flocking hydrodynamics with non-symmetric kernel k(x, y, t) = φ(|x − y|)/(φ * ρ)(x, t). The lower-bound shows that flocking holds for positive, bounded φ's, with exponential rate dictated by the condition number of φ but otherwise independent of the total mass, V (t) V (0)e −(ι φ /I φ )t .
Since the argument of lemma 1.2 does not use local integrability, it applies in the present setting with ι α = inf x φ α (|x|) > 0, leading to fast alignment (1.8).
We close this subsection by noting that the the extension of lemma 1.2 to the case of open space Ω = R was proved in [TT2014]. To this end one restricts attention to the dynamics over {supp ρ(·, t)}: the growth of the velocity diameter V (t) := max is coupled with the obvious bound on the growth of the density support, d dt D(t) V (t)D(t). Assume that φ is decreasing so that ι φ φ(D(t)). It implies a decreasing free energy E(t) := E 0 , and fast alignment follows with a finite diameter, D(t) D ∞ , dictated by Thus, in the case of open space, Ω = R, compactness of {supp ρ(·, t)} requires a finite velocity In particular, an unconditional flocking follows for global φ's with unbounded integrable tails. Of course, in the periodic settings, compactness of the support of ρ is automatic.
1.3. Statement of main results. Lemma 1.2 tells us that for positive φ's, the question of flocking is reduced to the question of global regularity. The latter question -the global regularity of (1.4), was addressed in our previous study [ST2016] in the larger context of three classes of interaction kernels. Namely, for bounded φ's, global regularity follows for sub-critical initial data such that u ′ 0 (x)+φ * ρ 0 (x) > 0. For singular kernels φ α (x) := |x| −(1+α) corresponding to L φ = −(−∆) α/2 , global regularity follows for α ∈ [1, 2). Finally, global regularity also holds in the limiting case α = 2 which corresponds to the Navier-Stokes equations with L φ = ∆, and we recall that the global regularity for the cases α ∈ [1, 2] is independent of a critical threshold requirement. A main feature of the forcing in all three cases is their commutator structure in (1.1) which yields is the conservative transport of the first-order quantity u x + L φ (ρ), [CCTT2016, ST2016] (1.10) e t + (ue) x = 0, e := u x + L φ (ρ).
In this paper, we complement our earlier study of global regularity with the flocking behavior for these classes of interaction kernels. In particular, we make a more precise flocking statement, where fast alignment max to an exponential decay of slope and curvature of the velocity |u x | ∞ + |u xx | ∞ e −δt , and the flocking itself is proved in the strong sense of exponential convergence to one of the flocking states F . We treat here the flocking behavior in the two cases of positive φ's, and of fractional φ α , α ∈ [1, 2). The limiting case of Navier-Stokes equations L φ = ∆ does not seem to satisfy fast alignment due to lack of non-local interactions. Its large-time behavior remains open.
We begin with the case of a bounded positive kernel, and general density with a possibility of vacuum. The result is proved in both periodic and open line domains.
Lemma 2.1. Suppose e 0 > 0 on Ω. There exist constants C, δ > 0 such that Proof. We rewrite (2.1) as a logistic equation along characteristicsẋ(t) = u(x(t), t) with non-autonomous threshold h := φ * ρ, Here and below, D Dt denotes differentiation along generic particle path Since e 0 is uniformly bounded from above and away from zero, it follows that there exists a time t 0 > 0 such that the quantity e(t) = e(t; x 0 ) remains likewise bounded from above and below uniformly for all initial conditions e 0 = e(t = 0; x 0 ), Let us now write the equation for φ * ρ by convolving the mass equation φ * ρ t + φ * (ρu) x = 0: where the commutator is given by In view of Lemma 1.2, and the fact that φ ∈ W 1,∞ we have In what follows we denote by E = E(t) a generic exponentially decaying quantity, so (2.4) reads D Dt h = E, uniformly over all initial conditions x 0 ∈ Ω. Let us rewrite this equation for Recall that for large enough time, t > t 0 we have the positive boundedness 0 < c 0 < e < C 0 uniformly over initial conditions. This readily implies uniformly over x 0 . Since at any time t characteristics cover all Ω we arrive at the desired bound |u x (t)| ∞ E(t).
Solving the density equation along characteristics we obtain So, in view of Lemma 2.1 the density enjoys a pointwise global bound (which is not given a priori) (2.6) sup t>0 |ρ(·, t)| ∞ < ∞.
Next we establish a second round of estimates in higher order regularity in order to get a control over ρ x and then prove flocking of the density.
Lemma 2.2. There exist C, δ > 0 such that Proof. Let us write the equations for e x and h x = φ ′ * ρ: Clearly, the right hand side of h x -equation is exponentially decaying, = E. Subtracting the two we obtain the equation for u xx : Note that clearly, h x is a bounded function, hence u x e x = Eu xx + E. So, Once again, the positive boundedness of e in (2.3), implies that for large enough time, (2.9) D Dt u xx = −u xx e + E, t > t 0 , and the lemma follows.
Let's write the equation for ρ x : This shows that ρ x is uniformly bounded. We are now ready to prove Theorem 1.5. We state the last computation in a lemma as it will also be used ad verbatim in the next section.
Proof. The velocity alignment goes to its natural limitū = P/M. Denote ρ(x, t) := ρ(x + tū, t). Then ρ satisfies where all the u's are evaluated at x + tū. According to the established bounds we have | ρ t | ∞ < Ce −δt . This proves that ρ(t) is Cauchy as t → ∞, and hence there exists a unique limiting state, ρ ∞ (x), such that Shifting x this can be expressed in terms of ρ andρ(·, t) = ρ ∞ (x − tū) We also haveρ ∈ W 1,∞ from weak-star compactness. The statement of the lemma follows by interpolation of this exponential bound with the C 1 -bound of ρ.

Flocking with singular kernels
3.1. Uniform bounds on density, and velocity alignment. The results of this section lead towards the first step in the proof of Theorem 1.6. However, we state them in such generality since they hold for a much broader class of kernels satisfying the following three properties.
We recall that due to the positivity (ii), lemma Lemma 1.2 applies -as noted in Example 1.4, fast flocking holds irrespective of (lack of) local integrability As before, our starting point is the conservative transport (1.10) involving L α ≡ L φα , Paired with the mass equation we find that the ratio q = e/ρ satisfies the transport equation Starting from sufficiently smooth initial condition with ρ 0 away from vacuum we can assume that Lemma 3.1. Fix α ∈ [1, 2) and consider the pair (u, ρ) which is a smooth solution to (1.1),(1.2) α subject to initial density ρ 0 away from vacuum. Then there are positive constants c = c(M, Q, φ), C = C(M, Q, φ) such that (3.6) 0 < c ρ(x, t) C, x ∈ T, t 0.
Remark 3.2. The upper bound and the weaker lower-bound ρ(·, t) 1/t was established for the CS hydrodynamics (1.3) along the line of the 1D decay for commutator forcing [ST2016]. We include both cases for completeness.
Proof. Let us recall that the density equation can be rewritten as Let us evaluate at a point where the maximum ρ + is reached at x + . We obtain d dt In view of assumption (iii) we can pick r large enough to satisfy rι φ (r) > Q + 1, while according to (i) ι φ (r) itself remains finite. We thus achieve inequality d dt ρ + −ρ 2 + + ι φ (r)Mρ + , which establishes the upper bound by integration.
As to the lower bound we argue similarly. Let ρ − and x − the minimum value of ρ and a point where such value is achieved. We have d dt In view of (ii) the linear term on the right hand side has a positive coefficient. This readily implies the uniform lower bound.
Subsequently we focus solely on the critical case α = 1. The case 1 < α < 2 requires the same prerequisites as established in this section, however, incorporating it into the existing proof below would actually require fewer steps due to excess of dissipation. We will skip those for the sake of brevity of what will already be a technical exposition.
3.2. An initial approach towards Theorem 1.6. In this section we specialize on the case of critical singular kernel, and establish uniform control on the first order quantities |ρ x | ∞ , |u x | ∞ . We assume that we start with initial condition (u 0 , ρ 0 ) ∈ H 3 (T). As shown in [ST2016] there exists a global solution to (1.1) in the same space. So, we can perform all of the computations below as classical.
First, let us recite one argument from [ST2016]. Recall that the density ρ satisfies the following parabolic form of the density equation, expressed in terms of L 1 ≡ −Λ, Similarly, one can write the equation for the momentum m = ρu: (3.9) m t + um x + em = −ρΛ(m).
In both cases the drift u and the forcing eρ or em are bounded a priori due to the maximum principle and Lemma 3.1. Moreover, the diffusion operator has kernel Using lower bound on the density from Lemma 3.1 we conclude that the kernel falls under the assumptions of Schwab and Silverstre [SS2016] uniformly on the time line. A direct application of [SS2016] implies that there exists an γ > 0 such that (3.10) holds for all t > 0. Since the right hand sides are uniformly bounded on the entire line we have obtained uniform bounds on C γ -norm starting from time t = 1. Since we are concerned with long time dynamics let us reset initial time to t = 1, and allow ourselves to assume that C γ -norms are bounded from time t = 0.
Lemma 3.3. We make the same assumptions stated in Theorem 1.6. Then the following uniform bound holds Proof. The argument goes verbatim as presented in [ST2016, Section 6.2] with all the constants involved being uniform in view of the established bounds above. We recall the penultimate inequality d dt where all the quantities are evaluated at a point of maximum of |ρ ′ |, and where Using the nonlinear maximum bound from [CV2012] Dρ we can further hide the quadratic term into dissipation to obtain This enables us to conclude the Lemma.
We turn to study the flocking behavior in the singular case. To this end, we first prepare preliminary estimates on the dissipation terms to be encountered in the sequel. Below is an improvement on nonlinear maximum principle bound of [CV2012] in the case of small amplitudes. As a byproduct we obtain a trilinear estimate that will be used in the sequel.
Lemma 3.4 (Enhancement of dissipation by small amplitudes). Let u ∈ C 1 (T) be a given function with amplitude V = u + − u − . There is an absolute constant c 1 > 0 such that the following pointwise estimate holds In addition, there is an absolute constant c 2 > 0 such that for all B > 0 one has (3.14) Proof. We start as in [CV2012]. Using smooth truncations in the integrals we obtain, for every r > 0: where we dropped the positive term with |u ′ (x+z)| 2 . Now, using that u ′ (x+z) = u z (x+z) = (u(x + z) − u(x)) z we can integrate by parts in the second term to obtain By picking r = 2c 0 V |u ′ (x)| we obtain (3.13). Picking r = 1 2B and using Young's inequality , and (3.14) follows.
We proceed to the exponential decay of u x and u xx which is quantified in the next two lemmas, in terms of constants C, δ > 0, depending on M, |u 0 | H 3 and additional parameters specified below.
Lemma 3.5. We make the same assumptions stated in Theorem 1.6. There exist constants C, δ > 0 such that for all t > 0 one has Proof. Differentiating the u-equation and evaluating at a point of maximum we obtain d dt First, as to the dissipation term, let us fist observe Thus, in view of Lemma 3.1, Let us turn to the remaining term T (ρ ′ , u)u ′ . We have Expressing Λρ = e − u x the first term, and using uniform bound on ρ x from Lemma 3.3 we obtain the bound To bound the integral we split it into the long range {|z| > π} and short range {|z| π} parts. For the short range we use the bound For the long range part, we apply the following argument (to be used several other times in the sequel). We have (3.16) Proceeding to the second part, (3.17) Putting the estimates together we obtain Altogether we have obtained, resetting the constant counter, In view of Lemma 3.4, we have Integrating we discover that |u ′ | → 0 at an exponential rate of at least 1 2 min{c 9 , cI φ }, the latter being the rate of decay of V . This finishes the proof.
The estimates established so far are sufficient to prove an initial version of Theorem 1.6 as stated in Lemma 2.3. The proof the lemma goes ad verbatim in the present case.
3.3. Completing the proof of Theorem 1.6. In this section we perform computations with the goal to show that the flocking proved in Lemma 2.3 takes place in all spaces up to H 3 , and that the limiting profileρ itself belongs to H 3 . We prove exponential flattening of u in terms of curvature |u ′′ | ∞ and third derivative u ′′′ in L 2 . This complements the statement of Lemma 2.3 to the full extent of Theorem 1.6.
We start by showing exponential decay of |u ′′ | ∞ . As before we denote by E = E(t) any quantity with an exponential decay, e.g. |u ′ | ∞ = E, or V = E. Thus, according to Lemma 3.4, we have pointwise bounds (3.18) Due to these bounds the dissipation term absorbs all cubic and quadratic terms with bounded coefficients. It does the latter at the cost of adding a free E-term unattached to either u or ρ. Finally, let us recall from [ST2016] that the quantity Q = (e/ρ) x /ρ is transported: Q t + uQ x = 0. As such, it remains bounded uniformly for all times. Expressing e ′ from Q, we see that e ′ is controlled by ρ ′ and ρ, which in view of Lemma 3.3 implies uniform bound on e ′ : Using this additional piece of information we are in a position to prove control of the curvature.
To bound the remaining term T (ρ ′ , u ′ )u ′′ we will make use of the dissipation.
We have |Hρ ′′ | ∞ = |Λρ ′ | ∞ |e ′ | ∞ + |u ′′ | ∞ . So, the corllary simply follows from Lemma 3.6 and (3.19). Since we are in the torus settings, this automatically implies uniform bound for all L p -norms of ρ ′′ , for 1 p < ∞: In what follows we tacitly use these bounds by simply replacing uniformly bounded quantity such as above by constants.
We are now in a position to perform final estimates in the top regularity class H 3 .