Contents Introduction and statement of main . ثڑ(ث†), [CCTT2016,ST2016] (1.10) e t+ (ue) x= 0;...

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Transcript of Contents Introduction and statement of main . ثڑ(ث†), [CCTT2016,ST2016] (1.10) e t+ (ue) x= 0;...

  • EULERIAN DYNAMICS WITH A COMMUTATOR FORCING II: FLOCKING

    ROMAN SHVYDKOY AND EITAN TADMOR

    Abstract. We continue our study of one-dimensional class of Euler equations, introduced in [ST2016], driven by a forcing with a commutator structure of the form [Lφ, u](ρ), where u is the velocity field and Lφ belongs to a rather general class of convolution operators depending on interaction kernels φ.

    In this paper we quantify the large-time behavior of such systems in terms of fast flocking for two prototypical sub-classes of kernels: bounded positive φ’s, and singular φ(r) = r−(1+α) of order α ∈ [1, 2) associated with the action of the fractional Laplacian Lφ = −(−∂xx)α/2. Specifically, we prove fast velocity alignment as the velocity u(·, t) ap- proaches a constant state, u→ ū, with exponentially decaying slope and curvature bounds |ux(·, t)|∞+|uxx(·, t)|∞ . e−δt. The alignment is accompanied by exponentially fast flocking of the density towards a fixed traveling state ρ(·, t)− ρ∞(x− ūt)→ 0.

    Contents

    1. Introduction and statement of main results. 1 2. Flocking with smooth positive kernels 5 3. Flocking with singular kernels 8 References 17

    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+ 7→ R is governed by the system of form

    (1.1)

    { ρt + (ρu)x = 0,

    ut + uux = [Lφ, u](ρ), where the commutator on the right, [Lφ, u](ρ) := Lφ(ρu) − Lφ(ρ)u, involves a convolution kernel

    (1.2) Lφ(f) := ∫ R φ(|x− y|)(f(y)− f(x))dy.

    Date: February 1, 2017. 1991 Mathematics Subject Classification. 92D25, 35Q35, 76N10. Key words and phrases. flocking, alignment, fractional dissipation, Cucker-Smale, Mortsch-Tadmor, crit-

    ical thresholds. Acknowledgment. Research was supported in part by NSF grants DMS16-13911, RNMS11-07444 (KI-

    Net) and ONR grant N00014-1512094 (ET) and by NSF grant DMS 1515705 (RS). RS thanks CSCAMM for the hospitality for his March 2016 visit which initiated this project. Both authors thank the Institute for Theoretical Studies (ITS) at ETH-Zurich for the hospitality.

    1

  • 2 ROMAN SHVYDKOY AND EITAN TADMOR

    The motivation for (1.1) comes from the hydrodynamic description of a large-crowd dynamics driven by Cucker-Smale agent-based model

    (1.3)

     ẋi = vi,

    v̇i = 1

    N

    N∑ j=1

    φ(|xi − xj|)(vj − vi), (xi, vi) ∈ Ω× R, i = 1, 2, . . . , N.

    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- tribution fN =

    1 N

    ∑N i=1 δxi(x) ⊗ δvi(v), so that its limiting moments lead to a density,

    ρ(x, t) = lim N→∞

    ∫ R fN(x, v, t)dv, and momentum, ρu(x, t) = lim

    N→∞

    ∫ R vfN(x, v, t)dv, governed

    by (1.1), [HT2008, CCP2017].

    (1.4)

     ρt + (ρu)x = 0,

    ut + uux =

    ∫ R φ(|x− y|)(u(y, t)− u(x, t))ρ(y, t) dy

    (x, t) : Ω× [0,∞).

    The other important limit of such systems — their large time behavior for t � 1, is described by the flocking phenomenon. To this end, let us introduce the set of flocking state solutions, consisting of constant velocities, ū, and traveling density waves ρ̄ = ρ∞(x− tū), (1.5) F = {(ū, ρ̄) : ū ≡ constant, ρ̄(x, t) = ρ∞(x− tū)}. We say that a solution (u(·, t), ρ(·, t)) converges to a flocking state (ū, ρ̄) ∈ F in space X×Y if

    ‖u(·, t)− ū‖X + ‖ρ(·, t)− ρ̄(·, t)‖Y → 0, as t→∞. This represents the process of alignment where the diameter of velocities tends to zero

    (1.6a) V (t) := max x,y∈supp ρ(·,t)

    |u(x, t)− u(y, t)| → 0, as t→∞.

    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 ≡M0, P(t) :=

    1

    ∫ T (ρu)(x, t)dx ≡ P0

    implies that a limiting flocking velocity, provided it exists, is given by ū = P0/M0.

    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 config- uration with finite initial velocity variation,

    D0 := max x,y∈supp ρ0

    |x− y|

  • EULERIAN DYNAMICS WITH A COMMUTATOR FORCING 3

    1.2. Smooth solutions must flock. The flocking hydrodynamics of (1.4) for bounded positive φ’s follows, as long as they admit global smooth solutions. Indeed, the statement that “smooth solutions must flock” holds in the general setup of positive kernels whether symmetric or not [ISV2016, Lemma 3.1], [TT2014, Theorem 2.1]. For the sake of complete- ness we include below the proof of flocking along the lines of [MT2014, theorem 2.3] which is stated in the following lemma for the periodic case Ω = T.

    Lemma 1.2 (Smooth solutions must flock). Let (ρ, u) be a smooth solution of the one- dimensional system

    (1.7)

     ρt + (ρu)x = 0,

    ut + uux =

    ∫ R k(x, y, t)(u(y, t)− u(x, t))ρ(y, t) dy

    (x, t) : T× [0,∞),

    with strictly positive kernel, ιk(t) = inf x,y∈T

    k(x, y, t) > 0. Then there is a flocking alignment

    V (t) 6 V (0)exp

    { −M

    ∫ t τ=0

    ιk(τ)dτ

    } , V (t) = max

    x,y∈T |u(x, t)− u(y, t)|.

    In particular, the case of symmetric interaction (1.4) admits fast alignment,

    (1.8) V (t) 6 V (0)e−Mιφt, ιφ := min x∈T

    φ(|x|).

    Proof. Let x−(t) be a point where u− = u(x−(t), t) = minu, and x+(t) be a point where u+ = u(x+(t), t) = maxu. Then the maximal value does not exceed,

    d

    dt u+ =

    ∫ T k(x+, y, t)(u(y)− u+)ρ(y, t) dy 6 ιk

    ∫ T (u(y)− u+)ρ(y, t) dy.

    Similarly, we have the lower bound

    d

    dt u− > ιk

    ∫ T (u(y)− u−)ρ(y, t) dy.

    Subtracting the latter implies that the velocity diameter V (t) = max x,y∈T

    |u(x, t)−u(y, t)| satisfies

    d

    dt V (t) 6 −ιkMV (t), V (t) = u+(t)− u−(t)

    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(|xi − xj|), leading to the flocking hydrodynamics with non-symmetric kernel

    k(x, y, t) = φ(|x− y|)/(φ ∗ ρ)(x, t). The lower-bound

    k(x, y, t) = φ(|x− y|)

    (φ ∗ ρ)(x, t) >

    ιφ IφM

    , Iφ = max x∈T

    φ(|x|).

    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) 6 V (0)e−(ιφ/Iφ)t.

  • 4 ROMAN SHVYDKOY AND EITAN TADMOR

    Example 1.4 (an example of unbounded kernels). The fractional Laplacian Lα := Lφα is associated with the singular periodized kernels

    (1.9) φα(x) = ∑ k∈Z

    1

    |x+ 2πk|1+α , for 0 < α < 2.

    Since the argument of lemma 1.2 does not use local integrability, it applies in the present setting with ια = infx φα(|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

    x,y∈supp ρ(·,t) |u(x, t)− u(y, t)|,

    d

    dt V (t) 6 −ιkMV (t), V (t) = max

    x∈supp ρ(·,t) u(x, t)− min

    y∈supp ρ(·,t) u(y, t),

    is coupled with the obvious bound on the growth of the density support, d dt D(t) 6 V (t)D(t).

    Assume that φ is decreasing so that ιφ > φ(D(t)). It implies a decreasing free energy E(t) := V (t) +

    ∫ D(t) τ=0

    φ(τ)dτ 6 E0, and fast alignment follows with a finite diameter, D(t) 6 D∞, dictated by

    D(t) 6 D∞, M ∫ D∞ D0

    φ(s)ds = V0.

    Thus, in the case of open space, Ω = R, compactness of {supp ρ(·, t)} requires a finite velocity

    variation V0 <

    ∫ ∞ D0

    φ(s)ds. 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∗ρ0(x) > 0. For singular kernels φα(x) := |x|−(1+α) corresponding to Lφ = −(−∆)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 equation