Global regularity of two-dimensional flocking hydrodynamics

TL;DR

This paper analyzes the conditions under which smooth solutions to two-dimensional flocking hydrodynamics remain globally regular and exhibit flocking behavior, based on initial configuration thresholds.

Contribution

It derives sharp critical thresholds in initial conditions that guarantee global regularity and flocking in 2D Euler systems with velocity alignment.

Findings

Global regularity persists under sub-critical initial conditions.

Initial divergence and spectral gap determine long-term behavior.

Flocking behavior is linked to initial phase space thresholds.

Abstract

We study the systems of Euler equations which arise from agent-based dynamics driven by velocity \emph{alignment}. It is known that smooth solutions of such systems must flock, namely -- the large time behavior of the velocity field approaches a limiting "flocking" velocity. To address the question of global regularity, we derive sharp critical thresholds in the phase space of initial configuration which characterize the global regularity and hence flocking behavior of such \emph{two-dimensional} systems. Specifically, we prove for that a large class of \emph{sub-critical} initial conditions such that the initial divergence is "not too negative" and the initial spectral gap is "not too large", global regularity persists for all time.

Equations

$$\left\{\begin{array}{rl}& {\rho }_t+\nabla \cdot (\rho \mathbf{u})=0,\\ & {\mathbf{u}}_t+\mathbf{u}\cdot \nabla \mathbf{u}=\int a(x,y,t)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (y,t)\text{d}y\end{array}\right\}(x,t)\in {\mathbb{R}}^n\times {\mathbb{R}}_{+}.$$

$$\displaystyle\left\{\begin{array}[]{rr}\begin{aligned} &\dot{x}_{i}={\mathbf{v}}_{i}\\ &\dot{{\mathbf{v}}}_{i}=\frac{1}{\text{deg}_{i}}\sum_{j=1}^{N}\phi(|x_{i}-x_{j}|)({\mathbf{v}}_{j}-{\mathbf{v}}_{i})\end{aligned}\end{array}\right.$$

$$\{\begin{array}{rl}{\rho }_t+\nabla \cdot (\rho \mathbf{u})& =0\\ (\rho \mathbf{u}{)}_t+\nabla (\rho \mathbf{u}\otimes \mathbf{u})& =\frac{\alpha (x,t)}{(\varphi \ast \rho )(x,t)}\int \varphi (|x-y|)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (x,t)\rho (y,t)\text{d}y.\end{array}$$

$$\{\begin{array}{rl}{\rho }_t+\nabla \cdot (\rho \mathbf{u})& =0,\\ {\mathbf{u}}_t+\mathbf{u}\cdot \nabla \mathbf{u}& =\frac{\alpha (x,t)}{(\varphi \ast \rho )(x,t)}\int \varphi (|x-y|)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (y,t)\text{d}y.\end{array}$$

$$\displaystyle{\mathbf{u}}_{t}+{\mathbf{u}}\cdot\nabla{\mathbf{u}}=\alpha(x,t)\big{(}\overline{{\mathbf{u}}}(x,t)-{\mathbf{u}}(x,t)\big{)},\qquad\overline{{\mathbf{u}}}(x,t):=\frac{\phi*(\rho{\mathbf{u}})(x,t)}{(\phi*\rho)(x,t)}.$$

$$D(t):=\underset{x,y\in \text{supp}\{\rho (\cdot ,t)\}}{\sup }|x-y|,V(t):=\underset{x,y\in \text{supp}\{\rho (\cdot ,t)\}}{\sup }|\mathbf{u}(x,t)-\mathbf{u}(y,t)|.$$

$$\underset{t⩾0}{\sup }D(t)⩽D_{\infty }\text{ and }V(t)\stackrel{t\to \infty }{⟶}0.$$

$$V_0<m_0{\int }_{D_0}^{\infty }\varphi (r)dr,m_0:=|{\rho }_0{|}_1,$$

$$\underset{t⩾0}{\sup }D(t)⩽D_{\infty }\text{where}{m}_0{\int }_{D_0}^{D_{\infty }}\varphi (s)\text{d}s=V_0,$$

$$\displaystyle V(t)\leqslant V_{0}e^{-\kappa t}\longrightarrow 0,\qquad\kappa:=\left\{\begin{array}[]{ll}m_{0}\phi_{\infty},&\mathrm{CS}\ \ \text{model},\\ \phi_{\infty},&\mathrm{MT}\ \text{model},\end{array}\right.\ \ \phi_{\infty}:=\phi(D_{\infty}).$$

$$(\partial_{t}+{\mathbf{u}}\cdot\nabla)\langle{\mathbf{u}}(x,t),{\mathbf{w}}\rangle=\int\phi(|x-y|)\Big{(}\langle{\mathbf{u}}(y,t),{\mathbf{w}}\rangle-\langle{\mathbf{u}}(x,t),{\mathbf{w}}\rangle\Big{)}\rho(y,t)\text{d}y.$$

$$\begin{split}\frac{\text{d}}{\text{d}t}u_{+}&=\int\phi(|x_{+}-y|)\Big{(}\langle{\mathbf{u}}(y,t),{\mathbf{w}}\rangle-\langle{\mathbf{u}}(x_{+},t),{\mathbf{w}}\rangle\Big{)}\rho(y,t)\text{d}y\\ &\leqslant\min_{x,y\in\text{supp}\,\rho(\cdot,t)}\phi(|x-y|)\int\Big{(}\langle{\mathbf{u}}(y,t),{\mathbf{w}}\rangle-\langle{\mathbf{u}}(x_{+},t),{\mathbf{w}}\rangle\Big{)}\rho(y,t)\text{d}y\end{split}$$

$$\frac{\text{d}}{\text{d}t}u_{-}\geqslant\min_{x,y\in\text{supp}\{\rho(\cdot,t)\}}\phi(|x-y|)\int\Big{(}\langle{\mathbf{u}}(y,t),{\mathbf{w}}\rangle-\langle{\mathbf{u}}(x_{-},t),{\mathbf{w}}\rangle\Big{)}\rho(y,t)\text{d}y$$

$$\frac{\text{d}}{\text{d}t}|u_{+}(t)-u_{-}(t)|⩽-\varphi (D_{\infty })m_0|u_{+}(t)-u_{-}(t)|,\varphi (D_{\infty })=\underset{x,y\in \text{supp}\{\rho (\cdot ,t)\}}{\min }\varphi (|x-y|).$$

$$(\varphi \ast \rho )(x,t)⩾m_0\varphi (d(x,t)+D_{\infty })>0,d(x,t)=\text{dist}\{x,\text{supp}\{\rho (\cdot ,t)\}\}$$

$$\varphi (|x-y|)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (x,t)\rho (y,t)\equiv 0,|x-y|>D_{\infty },$$

$$A(t):=\{x|0<\text{dist}\{x,\text{supp}\{\rho (\cdot ,t)\}\}<D_{\infty }\},$$

$$\left\{\begin{array}{rl}{\rho }_t+\nabla \cdot (\rho \mathbf{u})& =0,\\ {\mathbf{u}}_t+\mathbf{u}\cdot \nabla \mathbf{u}& =\frac{\alpha (x,t)}{\varphi \ast \rho }\int \varphi (|x-y|)(\mathbf{u}(y)-\mathbf{u}(x))\rho (y)\text{d}y\end{array}\right\}\text{dist}\{x,\text{supp}\{\rho (\cdot ,t)\}\}<D_{\infty }.$$

$$\left|\int \varphi (|x-y|)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (y,t)dy\right|⩽V(t)\cdot |\rho (\cdot ,t){|}_{\infty }\times {\int }_{y\in Y_x(t)}\varphi (|x-y|)dy.$$

$$\mathbf{u}(x,t)\equiv {\mathbf{u}}_{\infty },\text{for}\text{dist}\{x,\text{supp}\{\rho (\cdot ,t)\}\}>D_{\infty }.$$

$$\{\begin{array}{rl}& {\rho }_t+(\rho u{)}_x=0,\\ & u_t+u{u}_x={\int }_{\mathbb{R}}\varphi (|x-y|)(u(y,t)-u(x,t))\rho (y,t)\text{d}y\end{array}(x,t)\in (\mathbb{R},{\mathbb{R}}_{+}).$$

$${\partial }_x{u}_0(x)⩾-(\varphi \ast {\rho }_0)(x),\text{mbox}forallx\in \mathbb{R}.$$

$$\left\{\begin{array}{rl}& {\rho }_t+\nabla \cdot (\rho \mathbf{u})=0,\\ & {\mathbf{u}}_t+\mathbf{u}\cdot \nabla \mathbf{u}=\int \varphi (|x-y|)(\mathbf{u}(y,t)-\mathbf{u}(x,t))\rho (y,t)\text{d}y\end{array}\right\}x\in {\mathbb{R}}^2,t⩾0,$$

$$V_0⩽m_0\cdot \min \left\{|\varphi {|}_1,\frac{{\varphi }_{\infty }^2}{4|{\varphi }^{\prime }{|}_{\infty }}\right\},V_0={\max }_{x,y\in \text{supp}({\rho }_0)}|{\mathbf{u}}_0(x)-{\mathbf{u}}_0(y))|,{\varphi }_{\infty }=\varphi (D_{\infty }).$$

$$\mathrm{div}{\mathbf{u}}_0(x)⩾-\varphi \ast {\rho }_0(x)\text{mbox}forallx\in {\mathbb{R}}^2.$$

$$\max_{x}\big{|}\eta_{{}_{S_{0}}}(x)\big{|}\leqslant\frac{1}{2}m_{0}\phi_{\infty},\qquad\eta_{{}_{S}}=\mu_{2}(S(x,t))-\mu_{1}(S(x,t)).$$

$$|{\partial }_j{u}_i(x,0)|⩽\frac{1}{4\sqrt{2}}{m}_0{\varphi }_{\infty }.$$

$${\int }_{D_0}^{D_{\infty }}\varphi (s)ds⩽\frac{{\varphi }^2(D_{\infty })}{4{\max }_{s⩽D_{\infty }}|{\varphi }^{\prime }(s)|}.$$

$$M_t+\mathbf{u}\cdot \nabla M+M^2=-(\varphi \ast \rho )M+R,R_{ij}:={\partial }_j\varphi \ast (\rho u_i)-u_i{\partial }_j\varphi \ast \rho .$$

$$|R_{ij}|=\left|{\int }_{{\mathbb{R}}^n}{\partial }_j\varphi (|x-y|)(u_i(y,t)-u_i(x,t))\rho (y,t)\text{d}y\right|⩽|{\varphi }^{\prime }{|}_{\infty }{m}_0{V}_0{e}^{-\kappa t},\kappa =m_0{\varphi }_{\infty }.$$

$${\mathsf{d}}_t+\mathbf{u}\cdot \nabla \mathsf{d}+\mathrm{Tr}{M}^2=-(\varphi \ast \rho )\mathsf{d}+\mathrm{Tr}R.$$

$$\mathrm{Tr}R=\varphi \ast \nabla \cdot (\rho \mathbf{u})-\mathbf{u}\cdot \nabla \varphi \ast \rho =-(\varphi \ast \rho {)}_t-\mathbf{u}\cdot \nabla \varphi \ast \rho =-(\varphi \ast \rho {)}^{\prime },$$

Full text

Acknowledgment. Research was supported in part by NSF grants DMS16-13911, RNMS11-07444 (KI-Net) and ONR grant N00014-1512094. We thank the ETH Institute for Theoretical Studies (ETH-ITS) for the support and hospitality.

1. 1 Flocking hydrodynamics
2. 2 Cucker-Smale hydrodynamics
3. 3 Motsch-Tadmor hydrodynamics: global regularity and fast alignment

1. Flocking hydrodynamics

We consider the system of Eulerian dynamics where the density $\rho(x,t)$ and velocity field ${\mathbf{u}}(x,t)=(u_{1},\ldots u_{n}):{\mathbb{R}}^{n}\times{\mathbb{R}}_{+}\mapsto{\mathbb{R}}^{n}$ are driven by nonlocal alignment forcing,

[TABLE]

A solution $(\rho,{\mathbf{u}})$ is sought subject to the compactly supported initial density $\rho(x,0)=\rho_{0}(x)\in L_{+}^{1}(\mathbb{R}^{n})$ and uniformly bounded initial velocity ${\mathbf{u}}(x,0)={\mathbf{u}}_{0}(x)\in W^{1,\infty}(\mathbb{R}^{n})$. The alignment forcing on the right hand side of (1.1) involves the non-negative interaction kernel $a(x,y,t)$.

Such systems arise as macroscopic realization of agent-based dynamics which describes the collective motion of $N$ agents, each of which adjusts its velocity to a weighted average of velocities of its neighbors

[TABLE]

Here, the weighted average of the right of (1.3) is dictated by influence function $\phi(\cdot)$ which is assumed to be decreasing, and $\text{deg}_{i}$ is a weighting normalization factor. Different agent based models employ different $\text{deg}_{i}$’s, e.g., [CCP2017]. We focus here on two such models. The Cucker-Smale (CS) model [CS2007] sets a uniform averaging $\text{deg}_{i}\equiv N$ which leads to the symmetric interaction kernel $a(x,y)=\phi(|x-y|)$. The Motsch-Tadmor (MT) model [MT2011] uses an adaptive normalization $\text{deg}_{i}=\sum_{j}\phi(|x_{i}-x_{j}|)$ which leads to $\displaystyle a(x,y,t)=\frac{\phi(|x-y|)}{(\phi*\rho)(x,t)}$. The kernel is non-symmetric but normalized such that $\int a(x,y,t)\rho(y,t)\text{d}y=1$. The dynamics of (1.3) can be described in terms of the empirical distribution $f(x,{\mathbf{v}},t):=\frac{1}{N}\sum_{j}\delta_{x-x_{j}(t)}\otimes\delta_{{\mathbf{v}}-{\mathbf{v}}_{j}(t)}$. For large crowds of $N$ agents, $N\gg 1$, a limiting distribution of the approximate form $f(x,{\mathbf{v}},t)\approx\rho(x,t)\delta({\mathbf{v}}-{\mathbf{u}}(x,t))$ is captured by the first two velocity moments, namely – the density $\rho:=\langle f(x,{\mathbf{v}},t)\rangle$ and momentum $\rho{\mathbf{u}}:=\langle{\mathbf{v}}f(x,{\mathbf{v}},t)\rangle$ satisfy the conservative system [HT2008, CCR2009, CFRT2010, MOA2010]

[TABLE]

Here $\alpha(x,t)$ is the amplitude of alignment, $\alpha(x,t)=(\phi*\rho)(x,t)$ in the case of CS model, and $\alpha(x,t)\equiv 1$ in MT model. When classical solutions of these equations are restricted to the support of $\rho(\cdot,t)$, one ends with the equivalent system (1.1) with $a(x,y,t)=\alpha(x,t)\phi(|x-y|)/(\phi*\rho)(x,t)$, namely

[TABLE]

Since the alignment forcing on the right is non-local, dictated by the support of $\phi$, it acts even within the vacuum region where $\text{dist}\{x,\text{supp}\{\rho(\cdot,t)\}\}>0$, and (1.5) extends throughout $\mathbb{R}^{n}$. We elaborate on this issue in §1.3 below.

We note that the dynamics of both models can be interpreted in terms of the mean velocity $\overline{{\mathbf{u}}}(x,t)$

[TABLE]

This formulation reveals that system (1.5) (and in its general form (1.1)) is dynamically aligned towards the mean $\overline{u}(x,t)$, and its large time behavior is expected to approach a constant limiting velocity. This is the flocking hydrodynamics alluded to in the title, where a finite-size of non-vacuum state is approaching a limiting velocity as $t\rightarrow\infty$. Specifically, the dynamics can be characterized in terms of the diameters

[TABLE]

The system (1.1) converges to a flock if there exists a finite $D$ such that

[TABLE]

This corresponds to the flocking behavior at the level of agent-based description [HT2008], [MT2011, definition 1.1] where a cohesive flock of a finite diameter $\max_{i,j}|x_{i}(t)-x_{j}(t)|\leqslant D_{\infty}<\infty$, is approaching a limiting velocity, $\max_{i,j}|{\mathbf{v}}_{i}(t)-{\mathbf{v}}_{j}(t)|\rightarrow 0$ as $t\rightarrow\infty$.

1.1. Strong solutions must flock

In this work we focus on the case where $\phi$ is global. Since the agent based model (1.3) exhibit flocking behavior in this case, [MT2014], it is natural to to expect a similar result for its macroscopic description (1.5). This is the content of the following theorem.

Theorem 1.1** (Strong solutions must flock [TT2014]).**

Let $(\rho(\cdot,t),{\mathbf{u}}(\cdot,t))\in(L^{\infty}\cap L^{1})\times W^{1,\infty}$ be a global strong solution of the system (1.5) subject to a compactly supported initial density $\rho_{0}=\rho(\cdot,0)\geqslant 0$ and bounded initial velocity ${\mathbf{u}}_{0}={\mathbf{u}}(\cdot,0)\in W^{1,\infty}$. Assume that a monotonically decreasing influence function $\phi\leqslant\phi(0)=1$ is global in the sense that111We let $|\cdot|_{p}$ denote the usual $L^{p}$ norm.

[TABLE]

where $D_{0}$ and $V_{0}$ are the initial diameters of non-vacuum density and velocity. Then $(\rho,{\mathbf{u}})$ converges to a flock at exponential rate, namely — the support of $\rho(\cdot,t)$ remains within a finite diameter $D_{\infty}$ whose existence follows from assumption (1.7)

[TABLE]

In particular, if $|\phi|_{1}=\infty$ then there is an unconditional flocking in the sense that (1.8) holds for all finite $V_{0}$.

For the sake of completeness we provide below an alternative derivation of the exponential alignment in (1.8), as an a priori bound instead of the “propagation along characteristics” argument in [TT2014, Theorem 2.1]. To this end, we extend the scalar argument in [ST2017, Lemma 1.1] to general systems using a projection argument employed in [MT2014, Theorem 2.3]. Fix an arbitrary ${\mathbf{w}}\in\mathbb{R}^{n}$ and project the CS model (1.5) on ${\mathbf{w}}$ to find

[TABLE]

It follows that $\displaystyle u_{+}(t):=\max_{x\in\text{supp}\{\rho(\cdot,t)\}}\langle{\mathbf{u}}(x,t),{\mathbf{w}}\rangle$ satisfies

[TABLE]

Similarly, we have the lower bound on $\displaystyle u_{-}(t):=\mathop{\min}_{x\in\text{supp}\{\rho(\cdot,t)\}}\langle{\mathbf{u}}(x,t),{\mathbf{w}}\rangle$

[TABLE]

The difference of the last two inequalities implies

[TABLE]

It follows that the CS velocity diameter, $\displaystyle V(t)=\sup_{|{\mathbf{w}}|=1}|u_{+}(t)-u_{-}(t)|$, satisfies the bound (1.8d) with $\kappa=m_{0}\phi_{\infty}$. The same argument follows for MT model with $\kappa=\phi_{\infty}$, independently of $m_{0}$.

1.2. Critical thresholds

Theorem 1.1 raises the problem whether solutions of the hydrodynamic model (1.5) remain smooth for all time. This question was addressed in [TT2014, CCTT2016], proving that the compactly supported initial data stay below certain critical threshold in configuration space then initial smoothness propagates and as a result, the corresponding strong solutions will flock. Recall the finite-time blow-up of compactly supported density in the presence of local pressure [Si1985, LY1997] and even in the presence of global Poisson forcing [Ma1992]. In both cases, a positive lower-bound on the (potential of) the forcing — the pressure, Poisson, etc, over the finite $\text{supp}\{\rho(\cdot,t)\}$ leads to finite time blow up. In contrast, here the non-local character of the influence function $\phi$ guarantees global regularity, at least for sub-critical initial data. This type of conditional regularity for Eulerian dynamics depending on a critical threshold in configuration space, was advocated in a series of papers [ELT2001, LT2002, LT2003, LT2004, HT2008, LL2013]. Here, we pursue this approach to derive sharp critical thresholds for propagation of regularity of the two-dimensional flocking hydrodynamics.

1.3. Vacuum and the finite horizon alignment

According to (1.7), if the influence function is global in the sense that $\displaystyle\int^{\infty}\!\!\phi(r)dr=\infty$, then the alignment dynamics (1.5) admits unconditional flocking in the sense that (1.8) holds for all $V_{0}$’s. This holds for both the symmetric CS model and non-symmetric MT model [MT2014, proposition 2.9]. In this case, alignment in (1.5) is active throughout $\mathbb{R}^{n}$, inside and outside $\text{supp}\{\rho(\cdot,t)\}$. Indeed, one has a global lower-bound on the action of alignment for all $x\in\mathbb{R}^{n}$, [TT2014, proposition 6.1]

[TABLE]

The flocking behavior of such a global approach was pursued in [TT2014].

Another possible approach to study (1.5) is to focus on a specific initial configuration with finite velocity variation $V_{0}<\infty$. Then, since $\text{supp}\{\rho(\cdot,t)\}$ cannot grow beyond a maximal diameter of size $D_{\infty}$ dictated by (1.8a), it follows that the alignment term on the right of the underlying conservative formulation (1.4),

[TABLE]

independently of the values of $\{\phi(r),\ r>D_{\infty}\}$. Alternatively, we can fix a compactly support influence function $\phi$ and view (1.8a) as a restriction on initial velocities whose variation is “not too large”, so that they lead to flocking. With either one of these two points of view, the values of $\phi(r)$ for $r>D_{\infty}$ play no role in the dynamics. We therefore may set $\phi(r)_{|r>D_{\infty}}\equiv 0$ which in turn sets a finite horizon on the action of alignment. Namely, the alignment in (1.5) is still active in the vacuous annulus outside $\text{supp}\{\rho(\cdot,t)\}$,

[TABLE]

and (1.5) applies in $\text{supp}\{\rho(\cdot,t)\}\cup A(t)$,

[TABLE]

2. Cucker-Smale hydrodynamics

2.1. Global regularity

We begin by recalling the one-dimensional Cucker-Smale model for $(\rho,u):(\mathbb{R},\mathbb{R}_{+})\mapsto(\mathbb{R}_{+},\mathbb{R})$,

[TABLE]

In [CCTT2016] it was proved that (2.1) has a global classical solution if and only if the initial data satisfies

[TABLE]

Condition (2.2) separates the space of initial configurations into two distinct regimes: a sub-critical regime of initial data satisfying $\partial_{x}u_{0}(x)\geqslant-\phi*\rho_{0}(x),\forall x\in supp(\rho_{0})$, which guarantee global smooth solutions; and a supercritical regime of initial conditions such that $\partial_{x}u_{0}(x_{0})\leqslant-\phi*\rho_{0}(x_{0})$ for some $x_{0}\in\mathbb{R}$, which leads to a finite time blowup. This is a typical one-dimensional example for the critical threshold behavior. Condition (2.2) provides a sharp improvements to the earlier critical threshold results in [ST1992, LT2001, TT2014]. Recent results in [ST2016, DKRT2017] prove the global regularity of (2.1) for singular kernels $\phi(|x|)=|x|^{-(1+\alpha)}$ for $\alpha\in(0,2)$ independent of any finite critical threshold. Singularity helps!.

A first attempt to extend the study of critical threshold to the two-dimensional CS model was derived in [TT2014]. Here, we improve this result with a simplified derivation of a sharper critical threshold condition, leading to alignment decay of order $e^{-\kappa t}$. We recall (1.8d) which set $\kappa=m_{0}\phi_{\infty}$ in the present case of CS model.

Theorem 2.1** (Critical threshold for 2D Cucker-Smale hydrodynamics).**

Consider the two-dimensional CS model

[TABLE]

subject to initial conditions, $(\rho_{0},{\mathbf{u}}_{0})\in(L_{+}^{1}(\mathbb{R}^{2}),W^{1,\infty}(\mathbb{R}^{2}))$, with compactly supported density, $D_{0}<\infty$, and such that the variation of the initial velocity satisfies the strengthened bound

[TABLE]

Assume that the following critical threshold condition holds.

(i) The initial velocity divergence satisfies

[TABLE]

(ii) Let $S=\frac{1}{2}\{(\partial_{j}u_{i}+\partial_{i}u_{j})\}$ denote the symmetric part of the velocity gradient with eigenvalues $\mu_{i}=\mu_{i}(S)$. Then the initial spectral gap $\eta_{{}_{S_{0}}}:=\mu_{2}(S_{0})-\mu_{1}(S_{0})$ is bounded

[TABLE]

Then the class of such sub-critical initial conditions (2.5),(2.6) admit a classical solution

$(\rho(\cdot,t),{\mathbf{u}}(\cdot,t))\in C(\mathbb{R}^{+};L^{\infty}\cap L^{1}(\mathbb{R}^{2}))\times C(\mathbb{R}^{+};\dot{W}^{1,\infty}(\mathbb{R}^{2}))$* with large time hydrodynamics flocking behavior (1.8d), $\displaystyle\mathop{\max}_{x,y\in\text{supp}(\rho(\cdot,t))}|{\mathbf{u}}(x,t)-{\mathbf{u}}(y,t)|\lesssim e^{-\kappa t}$.*

Before turning to the proof of theorem 2.1, we comment on its assumptions.

Remark 2.1** (on the critical threshold (2.5),(2.6)).**

Theorem 2.1 recovers the one-dimensional critical threshold (2.2). It amplifies the same theme of critical threshold required for global regularity of other two-dimensional Eulerian dynamics found in restricted Euler-Poisson [LT2003], rotational Euler [LT2004],…, namely — if the initial divergence is “not too negative” as in (2.5), and the initial spectral gap is “not too large” as in (2.6), then global regularity persists for all time. In particular, since $\eta_{{}_{S}}=\sqrt{(\partial_{1}u_{1}-\partial_{2}u_{2})^{2}+(\partial_{1}u_{2}+\partial_{2}u_{1})^{2}}$ we find that both (2.5),(2.6) hold if

[TABLE]

Remark 2.2** (on the finite variation (2.4)).**

Observe that (2.4) places a restriction on the size of $V_{0}$ even in the case of unconditional flocking, $|\phi|_{1}=\infty$. Specifically, recall that $V_{0}$ dictates the maximal diameter of the flock in (1.8a) and thus, (2.4) amounts to

[TABLE]

Since the term on the left is increasing while the term on the right is decreasing as functions of $D_{\infty}$, it follows that (2.7) is satisfied for diameters $D_{\infty}$ up to some maximal finite size, that is — the condition made in (2.4) is met for finite $\displaystyle V_{0}=m_{0}\int^{D_{\infty}}\phi(s)ds$ depending on the influence function $\phi$. This finite restriction on $V_{0}$ can probably be improved, but unlike the one-dimensional case it cannot be completely removed. In fact, since $V_{0}\leqslant(\mu_{2}(S_{0})+\omega_{0})D_{\infty}$, the bound sought in (2.4) places a purely two-dimensional restriction on the size of initial vorticity.

Remark 2.3** (on the finite horizon).**

Observe that in the case of alignment with a finite horizon, the critical threshold (2.5) requires that $\operatorname{div}{{\mathbf{u}}_{0}}(x)\geqslant 0$ for $\text{dist}\{x,\text{supp}\{\rho_{0}\}\}>D_{\infty}$. This is precisely the critical threshold condition which rules out finite time blow-up in the pressure-less equations [Ta2017], which is satisfied when prescribing far-field constant velocity (1.9b). In this case, the critical threshold (2.5) needs to be verfied within the finite horizon $\text{dist}\{x,\text{supp}\{\rho_{0}\}\}<D_{\infty}$.

Proof.

Our purpose is to show that the derivative $\{\partial_{j}u_{i}\}$ are uniformly bounded. We proceed in four steps.

Step #1 — the dynamics of $\operatorname{div}{{\mathbf{u}}}+\phi*\rho$. Differentiation of (1.1) implies that the $2\times 2$ velocity gradient matrix, $M_{ij}:=\partial_{j}u_{i}$, satisfies

[TABLE]

The entries of the residual matrix $\{R_{ij}\}$ can bounded by the commutator estimate [TT2014, proposition 4.1] in terms of $\displaystyle V(t)=\sup_{\text{supp}(\rho)}|u_{i}(x,t)-u_{i}(y,t)|\leqslant V_{0}e^{-\kappa t}$,

[TABLE]

The first step is to bound the divergence: taking the trace of (2.8) we find that ${\sf{d}}:=\nabla\cdot{\mathbf{u}}$ satisfies

[TABLE]

Expressed in terms of the material derivative along particle path, $X^{\prime}:=(\partial_{t}+{\mathbf{u}}\cdot\nabla)X$, we have ${\sf{d}}^{\prime}+\operatorname{Tr}{M^{2}}=-(\phi*\rho){\sf{d}}+\operatorname{Tr}{R}$. We now make a key observation that $\operatorname{Tr}{R}$ is in fact an exact derivative along particle path. Indeed, as in [CCTT2016] we invoke the mass equation,

[TABLE]

and we end up with

[TABLE]

To proceed, we express $\displaystyle\operatorname{Tr}{M^{2}}\equiv\frac{{\sf{d}}^{2}+\eta_{{}_{M}}^{2}}{2}$ in terms of the spectral gap, $\eta_{{}_{M}}:=\lambda_{2}(M)-\lambda_{1}(M)$, associated with the eigenvalues of $M$,

[TABLE]

We need to follow the dynamics of the spectral gap $\eta_{M}$. To this end, one may try to use the spectral dynamics associated with $M$, [LT2002]: by (2.8) the $\lambda_{i}$’s satisfy

[TABLE]

where $\{{\boldsymbol{\ell}}_{i},{\mathbf{r}}_{i}\}$ are the left and right eigenvectors associated with $\lambda_{i}$, normalized such that $\langle{\boldsymbol{\ell}}_{i},{\mathbf{r}}_{i}\rangle=1$. Taking the difference of these two equations shows that the spectral gap $\eta_{{}_{M}}=\lambda_{2}-\lambda_{1}$, satisfies the transport equation

[TABLE]

Here one faces the difficulty which arises with the term on the right, namely — even with the control of the entries $\{R_{ij}\}$, we may still encounter an ill-conditioned $M$ with $|{\boldsymbol{\ell}}_{i}|\cdot|{\mathbf{r}}_{i}|\gg 1$ so that the magnitude of this term is left unchecked. To circumvent this difficulty, we proceed along the lines argued in [Ta2017]: we split $M$ into its symmetric and antisymmetric parts $M=S+\Omega$ and use the identity222Equating the trace of $M^{2}$ with that of $S^{2}+\Omega^{2}+S\Omega+\Omega S$ we find $\operatorname{Tr}{M^{2}}=\operatorname{Tr}{S^{2}}-2\omega^{2}$. Using $\operatorname{Tr}{X^{2}}=\frac{1}{2}({{\sf{d}}^{2}+\eta_{{}_{X}}^{2}})$ with $X=M$ on the left and $X=S$ on the right implies (2.11).

[TABLE]

where $\omega$ is the scaled vorticity333The use of such scaling simplifies the computation below. $\omega=\frac{1}{2}(\partial_{1}u_{2}-\partial_{2}u_{1})$. Expressed in terms of $\eta_{{}_{S}}$, the trace dynamics (2.10) now reads

[TABLE]

This calls for the introduction of the new “natural” variable ${\sf{e}}={\sf{d}}+\phi*\rho$, satisfying

[TABLE]

Our purpose is to show that $\{x\ |\ {\sf{e}}(x,t)\geqslant 0\}$ is invariant region of the dynamics (2.12).

Step #2 — bounding the spectral gap $\eta_{{}_{S}}$. Consider the dynamics of the symmetric part of (2.8)

[TABLE]

The spectral dynamics of its eigenvalues, $\mu_{2}(S)\geqslant\mu_{1}(S)$, is governed by

[TABLE]

driven by the orthonormal eigenpair $\{{\mathbf{s}}_{1},{\mathbf{s}}_{2}\}$ of the symmetric $S$. Taking the difference, we find that $\eta_{{}_{S}}:=\mu_{2}(S)-\mu_{1}(S)\geqslant 0$ satisfies,

[TABLE]

This is the same dynamics found with $\eta_{{}_{M}}$ except that the different residual on the right of (2.14) given by

[TABLE]

is now controlled by the size of $\{R_{ij}\}$: since ${\mathbf{s}}_{i}$ are normalized,

[TABLE]

Hence, as long as ${\sf{e}}(\cdot,t)$ remains positive then $\eta_{{}_{S}}$ remain uniformly bounded

[TABLE]

The first inequality on the right follows from integration of (2.14)-(2.15); the second follows from the $V_{0}$-bound in (2.4) and the third from the assumed bound on $\eta_{{}_{S_{0}}}$ in (2.6).

Step #3 — the invariance of ${\sf{e}}(\cdot,t)\geqslant 0$ . We return to (2.12): expressed in terms of $c(x,t):=\sqrt{(\phi*\rho)^{2}-\eta_{{}_{S}}^{2}}$ we have

[TABLE]

Observe that $c(\cdot)$ is well-defined in $\mathbb{R}$: the upper-bound (2.16) and the lower-bound $\phi*\rho\geqslant m_{0}\phi_{\infty}$ imply that as long as ${\sf{e}}\geqslant 0$, the right term on the right of (2.17) remains boundedly positive

[TABLE]

Since ${\sf{e}}^{\prime}\geqslant\frac{1}{2}(c_{\text{min}}^{2}-{\sf{e}}^{2})=\frac{1}{2}(c_{\text{min}}-{\sf{e}})(c_{\text{min}}+{\sf{e}})$, it follows that ${\sf{e}}$ is increasing whenever ${\sf{e}}\in(-c_{\text{min}},c_{\text{min}})$ and in particular, if ${\sf{e}}_{0}\geqslant 0$ then ${\sf{e}}(x,t)$ remains positive at later times. Thus, if the initial data are sub-critical in the sense that (2.5) holds

[TABLE]

then ${\sf{e}}(\cdot,t)\geqslant 0$ and $\eta_{{}_{S}}(\cdot,t)$ remains bounded.

Step #4 — an upper-bound of ${\sf{e}}(\cdot,t)$. The lower-bound ${\sf{e}}\geqslant 0$ implies that the vorticity is bounded. Indeed, the anti-symmetric part of (2.8) yields that the vorticity $\omega=\frac{1}{2}\operatorname{Tr}{JM}$ satisfies

[TABLE]

hence

[TABLE]

and we end up with same upper-bound on $\omega$ as with $\eta_{{}_{S}}$,

[TABLE]

Returning to (2.12) we have (recall $\phi\leqslant 1$)

[TABLE]

which implies that ${\sf{e}}(x,t)\leqslant{\sf{e}}_{\text{max}}<\infty$. The uniform bound on ${\sf{e}}$ implies that $\operatorname{div}{u}$ is uniformly bounded, $|\operatorname{div}{{\mathbf{u}}}|\leqslant|{\sf{e}}|_{\infty}+|\phi*\rho|_{\infty}\leqslant{\sf{e}}_{\text{max}}+m_{0}$, and together with the bound on the spectral gap (2.16), it follows that the symmetric part $\{S_{ij}\}$ is bounded. Finally, together with the vorticity bound (2.20) it follows that $\{\partial_{j}u_{i}\}$ are uniformly bounded which completes the proof. ∎

Remark 2.4**.**

Observe that the region of sub-critical configuration leading global regularity becomes larger for $|\omega_{0}|\gg 1$ in agreement with the statements made in [LT2004, CT2008] that rotation prevents or at least delays finite-time blow-up. Specifically, if $|\omega_{0}(\cdot)|\geqslant\omega_{\text{min}}>0$ then one can set a larger lower barrier $c=\sqrt{(\phi*\rho)^{2}+4\omega^{2}_{\text{min}}-\eta_{{}_{S}}^{2}}$ in (2.17) leading to the improved threshold $\operatorname{div}{{\mathbf{u}}_{0}}>-\phi*\rho_{0}-\omega_{\text{min}}$. In particular, if $\omega$ is large enough so that $4\omega^{2}-\eta^{2}_{S}>0$, that is — if $M$ has complex-valued eigenvalues, then the invariance of the positivity of ${\sf{e}}$ follows at once from the fact that (2.12) is dominated equation by ${\sf{e}}^{\prime}\geqslant\frac{1}{2}\left((\phi*\rho)^{2}-{\sf{e}}^{2}\right)$. As in the 2D restricted Euler-Poisson equations [LT2003], the difficulty lies with the case of real eigenvalues.

Remark 2.5**.**

The proof of theorem 2.1 reveals two main aspects. First, the commutator structure of the alignment term on the right of (2.3)2, expressed as $[\phi*,u](\rho)$, leads to the ‘residual terms’ $R_{ij}$ with exponentially decaying bound. The role of commutator structure was highlighted in our recent work [ST2016]. Second, the use of spectral dynamics, [LT2002, LT2003, LL2013], to trace the propagation of regularity for the remaining, non-residual terms in (2.8).

2.2. Fast alignment

We extend the one-dimensional arguments of [ST2016] which show that exponentially rapid convergence towards a flocking state, consisting of a constant 2-vector velocity $\{\bar{{\mathbf{u}}}\in\mathbb{R}^{2}$ and a traveling density profile $\bar{\rho}(x,t)=\rho_{\infty}(x-t\bar{{\mathbf{u}}})\}$. We only indicate the main aspects in the passage to the present system. We start by noting that the positivity of ${\sf{e}}$ implies more than the mere boundedness of the spectral gap $\eta_{S}$ and the vorticity $\omega$. Indeed, (2.14) and (2.19) imply that these quantities follow the exponential decay of $q$ in (2.15)

[TABLE]

This shows that modulo rapidly decaying error terms $E(t)$ of order $E(t)\lesssim e^{-\kappa t}$, equation (2.12) which governs ${\sf{e}}$ takes the form

[TABLE]

Moreover, convolving the mass equation with $\phi$ we find

[TABLE]

Observe that the quantity on the right of rapidly decaying, being upper-bounded by $\lesssim|\phi^{\prime}|_{\infty}V(t)\lesssim e^{-\kappa t}$. Hence, the difference ${\sf{d}}={\sf{e}}-{\sf{h}}$ satisfies

[TABLE]

The positivity of ${\sf{e}}+{\sf{h}}$ then implies the rapid decay of the divergence, $|\operatorname{div}{{\mathbf{u}}}(\cdot,t)|_{\infty}\lesssim e^{-\kappa t}$. The exponential decay of the divergence, the vorticity and the spectral gap imply that $|\partial_{j}u_{i}(\cdot,t)|_{\infty}\lesssim e^{-\kappa t}$. Let $\bar{{\mathbf{u}}}$ be a large-time limiting value of ${\mathbf{u}}(\cdot,t)$. The mass equation reads

[TABLE]

The term on the right is rapidly decaying because ${\sf{d}}$ and $(\bar{{\mathbf{u}}}-{\mathbf{u}})$ are, and one concludes along the lines of [ST2017], that there exists a traveling density profile such that $\rho(x,t)-\rho_{\infty}(x-t\bar{{\mathbf{u}}})\rightarrow 0$.

3. Motsch-Tadmor hydrodynamics: global regularity and fast alignment

In this section, we study the flocking hydrodynamics which arises from MT model (1) with $\kappa=\phi_{\infty}$. We begin by recalling the one-dimensional case

[TABLE]

System (3.1) was recently studied in [BRSW2015], as the hydrodynamic description for agent-based model of “emotional contagion”, and in [GG2017] in the context of stable swarming. In [CCTT2016] it was proved that (3.1) has a global classical solution for sub-critical initial data such that

[TABLE]

for a certain critical curve $\sigma_{+}\geqslant 0$. We now make a precise statement of the critical threshold for both the one - and two-dimensional MT model.

Theorem 3.1** (Critical threshold for 2D Motsch-Tadmor hydrodynamics).**

Consider the two-dimensional MT model in $(x,t)\in(\mathbb{R}^{2},\mathbb{R}_{+})$,

[TABLE]

subject to initial conditions $(\rho_{0},{\mathbf{u}}_{0})\in(L^{1},W^{1,\infty}(\mathbb{R}^{2}))$, with compactly supported density, $D_{0}<\infty$ and initial velocity of finite variation

[TABLE]

Assume that the following critical threshold condition holds.

(i) The initial velocity divergence satisfies

[TABLE]

(ii) Then the initial spectral gap $\eta_{{}_{S_{0}}}:=\mu_{2}(S_{0})-\mu_{1}(S_{0})$ is bounded

[TABLE]

Then the class of such sub-critical initial conditions (3.5),(3.6) give rise to a classical solution $(\rho(t),{\mathbf{u}}(t)\in C(\mathbb{R}^{+};L^{\infty}(\mathbb{R}^{2}))\times C(\mathbb{R}^{+};\dot{W}^{1,\infty}(\mathbb{R}^{2}))$ with large time hydrodynamics flocking behavior (1.8d) $\displaystyle\mathop{\max}_{x\in\text{supp}(\rho)}|{\mathbf{u}}(x,t)-{\mathbf{u}}(y,t)|\lesssim e^{-\kappa t}$.

Remark 3.1**.**

In the case of finite horizon alignment encoded in (1.9) with $\alpha=\phi*\rho$, the critical thresholds (3.5),(3.6) can be restricted to the finite set $\text{dist}\{x,\text{supp}\{\rho_{0}\}\}$.

Proof.

As before, we trace the dynamics of $M=\partial_{j}u_{i}$,

[TABLE]

where the entries of the residual matrix $\{R_{ij}\}$ are given by

[TABLE]

Expressed in terms of the operator $A(w):=\int_{y}a(x,y,t)w(y)dy$, the entries of $R$ have the commutator structure $R_{ij}=\partial_{j}[A,u_{i}](\rho)$ which can be estimated by the commutator bound [TT2014, proposition 7.1] in terms of $V(t)=\sup_{\text{supp}(\rho)}|u_{i}(x,t)-u_{i}(y,t)|$,

[TABLE]

We now proceed as before. As a first step, we follow the dynamics of the ${\sf{d}}=\operatorname{div}{{\mathbf{u}}}$: taking the trace of ${(\ref{eq:MTM})}$ we find

[TABLE]

This calls for the introduction of a new variable ${\sf{e}}:={\sf{d}}+1$ where the last equation recast into the Riaccti’s form

[TABLE]

Our purpose si to show that the $\{x\ |\ {\sf{e}}(x,t)\geqslant 0\}$ is invariant of the dynamics (3.9) and to this end we need to bound the spectral gap $\eta_{{}_{S}}$.

The second step is to follow the spectral dynamics associated with the symmetric part of (3.7)

[TABLE]

Taking the difference and recalling that ${\mathbf{s}}_{i}$ are the normalized eigenvectors of $S$ we find the dynamics of the spectral gap,

[TABLE]

It follows that as long as ${\sf{e}}(\cdot,t)$ is positive then

[TABLE]

and therefore $c:=\sqrt{1-\eta_{{}_{S}}^{2}+2r}$ has the lower bound $c(x,t)\geqslant c_{\text{min}}>0$, where

[TABLE]

This inequality follows from the assumed bounds on $V_{0}$ in (3.4) and on the initial spectral gap (3.6), and the bound of $r$ in (3.8). As a final step, we return to (3.9) to find, ${\sf{e}}^{\prime}\geqslant\frac{1}{2}(c^{2}_{\text{min}}-{\sf{e}}^{2})$, which guarantees that if the critical threshold (3.5) holds, i.e., if ${\sf{e}}_{0}\geqslant 0$ then ${\sf{e}}(x,t)\geqslant 0$ at later time. Moreover, since ${\sf{e}}(\cdot,t)\geqslant 0$, the vorticity equation, $\omega^{\prime}+{\sf{e}}\omega=\frac{1}{2}\operatorname{Tr}{JR}$, shows that $|\omega(\cdot,t)|$ remains bounded in terms of $\max_{x}|R_{ij}(x,t)|\lesssim r_{\text{max}}<\infty$. The transport equation (3.9) implies

[TABLE]

and a uniform upper-bound of ${\sf{e}}(\cdot,t)\leqslant{\sf{e}}_{\text{max}}<\infty$ follows. ∎

Remark 3.2**.**

In the one-dimensional case, $\eta_{{}_{S}}=\omega\equiv 0$ and the dynamics of ${\sf{e}}={\sf{d}}+1$ in (3.9) simplifies into ${\sf{e}}^{\prime}=\frac{1}{2}(1+2r-{\sf{e}}^{2})$. Hence, the variation bound (3.4) can be related to

[TABLE]

so that $1+2r\geqslant c_{\text{min}}>0$ and ${\sf{e}}^{\prime}>\frac{1}{2}(c_{\text{min}}-{\sf{e}}^{2})$ implies global smoothness under the critical threshold condition $\partial_{x}u_{0}(x)\geqslant-1$.

Remark 3.3**.**

One can follow the argument in section 2.2 to conclude that the same rapid alignment holds for MT model. Indeed, the MT model enhances the convergence rate towards a limiting flocking state.