Unconditional flocking for Weak solutions to self-organized systems of Euler-type
with all-to-all interaction kernel

The study of hydrodynamic models that emerged in the area of self-organization has received alot of attention in the recent years and many new challenges in partial differential equations have arisen that yield interesting questions in the mathematical community. In this paper, we set up a problem in this context and study solutions in the weak framework that are appropriate to capture asymptotic flocking. To begin with, we consider the Cauchy problem for the system

with $(x,t)\in\mathbb{R}\times[0,+\infty)$. Here $\rho\geq 0$ stands for the density, $\mathtt{v}$ for the velocity, $p$ for the pressure, given by

and $K>0$ is a given constant. Having set $\mathtt{m}:=\rho\mathtt{v}$ as the momentum variable, let the initial condition be

and our aim is to formulate a problem to (1.1)–(1.3) with conditions appropriate for the models of self-organized systems and then seek weak solutions. The pioneering work of Cucker and Smale [7] led a major part of the mathematical community to conduct research intensively on this topic. Many mathematical models have arised and most work so far is on the behavior of the particle models, the kinetic equation and the hydrodynamic formulation. However, very little is done in this area on weak solutions and the scope of this paper is to contribute in this direction of weak solutions to the an Euler-type flocking system that is derived in a hydrodynamic formulation. We refer the reader to the reviews [8, 18, 19] and the references therein.

Karper, Mellet and Trivisa in [14] prove the convergence of weak solutions to the kinetic equation Cucker-Smale flocking model on $(x,t,\omega)\in(0,T)\times\mathbb{R}^{d}\times\mathbb{R}^{d}$

to strong solutions of an Euler-type flocking system of the form (1.1). Here, $f^{\epsilon}\dot{=}f^{\epsilon}(x,t,\omega)$ stands for the scalar density of individuals and $L$ the alignment operator that is the usual Cucker-Smale operator given by

where $K$ is a smooth symmetric kernel and $\epsilon>0$ a small positive parameter. Moreover, on the right hand side of (1.4), the first term is due to the presence of a stochastic forcing at the particle level, see [5] and the last term in (1.4) is the strong local alignment interaction with $\mathtt{v}^{\epsilon}$ to be the average local velocity. This alignment term was derived in [13] from the Motsch-Tadmor [16] alignment operator (MT) as a singular limit and it is known that the MT operator is an improvement of the standard Cucker-Smale model at small scales. In [14], they study the singular limit corresponding to strong noise and strong local alignment, i.e. $\epsilon\to 0^{+}$, and show the convergence

while the macroscopic variables $\rho,\,\mathtt{v}$, which are the $\epsilon\to 0^{+}$ limits of

satisfy the Euler-type flocking system (1.1) with pressure (1.2). However, in the literature, most hydrodynamic models for flocking so far are described by a pressureless Euler system. Actually, the pressureless systems arise because the microscopic description of the particles motion, considered in those works, does not contain a stochastic forcing, hence the kinetic equation does not contain the diffusion term and the momentum equation can be closed by the mono-kinetic ansatz that renders it different from (1.5). It is true that the system with pressure received less attention than the pressureless one and especially in view of weak solutions. We refer to [6] for a result on smooth, space-periodic solutions to this model with pressure and our previous result [2] for (1.1) with $K=1$ on weak solutions and conditional flocking. We will elaborate on the results in [2] after setting up the problem.

To set up our problem, we assume that the initial mass $\rho_{0}$ is confined in a bounded interval, and is uniformly positive in there i.e. there exist $a_{0}<b_{0}$ such that, for $I_{0}~{}\dot{=}~{}[a_{0},b_{0}]$:

This assumption is imposed in order our problem to be suitable model for flocking. For this reason, we will also seek solutions having a bounded support for every $t>0$, that is, there exists two continuous curves $t\mapsto a(t)$, $b(t)$, $t\in[0,+\infty)$ with

such that the support satisfies

Actually, we require that the two interfaces $a(t)$ and $b(t)$ correspond to particle trajectories:

Having these, we set

as the non-vacuum region and consider the velocity $\mathtt{v}=\mathtt{m}/\rho$ on $\Omega$. Then system (1.1) can be considered in the sense of distributions on $\Omega$ and rewrites as

We aim for a notion of entropy weak solutions with concentration that is motivated by the ad-hoc boundary condition:

This motivation has as a target to capture a sharp front with finite speed around the non-vacuum region $\Omega$ that is expected to arise in flocking. In this way, we exclude the case of a rarefaction connecting a vacuum region with a non-vacuum since in such a case, due to the pressure law (1.2), the front would not have a proper interpretation in terms of flocking because of the unbounded maximal speed, having that $\int_{0}^{\rho}\frac{\sqrt{p^{\prime}(s)}}{s}\,ds=+\infty$ for $\rho>0$.

The definition of the entropy weak solution with concentration is stated in Section 2 and in short, we can say that such solution is entropy weak to (1.11) in $\Omega$ but delta point masses are present along the interfaces in the momentum equation when the solution is tested on the whole half plane. The point masses correspond to the values of pressure evaluated within the non-vacuum region and render the solution conserving mass and momentum in the following sense: From the integral identity of (1.11)₁, that is

and the Rankine-Hugoniot condition $[\mathtt{m}]=\dot{x}[\rho]$ that hold along the interfaces $a(t)$ and $b(t)$ thanks to (1.9), we get immediately that conservation of total mass holds true:

Here $[\cdot]$ denotes as usual the jump of the values. In addition, the concept of total extended momentum defined as

where

would be shown to be conserved as well from our notion of solution (see Prop. 2.1).

As it is also discussed in [2], the occurrence of a singularity for the solution in the sense that the momentum exhibits concentration at the interfaces $a(t)$ and $b(t)$ as Dirac deltas leads to the definition of the extended momentum $\widehat{\mathtt{m}}$ as the following distribution:

where $\delta_{x_{0}}$ denotes the Dirac delta function at $x_{0}\in\mathbb{R}$. This new singularity of the extended momentum $\widehat{m}$ along the interfaces $a(t)$ and $b(t)$ is known as delta shock and references can be found in [9], Chapter 9.

We further use the following standard notation $<\cdot,\cdot>$:

as the value of the functional $\widehat{\mathtt{m}}$ over $C_{0}^{\infty}$, for all test functions $\phi\in C_{0}^{\infty}(\mathbb{R}\times\mathbb{R}_{+})$. Note here that $\mathtt{m}=0$ for $x\notin I(t)$. It is shown in Proposition 2.1 that solutions in the sense of Definition 2.1 satisfy the conservation of total mass (1.12) and total extended momentum:

and at this point it is evident that concentration terms in the total momentum are the appropriate fit for the set up of this problem.

The aim of this paper is to show unconditional time asymptotic flocking for entropy weak solutions with concentration  (1.1), (1.3) with (1.2) that is a significant improvement of our previous result in [2], in which a condition associated with the initial data is imposed.

Before we state the main result, we set the average velocity to be

and provide the definition of time-asymptotic flocking.

Here we state the main theorem of this paper.

Global existence of such weak solutions is shown in [2] for $K=1$ and without much effort this is extended to any constant kernel $K>0$ in our setting by scaling the variables. However, the time asymptotic flocking needs careful treatment different from [2]. Let us point out that in our previous result [2], time asymptotic flocking is achieved under a special condition on the bulk of the initial data. More precisely, by means of Theorem 1.2 in [2] and the case $K=1$, one obtains a sufficient condition for time-asymptotic flocking that involves the parameter $\alpha$ in (1.2), the total mass $\mathsf{M}$, pointwise values of $\rho_{0}$ at the endpoints $a_{0}+$ and $b_{0}-$ and the quantity

Moreover, the oscillation of the velocity decays exponentially fast to zero. In the present work, we prove that the time-asymptotic flocking occurs unconditionally, i.e. for any initial data $(\rho_{0},\mathtt{m}_{0})\in BV(\mathbb{R})$ that satisfy (1.6) without any further condition and also capture exponential decay again. Furthermore, it is immediate from our analysis that the density $\rho$ decays exponentially fast to a constant state as obtained in (1.18) while this is not a requirement for flocking according to Definition 1.1.

Our analysis relies alot on the construction in [2] of an approximate solutions that converge up to a subsequence in $L^{1}_{loc}$ to entropy weak solutions with concentration along the interfaces $a(t)$ and $b(t)$, but it requires different tools and strategy to study the wave decay and control the total variation as $t\to\infty$ without any further conditions. Let’s introduce the construction in  [2]. First, by conservation of total mass and total extended momentum, the system can be reduced to a local one. Indeed, by means of (1.12) and (1.16), the integral term on the right hand side of (1.1) can be rewritten as

where $\phi_{1}$ is a test function equal to 1 for $x\in I(t)$ while $\bar{\mathtt{v}}$ is the average speed given at (1.17). Thus, system (1.1) or (1.11) rewrites as

and the equivalence between the nonlocal and the local system relies on the conservation of $\mathsf{M}$ and $\mathsf{M}_{1}$, hence it is suitable in the setting of our notion of weak solution with concentration.

Now, to construct a convergent approximate sequence to systems of conservation laws, we require strict hyperbolicity. Since the Cauchy problem (1.20), (1.3) is not strictly hyperbolic on $\mathbb{R}\times(0,\infty)$, we transform the problem into Lagrangian coordinates in the spirit of Wagner [20]. However, the equivalence from Eulerian $(\rho(x,t),\mathtt{m}(x,t))$ into the Lagrangian variables $(u(y,t),v(y,t))$ needs to be verified in our set up because in [20], it is assumed infinite total mass. Actually, our finite total mass condition has as an implication that the problem in Lagrangian coordinates is an initial boundary problem with fixed boundaries at $y=0$ and $y=\mathsf{M}$. More precisely, by recasting system (1.20) from Eulerian $(\rho(x,t),\mathtt{m}(x,t))$ into the Lagrangian variables $(u(y,t),v(y,t))$, we obtain the equations

with the domain $\{(y,t);\ t\geq 0\,,\ y\in(0,\mathsf{M})\}$ . It should be mentioned that global existence of weak solutions to the homogeneous system ($\mathsf{M}=0$) corresponding to (1.21) was first obtained by Nishida [17] using the random choice method for initial data of large $BV$ for both the Cauchy and the boundary case. For the non-homogeneous system (1.21), global weak solutions were constructed in [10, 15, 4], using either the random choice or the front tracking methods, but only for the Cauchy problem. Also, Frid [11] studied existence under certain initial-boundary value problems on a bounded domain for systems arising in isentropic gas dynamics and elasticity theory, which are different type from ours. Therefore, the aformentioned results do not apply in our setting and for this reason the machinery of front tracking algorithm was incorporated to establish the existence in [2] for $K=1$. This result is trivially extended to any constant $K>0$.

The heart of the matter is to prove that the total variation decays to zero as time tends to infinity without any further condition on the initial data. This requires delicate estimates on the cancellation properties of the wave strengths, that leads to wave decay. The analysis is performed around interaction of waves and time steps at the level of the approximate solutions using a novel approach to quantify the change of the total variation and get sharp bounds. Our approach relies on the interplay between the rarefaction part and the total variation and a chase to reveal the hidden relation between the two. The crucial point that uncovers the vanishing property of the total variation is that the rarefaction part of the solution is controlled by the variation of the linear functional. Since the linear functional is non-increasing, if we wait long enough, the rarefaction part can become arbitrarily small. The key idea to show that the rarefaction part can be controlled in this way relies on a region that serves as a trapped area for each family of rarefaction waves (see Lemma 4.1 and Figure 1). This region is actually triangular in Lagrangian coordinates and with an appropriate weighted functional we study the possible cases that can occur. As a consequence, also the total amount of shocks must vanish as $t\to\infty$: indeed, we show in Lemma 4.2 that a persisting positive amount of shocks would produce a uniformly positive amount of rarefactions, which contradicts the property that rarefactions vanish as $t\to\infty$. Finally, again by means of wave decay, we show in Lemma 5.1 that the total variation cannot stay uniformly positive for arbitrarily large times and then that there exists a time $T^{*}$ such that the total variation of the unknowns $\rho$, $\mathtt{v}$ reduces by a geometric rate when passing from a time $\tau$ to time $\tau+T^{*}$ within the non-vacuum region $\Omega$.

The paper is organized as follows: We first state the definition of entropy weak solution with concentration in Section 2 and show conservation of total mass and total extended momentum for such solutions. Then in Section 3, we describe the strategy of our proof that is postponed for the next sections. Actually, in Section 4, we show that the solution admits time asymptotic flocking and this is proven to happen exponentially fast in Section 5.

In this section, we first present the definition of entropy weak solution with concentration and then, we exploit it in terms of the conserved quantities. As it is shown, the concentration terms in (1.15) are crucial to establish conservation of total momentum, in the sense of (1.16).

First, we consider the functions

defined on $(0,+\infty)\times\mathbb{R}$, in terms of $\rho>0$ and $\mathtt{m}$, that constitute a pair of entropy-entropy flux functions for the system (1.1) i.e. they are differentiable on $(0,+\infty)\times\mathbb{R}$, $\eta$ is convex and the following relations hold

Let us now introduce the appropriate notion of entropy weak solution for (1.1), or equivalently (1.11), together with (1.3) and (1.2), which we call entropy weak solution with concentration.

From Definition 2.1, taking test functions supported in $\Omega$, it is immediate that this notion of solution considered is an entropy weak solution to the system

for $(x,t)\in\Omega$ rather than $\mathbb{R}\times\mathbb{R}_{+}$, where $M_{1}(t)$ is given at (1.13). However, for test functions continuous up to the interfaces $a(t)$ and $b(t)$, it is necessary to take into account boundary terms that arise from the term $\mathtt{m}^{2}/\rho$. The presence of these boundary contributions in the notion of solution was motivated in [2, Sec. 2.1] by studying the Riemann problem around vacuum. More precisely, consider the homogeneous system (1.11), i.e.

with Riemann data

denote by $(\rho_{\delta},\mathtt{m}_{\delta})$ its Riemann solution that is a $2$-shock and then take the limit $\delta\to 0+$ to capture the $2$-shock solution $(\rho,\mathtt{m})$ around vacuum at the point $x=b_{0}=0$. Here the left state $(\rho_{\ell},\mathtt{m}_{\ell})$ corresponds to a fixed state with $\rho_{\ell}>0$, while the right state $(\delta,\mathtt{m}_{r}(\delta))$, where $\mathtt{m}_{r}(\delta)=\delta\,\mathtt{v}_{r}(\delta)$, varies in terms of the parameter $\delta>0$. Using the Rankine-Hugoniot conditions

and the shock wave curve of the second family

we can determine this solution

for $\delta\in(0,\rho_{\ell})$ where $\mathtt{m}_{r}(\delta)=\delta\,\mathtt{v}_{r}(\delta)$ is obtained from (2.10) and the shock speed $\sigma$ is given by (2.8), (2.9) noting that $\sigma$ also depends on $\delta$. As $\delta\to 0+$, we have $\sigma\to\mathtt{v}_{\ell}$ and $(\rho_{\delta},\mathtt{m}_{\delta})\to(\rho,\mathtt{m})$ and it is immediate that the limit satisfies

Now, the weak formulation of the homogeneous system (1.11)₂, for $\delta>0$ reads as

for all $\phi\in C_{0}^{\infty}(\mathbb{R}\times(0,+\infty))$ and as $\delta\to 0+$, one can check that the term $(\mathtt{m}_{\delta})^{2}/\rho_{\delta}$ in the region $x>\sigma t$ does not vanish, but it gives a non trivial contribution

along the interface $x=\mathtt{v}_{\ell}t$. In fact, this results into a concentration term $p(\rho_{\ell})$ centered at $x=b(t)=\mathtt{v}_{\ell}\,t$. Therefore, the limit function $(\rho,\mathtt{m})$ satisfies the integral identities (2.2) and

We point out that the above computation is performed for the homogeneous system (2.6) and one can reach the integral identity (2.3) if the source term is included and both endpoints of the support $(a(t),b(t))$ are involved. Another point is that one could express the weak formulation for the extended momentum $\hat{m}$ using a Dirac delta that is concentrated along the discontinuity $\gamma=\{(\mathtt{v}_{\ell}t,t);\ t\geq 0\}$.

Regarding now the conserved quantities, it is obvious that the conservation of total mass as mentioned in (1.12) holds. Since there are concentration terms through the values of the pressure along the two interfaces, we confirm in the next proposition that the extended momentum $\hat{\mathtt{m}}$ given in (1.15) yields conservation of total momentum. It is of course important that the absolutely continuous functions $a(t)<b(t)$, which are also unknowns of the problem, correspond to particle paths, i.e. (1.9), and as it is shown below from here and on $M_{1}(t)$ is constant in time equal to $\mathsf{M}_{1}$, i.e. (1.16). Thus, the equivalence of the local system (1.20) with the nonlocal (1.1) or (1.11) is verified for entropy weak solutions with concentration.

In this section, we outline the strategy of the proof of Theorem 1.1 and for better readability, we organize it into four steps.

$\bullet$ Step 1: Rescaling system (1.20) to the case $K=1$.

First, we reduce the problem to the case that the constant interaction kernel is $1$ by rescaling the coordinates as follows: let $\lambda>0$ a parameter to be determined and consider the map

with inverse:

Now, define the scaled density $\tilde{\rho}$ as function of $(x^{\prime},t^{\prime})$:

or equivalently,

and similarly $\tilde{\mathtt{v}}$ and $\tilde{\mathtt{m}}$. Therefore, in the new coordinate system, we have the equations

where $\widetilde{\mathsf{M}}$ and $\widetilde{\mathsf{M}}_{1}$ are the total mass and total extended momentum for the scaled variables $(\tilde{\rho}(x^{\prime},t^{\prime}),\tilde{\mathtt{m}}(x^{\prime},t^{\prime}))$. It is straightforward to check that $\widetilde{\mathsf{M}}=\lambda\mathsf{M}$ and $\widetilde{\mathsf{M}}_{1}=\lambda\mathsf{M}_{1}$ while $\bar{\tilde{\mathtt{v}}}=\bar{\mathtt{v}}$ and the interval of the support is $[\tilde{a}(t),\tilde{b}(t)]=[\lambda\,a(t),\lambda\,b(t)]$. Selecting $\lambda^{2}=K$, we reduce to (3.1) with kernel being $1$. To simplify the notation, from here and on, we regard that $(\rho,\mathtt{m})$ satisfies  (3.1) for $K=1$ and without loss of generality we use $x$ and $t$, $\rho$ and $\mathtt{m}$ for the rescaled variables.

Furthermore, we reduce the problem to zero average velocity. More precisely, by the definition of the average velocity $\bar{\mathtt{v}}$ at (1.17), we perform the change of variables

that allows us to reduce to the case of $\mathsf{M}_{1}=0$, with $\mathsf{M}_{1}$ being defined at (1.16). Indeed, in the new variables called again $(x,t)$, the total extended momentum becomes zero:

and system (1.1) or (3.1) takes the form

Therefore, from here and on, we can assume that $\mathsf{M}_{1}=\bar{\mathtt{v}}=0$ and consider system (3.3). We will return to the original notation in § 5.1.

$\bullet$ Step 2: Existence of weak solutions.

From Theorem 1.1 in [2], there exists an entropy weak solution $(\rho,\mathtt{m})$ with concentration along $a(t)$ and $b(t)$ to (3.3). According to Definition 2.1 and with $\mathsf{M}_{1}=0$, the integral identity (2.3) of the entropy weak solution with concentration rewrites as:

for all test functions $\phi\in C^{\infty}_{0}(\mathbb{R}\times(0,\infty))$.

Following [2, Subsect. 2.2], we transform the problem from Eulerian into Lagrangian coordinates $(y,t)$ as follows: consider the map