Pressureless Euler Alignment System

• The topic is a hydrodynamic Cucker–Smale model characterized by pressureless, monokinetic dynamics and nonlocal velocity alignment that drive its singular and regular regimes.
• Its analysis employs scalar reduction, entropy selection, and critical-threshold methods to rigorously study global regularity, finite-time blowup, and related weak solution behavior.
• The model underpins diverse regimes—from sticky particle dynamics to overdamped limits—bridging kinetic theory, hydrodynamics, and optimal transport techniques.

Searching arXiv for the specified papers and related work on the pressureless Euler alignment system. The pressureless Euler alignment system is a hydrodynamic Cucker–Smale model in which the unknowns are a density $\rho$ and a velocity field $u$, the momentum flux is purely monokinetic, and the only forcing is nonlocal velocity alignment. In one space dimension, a standard form is

$\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$

with equivalent velocity form

$$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$

In several dimensions, the corresponding conservative law is

$$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$

The system is pressureless because there is no pressure tensor or scalar pressure term in the momentum equation; the monokinetic closure is inherited from the ansatz $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$ in kinetic Cucker–Smale theory (Leslie et al., 2021, Figalli et al., 2017).

1. Model class and hydrodynamic meaning

The defining structural feature is the nonlocal alignment operator

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$

or, in conservative form,

$\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$

This term relaxes velocity discrepancies through averaging against a communication kernel. In the one-dimensional low-regularity theory, the assumptions are that total mass is normalized to one, $\rho^0\in \mathcal P_c(\mathbb R)$, $u^0\in L^\infty(d\rho^0)$, and $u$0 is nonnegative, locally integrable, and radially decreasing; this includes bounded or Lipschitz kernels and weakly singular kernels with an integrable singularity at the origin (Leslie et al., 2021).

The kernel regime strongly influences the PDE. In the weakly singular setting, the model assumption

$u$1

places the equation between bounded-kernel threshold theory and strongly singular unconditional regularization (Tan, 2019). In the singular periodic model with

$u$2

the alignment acts as a density-modulated nonlocal dissipation, and the corresponding one-dimensional periodic system is globally smooth for all smooth strictly positive initial data (Do et al., 2017).

From the kinetic viewpoint, the pressureless Euler alignment system is the monokinetic limit of flocking equations. A rigorous derivation starts from a kinetic Cucker–Smale equation with standard nonlocal alignment plus a strongly scaled local alignment term and proves convergence to the macroscopic pressureless Euler system with nonlocal alignment. The limiting entropy is

$u$3

which is convex but not strictly convex in $u$4; this degeneracy is one reason Wasserstein control becomes essential in pressureless hydrodynamic limits (Figalli et al., 2017).

2. One-dimensional scalarization and entropic selection

The modern one-dimensional theory is built on an exact scalar reduction. A pivotal variable is

$u$5

where $u$6 is the odd antiderivative of $u$7. Then $u$8 satisfy the same continuity equation, and the cumulative primitives

$u$9

are transported by the same velocity field. If $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$0, then formally $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$1, which yields the scalar nonlocal balance law

$\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$2

This reformulation is the central mechanism behind the global weak theory in one dimension (Leslie et al., 2021).

The admissibility notion is not an Eulerian entropy inequality for $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$3 directly, but the entropy solution of the scalar law. For convex Lipschitz $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$4 and entropy flux $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$5 defined by $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$6,

$\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$7

in distributions. Kružkov entropies,

$\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$8

yield uniqueness and $\begin{cases} \partial_t \rho + \partial_x(\rho u)=0,\[2mm] \partial_t(\rho u)+\partial_x(\rho u^2) = \rho\,(\phi * (\rho u))-(\rho u)\,(\phi * \rho), \end{cases}$9-stability. Reconstruction is then canonical: $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$0 with $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$1 (Leslie et al., 2021).

This structure persists under additional nonlocal forcing. For the one-dimensional Euler–Poisson–alignment system, the transported quantity becomes

$$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$2

and the scalar law becomes

$$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$3

The time-dependent quadratic flux is the new analytical difficulty; when $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$4, the pure pressureless Euler alignment scalar law is recovered exactly (Leslie et al., 3 Jun 2026).

3. Critical thresholds and regularity regimes

A second organizing principle is the nonlinear slope variable

$$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$5

or, in the singular fractional setting,

$$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$6

For integrable alignment kernels, $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$7 obeys a transport or continuity equation together with $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$8, and the sign of $$\partial_t u + u\,\partial_x u = \int_{\mathbb R}\phi(x-y)\,\rho(y,t)\,\big(u(y,t)-u(x,t)\big)\,dy.$$9 controls the onset of compression (Tan, 2019).

In the one-dimensional weakly singular regime $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$0, $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$1, the threshold picture is sharp away from the borderline. If $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$2, then finite-time blow-up occurs; if $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$3, the solution is globally regular. The critical set $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$4 is qualitatively different: if $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$5 on an interval carrying positive density, finite-time blow-up occurs, whereas global critical solutions also exist when $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$6 only in vacuum and $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$7 is bounded (Tan, 2019). This distinguishes weakly singular kernels from the bounded-kernel theory, where $$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$8 is benign.

For the singular density-modulated model on the torus,

$$\partial_t \rho + \nabla_x\cdot(\rho u)=0,\qquad \partial_t(\rho u)+\nabla_x\cdot(\rho u\otimes u) = \int_{\mathbb T^d}\psi(x-y)\rho(x)\rho(y)\big(u(y)-u(x)\big)\,dy.$$9

the situation is different. The alignment nonlinearity enhances dissipation, and the one-dimensional periodic system with smooth strictly positive initial density has a unique global smooth solution for every $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$0. The mechanism combines the transport structure of $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$1, a nonlinear maximum principle for $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$2, and a modulus-of-continuity argument; the corresponding fractional Burgers equation is supercritical, but the Euler alignment model remains globally regular (Do et al., 2017).

The same threshold philosophy extends to larger coupled systems. In the repulsive Euler–Poisson–alignment model with variable background,

$f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$3

the characteristic dynamics reduce to

$f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$4

This produces a phase-plane decomposition into four regions and a critical-threshold theory assembled from localized Lyapunov curves. In the pure alignment limit $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$5, $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$6 is constant along characteristics, recovering the invariant underlying one-dimensional Euler-alignment threshold theory (Luan et al., 5 May 2025).

4. Sticky particles, gradient flows, and admissible weak solutions

Finite-time loss of smoothness leads to atomic densities, crossing characteristics, and nonuniqueness of distributional weak solutions. In one dimension, the distinguished continuation is encoded by sticky particle Cucker–Smale dynamics. For particles of masses $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$7, positions $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$8, and velocities $f(x,v,t)=\rho(x,t)\delta(v-u(x,t))$9,

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$0

between collisions, and colliding particles stick thereafter, preserving momentum. The discrete transported quantity

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$1

is constant between collisions and undergoes mass averaging at collision times. The corresponding empirical cumulative function $\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$2 is the entropy solution of a discretized scalar balance law and converges to the continuum entropy solution (Leslie et al., 2021).

A variational reformulation sharpens this picture. In Lagrangian coordinates, with monotone rearrangement $\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$3, the sticky dynamics arise as the $\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$4-gradient flow of

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$5

where

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$6

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$7 is the initial natural velocity, and $\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$8. The evolution is

$\int \phi(x-y)\big(u(y)-u(x)\big)\rho(y)\,dy,$9

This yields a unique Lagrangian solution, a distributional Eulerian solution in $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$0, and an entropy solution of the Leslie–Tan scalar balance law. Cluster formation is governed by the convex envelope $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$1 of the primitive $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$2 (Galtung, 2024).

Sticky admissibility is not universal once other nonlocal forces are added. In the attractive Euler–Poisson–alignment regime $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$3, sticky particle dynamics generates the entropy solution. In the repulsive regime $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$4, sticky continuation fails the Oleinik condition; atomic states may instead disperse, and the entropy solution can be a rarefaction. For the repulsive Euler–Poisson equation with $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$5, $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$6, and $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$7, the entropy solution instantly spreads the atom into the absolutely continuous density

$\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$8

(Leslie et al., 3 Jun 2026).

5. Multidimensional unidirectional theory and concentration geometry

Beyond one dimension, the complete theory remains much less developed, but two structured regimes are known. The first is the unidirectional class $\rho\,(\phi * (\rho u))-(\rho u)(\phi * \rho).$9, where $\rho^0\in \mathcal P_c(\mathbb R)$0 and only the first coordinate transports mass. In this setting, the system can be recast as a family of coupled scalar balance laws, one for each horizontal slice: $\rho^0\in \mathcal P_c(\mathbb R)$1 with time-independent transverse measure $\rho^0\in \mathcal P_c(\mathbb R)$2. Existence, uniqueness, and stability of entropy solutions hold under an even bounded kernel with bounded transverse Lipschitz derivative, and the construction proceeds by sticky particle Cucker–Smale approximations, first transverse to the flow and then along it (Adeleke et al., 9 Jun 2026).

The main technical novelty is that comparison between two solutions uses an optimal coupling $\rho^0\in \mathcal P_c(\mathbb R)$3 between their transverse marginals. The natural error functional is

$\rho^0\in \mathcal P_c(\mathbb R)$4

which is an averaged slice-wise Wasserstein distance. Two complementary stability estimates are proved: one depending on the Lipschitz seminorm of the flux difference, and a lower-regularity estimate depending only on its $\rho^0\in \mathcal P_c(\mathbb R)$5-norm in $\rho^0\in \mathcal P_c(\mathbb R)$6, inspired by Bouchut–Perthame and adapted to the nonlocal slice coupling (Adeleke et al., 9 Jun 2026).

The second structured multidimensional regime concerns long-time mass concentration under unidirectional velocity and smooth heavy-tailed kernels. There the entropy variable

$\rho^0\in \mathcal P_c(\mathbb R)$7

satisfies a scalar conservation law, and the zero set

$\rho^0\in \mathcal P_c(\mathbb R)$8

governs asymptotic singular concentration. If $\rho^0\in \mathcal P_c(\mathbb R)$9 is the limiting density measure, then its singular part is the pushforward of the initial mass on $u^0\in L^\infty(d\rho^0)$0, whereas the absolutely continuous part comes from $u^0\in L^\infty(d\rho^0)$1. In typical situations, the concentration set is a union of $u^0\in L^\infty(d\rho^0)$2 hypersurfaces, and in one dimension the box dimension of $u^0\in L^\infty(d\rho^0)$3 satisfies

$u^0\in L^\infty(d\rho^0)$4

when $u^0\in L^\infty(d\rho^0)$5 (Lear et al., 2020).

6. Vacuum, asymptotics, and related limits

A distinct weak-solution theory appears for the strongly singular one-dimensional system on $u^0\in L^\infty(d\rho^0)$6 with kernel $u^0\in L^\infty(d\rho^0)$7, $u^0\in L^\infty(d\rho^0)$8, allowing vacuum and non-smooth solutions. The structural variable is

$u^0\in L^\infty(d\rho^0)$9

and the augmented system

$u$00

supports global weak existence for data satisfying

$u$01

with $u$02. Under the stronger two-sided comparability $u$03, the hyperbolically rescaled solution converges to a Burgers rarefaction wave, and the density approaches a uniform profile on an expanding interval: $u$04 This identifies a rarefaction asymptotic universality class distinct from the $u$05 nonlocal porous-medium reduction (Cygan et al., 2024).

Another asymptotic regime is the overdamped limit of pressureless Euler equations with alignment, attraction, repulsion, and linear friction: $u$06 As $u$07, the system converges quantitatively to the first-order continuity equation

$u$08

with velocity determined implicitly by

$u$09

The proof combines relative entropy with a $u$10-Wasserstein estimate, replacing the pressure control unavailable in the pressureless setting (Choi, 2020).

Several extensions place the pressureless Euler alignment system inside broader families. A controlled version adds the feedback

$u$11

which shifts the one-dimensional threshold from $u$12 to $u$13 and enlarges the subcritical region (Albi et al., 2018). Euler–Poisson–alignment models show that the scalar-reduction and entropy-selection machinery is robust under added Poisson forcing, but also that attractive and repulsive post-singularity dynamics are fundamentally different (Leslie et al., 3 Jun 2026). Taken together, these developments indicate that the pressureless Euler alignment system is not a single equation but a structural class: a pressureless monokinetic transport law whose nonlocal slope variables, scalar reductions, and sticky or entropic admissibility principles remain central across singular-kernel, controlled, Poisson-coupled, and overdamped regimes.