3D Pressureless Compressible Navier–Stokes

• 3D pressureless compressible Navier–Stokes equations are a viscous flow model without pressure, emerging as a high-Mach-number limit of classical systems.
• The critical Besov space framework enables global existence and uniqueness even for discontinuous density profiles with large variations.
• Advanced methods like time-weighted energy estimates, dyadic decompositions, and Lagrangian uniqueness yield rigorous O(ε) convergence over finite time intervals.

Searching arXiv for the cited papers and closely related work on the 3D compressible pressureless Navier–Stokes equations. The three-dimensional compressible pressureless Navier–Stokes equations are a viscous compressible flow system in which the pressure force is absent. In the form studied by Wang–Wu–Xu, the model arises from collective-behavior equations and can be obtained formally as a high-Mach-number limit of the classical compressible Navier–Stokes system. Recent work establishes a global-in-time unique Fujita–Kato solution in dimension three for initial densities that may be discontinuous and have large variation, provided only that they are bounded above and below by positive constants and that the initial velocity is small in the critical Besov space $\dot B^{1/2}_{2,1}$ (Xu, 13 Aug 2025). Complementary critical-space analysis of the scaled compressible system justifies the pressureless limit and proves quantitative $O(\varepsilon)$ convergence on finite time intervals for $d\geq 3$ (Ni et al., 8 Jun 2026).

1. Governing equations and equivalent formulations

In Eulerian coordinates $(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$, the Cauchy problem is

$\partial_t \rho+\operatorname{div}(\rho u)=0,$

$$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$

with initial data

$(\rho,u)\big|_{t=0}=(\rho_0,u_0).$

Here $\rho\ge 0$ is the density, $u\in\mathbb{R}^3$ is the velocity, and the viscosity coefficients satisfy

$\mu>0,\qquad 2\mu+3\lambda\ge 0.$

This is the pressureless viscous compressible system analyzed in the three-dimensional global theory of Wang–Wu–Xu (Xu, 13 Aug 2025).

A perturbative representation is obtained by writing $O(\varepsilon)$0. In that form, the limiting pressureless system becomes

$O(\varepsilon)$1

$O(\varepsilon)$2

This formulation is central in the critical Besov treatment of the high-Mach-number limit, because it makes explicit the transport character of the density equation and the highest-order coupling generated by the viscous terms (Ni et al., 8 Jun 2026).

The system combines a continuity equation with a viscous momentum equation but no pressure gradient. That absence is analytically decisive: there is no density dissipation mechanism, and the density evolves by transport alone. The recent literature therefore treats the pressureless model not as a minor variant of the classical compressible system, but as a distinct critical PDE problem with its own structural difficulties.

2. High-Mach-number derivation and asymptotic origin

The pressureless system is obtained formally from the barotropic compressible Navier–Stokes equations

$O(\varepsilon)$3

One derivation introduces the Mach number $O(\varepsilon)$4 and rescales

$O(\varepsilon)$5

In these variables, the pressure gradient carries a factor $O(\varepsilon)$6 and vanishes formally as $O(\varepsilon)$7, yielding the pressureless equations (Xu, 13 Aug 2025).

An equivalent convention, used in the quantitative limit theory, takes $O(\varepsilon)$8 to be the inverse Mach number. After normalizing the pressure law to $O(\varepsilon)$9 and writing $d\geq 3$0, the scaled system is

$d\geq 3$1

$d\geq 3$2

As $d\geq 3$3, the pressure term $d\geq 3$4 disappears formally, and the pressureless perturbation system is recovered (Ni et al., 8 Jun 2026).

These two conventions describe the same asymptotic regime. The first emphasizes the vanishing of pressure under high Mach number, while the second is adapted to uniform estimates and convergence rates. In both settings, the pressureless equations are not postulated independently but arise as a limit model.

3. Critical functional framework and the Fujita–Kato notion

The relevant three-dimensional critical spaces are built from homogeneous Besov norms. For the velocity, the Fujita–Kato class is

$d\geq 3$5

where $d\geq 3$6 is the homogeneous Besov space induced by the Littlewood–Paley decomposition, and

$d\geq 3$7

is the Chemin–Lerner norm. In the high-Mach-number analysis, the density fluctuation and velocity are placed in the scale-invariant spaces

$d\geq 3$8

with embeddings

$d\geq 3$9

and the real interpolation relation

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$0

whenever $(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$1 (Xu, 13 Aug 2025, Ni et al., 8 Jun 2026).

In the three-dimensional pressureless theory, “Fujita–Kato solution” means a strong solution whose velocity lies in the same critical class as in the classical Fujita–Kato theory for incompressible Navier–Stokes. The initial data are taken in the “critical” scale so that the scaling invariance

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$2

leaves the norm of $(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$3 unchanged. No more than boundedness, above and below away from zero, is required on $(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$4; the key smallness condition is

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$5

(Xu, 13 Aug 2025).

This framework separates the roles of density and velocity. The density need not be close to a constant in amplitude, while the velocity must be small in the critical norm. That distinction is one of the defining features of the current three-dimensional theory.

4. Global existence and uniqueness with discontinuous large-variation density

The main global theorem for the pressureless system assumes

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$6

Under these hypotheses, there exists a unique global solution $(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$7 such that

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$8

and

$(t,x)\in \mathbb{R}^+\times \mathbb{R}^3$9

Moreover, for all $\partial_t \rho+\operatorname{div}(\rho u)=0,$0,

$\partial_t \rho+\operatorname{div}(\rho u)=0,$1

This is the core global existence and uniqueness statement for the three-dimensional Cauchy problem (Xu, 13 Aug 2025).

The result is notable because the initial density may be discontinuous and may have large variation. A recurrent assumption in earlier works, as summarized in the same source, was that the density should be a small fluctuation of a constant or should belong to a Besov multiplier space. The new theorem shows that, for this model, it is enough to assume only boundedness above and below by positive constants, regardless of the size of jump discontinuities or variations (Xu, 13 Aug 2025).

The statement is global in time rather than merely local, and uniqueness is part of the theorem rather than a separate conditional result. The conclusion therefore places the three-dimensional pressureless system within a fully critical small-velocity well-posedness theory despite the roughness of the density profile.

5. Analytical mechanisms: weighted estimates, dyadic decomposition, and Lagrangian uniqueness

The proof architecture in the Wang–Wu–Xu analysis rests first on time-weighted energy estimates in critical Besov norms. The authors study auxiliary heat-type equations with inhomogeneous variable-density coefficient and use a decomposition à la Hmidi–Keraani, stated as Lemma 2.64 in [BCD], to propagate $\partial_t \rho+\operatorname{div}(\rho u)=0,$2 norms of $\partial_t \rho+\operatorname{div}(\rho u)=0,$3 without any smallness on $\partial_t \rho+\operatorname{div}(\rho u)=0,$4. The introduction of weights $\partial_t \rho+\operatorname{div}(\rho u)=0,$5 yields additional time integrability and is essential for obtaining the global Lipschitz control

$\partial_t \rho+\operatorname{div}(\rho u)=0,$6

This integrability of the Lipschitz norm is the pivotal bridge between a priori bounds and uniqueness (Xu, 13 Aug 2025).

The same analysis identifies several structural estimates. Proposition 3.1 gives a global a priori estimate in the scale

$\partial_t \rho+\operatorname{div}(\rho u)=0,$7

for $\partial_t \rho+\operatorname{div}(\rho u)=0,$8 under only $\partial_t \rho+\operatorname{div}(\rho u)=0,$9. Propositions 3.2–3.5 provide a hierarchy of time-weighted bounds on quantities such as $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$0, $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$1, and $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$2 in appropriate Besov or Lebesgue–Lorentz norms. Proposition 2.3 uses Lorentz-space interpolation; specifically, to prove

$$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$3

the argument interpolates between bounds on $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$4 and $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$5 via $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$6–$$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$7 duality (Xu, 13 Aug 2025).

Uniqueness is then derived in Lagrangian coordinates. Once the global Lipschitz bound is available, the flow map is well defined, the continuity equation becomes trivial in the form $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$8, and the momentum equation becomes a purely parabolic-type system for $$\partial_t(\rho u)+\operatorname{div}(\rho u\otimes u)-\mu\Delta u-(\lambda+\mu)\nabla\operatorname{div}u=0,$$9. This removes the hyperbolic loss of derivative and permits a standard energy estimate for the difference of two solutions. The method is therefore not a purely Eulerian perturbative argument; it combines Eulerian critical estimates with a Lagrangian uniqueness mechanism adapted to rough density transport.

6. High-Mach-number limit, quantitative convergence, and open directions

The companion high-Mach-number study proves global well-posedness for the scaled compressible Navier–Stokes system in critical Besov spaces for small initial data, uniformly in the inverse Mach number $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$0. In dimension $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$1, there exists $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$2 such that if

$(\rho,u)\big|_{t=0}=(\rho_0,u_0).$3

then the scaled system admits a unique global solution $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$4 with $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$5 and a uniform estimate

$(\rho,u)\big|_{t=0}=(\rho_0,u_0).$6

where $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$7 is a hybrid Besov norm involving low, medium, and high frequency pieces defined by the thresholds

$(\rho,u)\big|_{t=0}=(\rho_0,u_0).$8

A crucial ingredient is the low-frequency estimate for $(\rho,u)\big|_{t=0}=(\rho_0,u_0).$9,

$\rho\ge 0$0

which compensates for the lack of intrinsic density dissipation. In the high-frequency regime, the effective velocity

$\rho\ge 0$1

is introduced to close the uniform estimates (Ni et al., 8 Jun 2026).

The same paper proves a quantitative finite-time convergence rate. If the initial discrepancy satisfies

$\rho\ge 0$2

then for any fixed $\rho\ge 0$3 there exists $\rho\ge 0$4 such that, for all $\rho\ge 0$5,

$\rho\ge 0$6

Thus the pressureless system is not only a formal limit but a quantitatively justified one in the three-dimensional critical Besov framework (Ni et al., 8 Jun 2026).

Several extensions and open questions are explicitly identified in the pressureless theory. The method might be adaptable to more general viscous pressureless models, including multi-fluid systems, to other critical spaces such as $\rho\ge 0$7-based Besov spaces, or to fractional dissipation. Important open problems include the truly vacuum case $\rho\ge 0$8, extension to the full compressible Navier–Stokes equations with small but rough pressure fluctuations in the high-Mach regime, and the incorporation of nonlocal alignment operators from collective-behavior models such as Euler–alignment and Cucker–Smale in order to obtain global regularity with discontinuous density profiles (Xu, 13 Aug 2025).

The resulting picture is technically specific. On the one hand, the three-dimensional pressureless system now has a global Fujita–Kato theory with discontinuous and large-variation density under positive lower and upper bounds. On the other hand, the model is rigorously connected to the compressible Navier–Stokes equations through a critical-space high-Mach-number limit with uniform estimates and finite-time $\rho\ge 0$9 error control. Together, these results define the current analytical baseline for the 3D compressible pressureless Navier–Stokes equations (Xu, 13 Aug 2025, Ni et al., 8 Jun 2026).