Local Pressure Expansion in Diverse Fields

• Local Pressure Expansion is a set of methodologies that locally decompose pressure fields to link observable measurements with global phenomena across various scientific domains.
• These techniques include cover refinements, singular-integral decompositions, and asymptotic expansions, providing a unified but framework-dependent approach across disciplines.
• They enable precise diagnostics and modeling in fields such as fluid dynamics, molecular simulation, geomechanics, and nonlinear elasticity by isolating local pressure contributions.

Searching arXiv for recent and relevant papers on “local pressure expansion” and closely related formulations across the domains represented here. Local pressure expansion is a domain-dependent concept rather than a single unified construction. Across the cited literature, it denotes localizations or asymptotic expansions of pressure in settings as different as topological dynamical systems, inhomogeneous fluids, molecular simulation, Navier–Stokes theory, heliospheric plasmas, metrology, geomechanics, and nonlinear elasticity. In some contexts the term refers to a genuinely local pressure functional built from covers, control volumes, or wall deformations; in others it refers to a pressure-driven expansion process diagnosed from local observables, or to a local pressure–amplitude expansion near a critical state (Zhu et al., 21 Jun 2025, Dufty et al., 2020, Bradshaw et al., 2020, Lugaz et al., 2020, Bock et al., 2020).

1. Terminological scope

The following usages are explicitly represented in the literature.

Domain	Local quantity	Meaning of “expansion”
Topological dynamics	$P(T,\mathcal{E};\mathcal{U})$ on open covers	refined-cover asymptotics recovering global nonlinear pressure
Inhomogeneous fluids and MD	$p_0^\ell(r)$, $p_{\mathrm{mech}}^\ell(r)$, $P(\mathbf r)$	gradient, virial, contact, or plane-based decomposition
Navier–Stokes theory	local pressure expansion / DLPE	localized singular-integral representation plus spatially constant term
Space and plasma physics	$\zeta$, $V_{\exp}$, $P_B$, $\nabla P_B$	pressure-controlled local expansion of magnetic structures or flux tubes
Engineering and solids	$\Delta p$, $p_i(\zeta)$, $p_0^\ell(r)$0	local measurement correction or pressure–deformation asymptotics

This suggests that the phrase is best understood as a family of local constructions unified by a common methodological move: pressure is first resolved locally, then related either to a limiting process, a cover refinement, a singular-integral decomposition, or an observed expansion law. The technical meaning is therefore set entirely by the governing framework and observables.

2. Nonlinear local topological pressure

In topological dynamics, local pressure expansion is formalized for a topological dynamical system $p_0^\ell(r)$1, an energy $p_0^\ell(r)$2, and an open cover $p_0^\ell(r)$3. The basic object is

$p_0^\ell(r)$4

with empirical measure $p_0^\ell(r)$5. The nonlinear local pressure and lower local pressure are then

$p_0^\ell(r)$6

A central technical point is that $p_0^\ell(r)$7 is generally not sub-additive, so $p_0^\ell(r)$8 may fail to converge (Zhu et al., 21 Jun 2025).

The decisive regularity assumption is “abundance of ergodic measures”: for every $p_0^\ell(r)$9 and $p_{\mathrm{mech}}^\ell(r)$0, there is $p_{\mathrm{mech}}^\ell(r)$1 such that

$p_{\mathrm{mech}}^\ell(r)$2

Under that hypothesis, the local variational principle states

$p_{\mathrm{mech}}^\ell(r)$3

The same work establishes equivalent formulations through separated sets, spanning sets, cover refinements with sup or inf weights, and subcover-based variants $p_{\mathrm{mech}}^\ell(r)$4, and shows that refined local formulations recover the global nonlinear pressure: $p_{\mathrm{mech}}^\ell(r)$5 The equivalence is not completely uniform across all local constructions: a specific example with $p_{\mathrm{mech}}^\ell(r)$6 gives $p_{\mathrm{mech}}^\ell(r)$7 but $p_{\mathrm{mech}}^\ell(r)$8. This sharply distinguishes the variationally well-behaved local pressure $p_{\mathrm{mech}}^\ell(r)$9 from certain subcover-based surrogates (Zhu et al., 21 Jun 2025).

3. Inhomogeneous fluids, confined systems, and molecular simulation

For inhomogeneous fluids, the literature distinguishes thermodynamic and mechanical local pressures. In equilibrium, a local pressure density $P(\mathbf r)$0 is defined by

$P(\mathbf r)$1

while in local equilibrium the thermodynamic pressure is

$P(\mathbf r)$2

The mechanical pressure is extracted from the momentum-flux tensor, with scalar part $P(\mathbf r)$3. These are not generally equal pointwise: $P(\mathbf r)$4 with $P(\mathbf r)$5 but $P(\mathbf r)$6. The paper resolves this by exploiting the gauge freedom of the stress tensor, adding a double-curl term and choosing $P(\mathbf r)$7 from

$P(\mathbf r)$8

so that the modified mechanical pressure equals the thermodynamic one. In weakly inhomogeneous states, coarse-graining yields the square-gradient expansion

$P(\mathbf r)$9

This formulation reconciles local equilibrium thermodynamics with hydrodynamic force balance, while also making clear that local pressure is not unique until a gauge is fixed (Dufty et al., 2020).

In atomistic molecular dynamics, local pressure in a finite region $\zeta$0 is derived from the Schweitz virial relation for open systems. The volume form is

$\zeta$1

where $\zeta$2 indicates whether particle $\zeta$3 lies in $\zeta$4 and $\zeta$5 is the fraction of the $\zeta$6–$\zeta$7 segment داخل $\zeta$8. An exactly equivalent boundary form resolves momentum flux and force transmission across $\zeta$9. The two expressions remain accurate even when the measurement region is very small; in the reported validation they work when $V_{\exp}$0 contains only $V_{\exp}$1 particles on average, provided the correction term $V_{\exp}$2 is retained. Omitting that term produces large systematic errors that increase as the region shrinks (Lion et al., 2011).

For confined fluids at corrugated hard walls, density-functional theory yields an exact local pressure as a functional derivative of the effective wall Hamiltonian: $V_{\exp}$3 For noninteracting hard particles this reduces to

$V_{\exp}$4

With finite-range interactions, the small-curvature expansion acquires a $V_{\exp}$5 correction, explicitly showing that morphometric thermodynamics and the surface-of-tension construction fail once wall curvature is not small compared with the interaction range (1711.02028).

Away from equilibrium, the method of planes provides a strictly mechanical local pressure as traction across a plane. For MACE machine-learning potentials, the configurational force decomposition

$V_{\exp}$6

permits a plane-resolved pressure

$V_{\exp}$7

supplemented by the kinetic plane flux. In the water–$V_{\exp}$8 test case, method-of-planes pressure satisfies exact control-volume balance at every timestep, whereas the local virial form develops near-wall peaks and fails the local force balance (Smith, 17 Sep 2025).

4. Navier–Stokes local pressure expansion

In the whole-space incompressible Navier–Stokes equations,

$V_{\exp}$9

the pressure formally satisfies

$P_B$0

For nondecaying solutions, the pressure is localized on each ball $P_B$1 by splitting the singular integral into near-field and far-field parts. The functional local pressure expansion takes the form

$P_B$2

while the distributional local pressure expansion replaces pointwise equality by a test-function identity with a localized double Riesz transform and a far-field correction involving $P_B$3 (Bradshaw et al., 2020).

The fundamental result is equivalence: for solutions in the uniformly local class, the pressure satisfies the distributional local pressure expansion if and only if the velocity field is mild. This recasts Lemarié–Rieusset’s equivalence theorem without Littlewood–Paley machinery and defines $P_B$4 through Poisson/BMO theory rather than dyadic decomposition (Bradshaw et al., 2020).

A later structural theorem shows that this equivalence is not automatic for all weak solutions with uniformly locally bounded energy. In the parabolic uniformly local $P_B$5 class, every weak solution is a transgalilean transformation of a solution whose pressure does satisfy the local pressure expansion in the distributional sense. The obstruction is an affine harmonic remainder,

$P_B$6

and the transformation

$P_B$7

removes that obstruction. A sufficient condition for the distributional local pressure expansion is origin-centered decay: $P_B$8 This places local pressure expansion at the center of the distinction between “genuine” and “superfluous” whole-space weak solutions (Bradshaw et al., 1 Aug 2025).

5. Pressure-driven local expansion in space and astrophysical plasmas

In magnetic clouds and coronal mass ejections, local expansion is commonly diagnosed from the in situ velocity gradient. For Helios magnetic-cloud data, the nondimensional local expansion rate is

$P_B$9

Non-perturbed magnetic clouds have $\nabla P_B$0, whereas perturbed clouds have $\nabla P_B$1. The total solar-wind pressure used in that study is

$\nabla P_B$2

and the force-free estimate $\nabla P_B$3 gives $\nabla P_B$4. The interpretation is that the smooth radial decrease of solar-wind pressure sets the near-universal expansion of non-perturbed events, while fast streams and other local perturbations alter the local pressure balance and therefore the observed $\nabla P_B$5 (Gulisano et al., 2012).

Two-spacecraft conjunction measurements show that this local picture near $\nabla P_B$6 does not directly encode global expansion history. For 42 CMEs, the global magnetic-field exponents were

$\nabla P_B$7

while local measures near $\nabla P_B$8 gave

$\nabla P_B$9

Although self-similar expansion would suggest $\Delta p$0, the observed $\Delta p$1–$\Delta p$2 correlations were absent or very weak. Global expansion correlates strongly with the inner-heliosphere magnetic field, but not with the field measured near $\Delta p$3. The resulting interpretation is two-stage: early expansion is driven by internal magnetic overpressure, while by $\Delta p$4 expansion is largely governed by the radial decline of solar-wind dynamic pressure (Lugaz et al., 2020).

A distinct plasma example appears in the dayside magnetosheath during an ICME magnetic-cloud passage. There, a localized magnetic enhancement created a magnetic-pressure maximum and a subsequent pressure-gradient force

$\Delta p$5

that drove sunward flow. Reported peak sunward speeds were $\Delta p$6, $\Delta p$7, and $\Delta p$8 at TH-E, TH-A, and TH-D, with fitted accelerations $\Delta p$9, $p_i(\zeta)$0, and $p_i(\zeta)$1. The average acceleration estimated from $p_i(\zeta)$2 was $p_i(\zeta)$3, and both the dense core and high-energy tail of the ion distribution moved sunward. In this usage, local pressure expansion means expansion of magnetosheath field lines under magnetic-pressure-gradient forcing in very low-$p_i(\zeta)$4 plasma (Madanian et al., 11 Apr 2025).

A more formal local model of spherical collapse and expansion gives an anisotropic FRW-like metric

$p_i(\zeta)$5

with pressure gradients rescaled by $p_i(\zeta)$6 horizontally and $p_i(\zeta)$7 vertically. The energy equation contains the geometric work term $p_i(\zeta)$8, and the mean pressure scales as

$p_i(\zeta)$9

This suggests a local-pressure-expansion viewpoint in which geometry itself drives adiabatic amplification or attenuation of pressure through $p_0^\ell(r)$00 work (Lynch et al., 2023).

6. Metrological, geomechanical, and elastic-pressure expansions

In static expansion metrology, local pressure expansion refers to a calibration bias created by valve closure. The locally measured filling pressure $p_0^\ell(r)$01 differs from the true pressure in the isolated starting volume by $p_0^\ell(r)$02, so that

$p_0^\ell(r)$03

The effect is strongest for small starting volumes and depends on valve geometry, conductance, closing speed, gas species, and pressure. For the SE3-sized case, the empirical correction was

$p_0^\ell(r)$04

Here the “local” aspect is literal: the pressure entering the expansion law is measured at a gauge position rather than inside the isolated starting volume itself (Bock et al., 2020).

In drained cylindrical cavity expansion for non-associated Mohr–Coulomb geomaterials, the wall pressure $p_0^\ell(r)$05 satisfies a first-order ODE in the Lagrangian auxiliary variable $p_0^\ell(r)$06, with different coefficients depending on the active yield face or corner. The full theory covers arbitrary $p_0^\ell(r)$07 and allows the principal-stress trajectory to move across distinct sextants of the Mohr–Coulomb hexagon. The limit cavity pressure is defined asymptotically by

$p_0^\ell(r)$08

with $p_0^\ell(r)$09. In this setting, local pressure expansion means reduction of the cavity boundary-value problem to a local wall-pressure evolution law parameterized by the cavity expansion variable (Chen, 2022).

For an inflated hyperelastic tube with fixed ends, the relevant local quantity is the bulge-center amplitude $p_0^\ell(r)$10. Near the pressure minimum, the weakly nonlinear amplitude equation reduces to a kink form and yields the local pressure expansion

$p_0^\ell(r)$11

with

$p_0^\ell(r)$12

For the Gent model quoted in the study, $p_0^\ell(r)$13 in scaled pressure units. Under fixed ends, the pressure–bulge-amplitude curve has both a maximum and a minimum, and all right ascending branches converge to a single curve depending only on the thickness-to-outer-radius ratio $p_0^\ell(r)$14. The corresponding Maxwell-like state is pressure-dependent, unlike the constant Maxwell pressure for free ends. This provides a precise example in which “local pressure expansion” means a near-critical pressure–amplitude asymptotic law rather than a field-theoretic localization of pressure (Guo et al., 2022).