DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pendingverified 2026-10-23 (gpt-6.1-sol)

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Incompressible inviscid Euler model with pressure multiplier

Physical scope

Smooth ideal inviscid incompressible continuum with stated boundary/initial data; no empirical derivation of zero viscosity.

Definition

Let $I \subseteq R$ be a nonempty open time interval measured in seconds and $Ω \subseteq R^{3}$ an open spatial region measured in metres in a Newtonian Cartesian inertial chart. Let $ρ_{0} > 0$ be constant in $k g m^{- 3}$ , $u \in C^{2} (I \times Ω; R^{3})$ velocity in $m s^{- 1}$ , and $p \in C^{1} (I \times Ω)$ pressure in pascals. Define $(div u) = \sum_{i = 1}^{3} \partial_{x_{i}} u_{i}$ and $D_{t} u = \partial_{t} u + (u \cdot \nabla) u$ . The selected unforced incompressible Euler branch is $ρ = ρ_{0}, div u = 0, ρ_{0} D_{t} u + \nabla p = 0.$ Here $p$ is a constraint multiplier. No viscous stress or equation of state is included. Initial, wall, periodic and other boundary data are separate inputs. This is an adopted model formulation, not a derivation of empirical validity or an assertion of solution existence.

Definition references

Stewart, MIT 8.09, Fluid Mechanics, §6.4, printed pp. 120–123

Fitzpatrick, Fluid Mechanics, UT Austin, nodes 62–64 and 102

Facts & Assumptions

Given: The open domains, regularity, positive constant density, inertial chart and adopted continuum framework stated above.

[A1]

Single-component nonpolar continuum and material balances: the smooth single-component continuum adopts integral mass, momentum, angular-momentum, energy and entropy balances; these are physical postulates, not proof of empirical validity.

[A2]

Specified wall, thermal, periodic and initial data: initial and boundary data are separately selected and imply no existence or uniqueness result.

[F1]

Formal constant-density incompressible Euler system: the mathematical incompressible Euler system is the constant-density divergence-free local equation with pressure multiplier.

[F2]

Classical frames, primitive quantities and SI units: Newtonian spacetime is $I \times E$ , with absolute time in seconds and $E$ an oriented Euclidean affine three-space. Each fixed-time spatial slice of a Cartesian inertial chart is an orientation-preserving affine isometry $x_{t} : E \to R^{3}$ with spatial coordinates in metres; together with the absolute-time coordinate these slices form the event-coordinate map; inertial charts are related by $t^{*} = t + t_{0}$ and $x^{*} = Q x + v t + a$ , where $Q \in S O (3)$ , $v$ is constant in metres per second, and $a$ is constant in metres. A frame here means this adapted chart, not an observer congruence or a basis field.

Proof

1.1givenA1F1F2algebra

Interpret the branch in the adapted Cartesian inertial chart of [F2]. The definition selects the local Euler branch from the adopted continuum framework and fixes $ρ = ρ_{0}$ , $div u = 0$ , and $ρ_{0} D_{t} u + \nabla p = 0$ ; this is a formulation choice, not a constitutive derivation.

2.1givenA2step 1.1algebra

No initial or boundary values occur in these equations, so those data must be selected separately under [A2]; neither formulation nor data imply a solution exists.

3.1givenstep 1.1algebra∎

Both $ρ_{0} D_{t} u$ and $\nabla p$ have SI units $k g m^{- 2} s^{- 2}$ . This dimensional check does not validate the model empirically.

Incompressible inviscid Euler model with pressure multiplier

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources