Math 2.0

Partial differential equations Definition

Euler equations

Builds on Partial differential equationVector fieldGradientDivergence

Definition

Let Ω⊆ℝ3\Omega \subseteq \R^3 be an open set and let T>0T > 0. The incompressible Euler equations are the system ∂tu+(u⋅∇)u=−∇p,∇⋅u=0,\partial_t u + (u \cdot \nabla) u = - \nabla p, \qquad \nabla \cdot u = 0, for an unknown vector field u ⁣:Ω×(0,T)→ℝ3u \colon \Omega \times (0, T) \to \R^3, the velocity, and an unknown function p ⁣:Ω×(0,T)→ℝp \colon \Omega \times (0, T) \to \R, the pressure. Here ∇p\nabla p is the gradient of pp in xx, ∇⋅u\nabla \cdot u is the divergence of uu, and (u⋅∇)u(u \cdot \nabla) u has components ∑j=13uj∂jui\sum_{j=1}^3 u_j \partial_j u_i. The density is normalised to 11. On a bounded domain one adds the boundary condition u⋅n=0u \cdot n = 0, where nn is the outer unit normal.

Remarks

Example: on Ω=ℝ3\Omega = \R^3, the rigid rotation u=(−y,x,0)u = (-y, x, 0) with pressure p=(x2+y2)/2p = (x^2 + y^2)/2 is a solution. Indeed ∇⋅u=0\nabla \cdot u = 0, (u⋅∇)u=(−x,−y,0)(u \cdot \nabla) u = (-x, -y, 0) and −∇p=(−x,−y,0)-\nabla p = (-x, -y, 0).

Non-example: u=(x,0,0)u = (x, 0, 0) is not a solution for any pressure, because ∇⋅u=1≠0\nabla \cdot u = 1 \ne 0.

The compressible Euler equations have the density ρ\rho and the pressure as further unknowns: ∂tρ+∇⋅(ρu)=0,∂t(ρu)+∇⋅(ρu⊗u)+∇p=0,\partial_t \rho + \nabla \cdot (\rho u) = 0, \qquad \partial_t (\rho u) + \nabla \cdot (\rho u \otimes u) + \nabla p = 0, together with an energy equation or an equation of state that relates pp to ρ\rho. They form a system of hyperbolic conservation laws, whose solutions can develop shock waves.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptPartial differential equation, concept, no evidence levelPartial differential …conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptEuler equations, definition, no evidence levelEuler equationsdefinition
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  1. Euclidean space object
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  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Euler equations definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)
About this page
Tag9ZBWK5WSC7
URIm2://9ZBWK5WSC7
Kinddefinition
WritingW0 stub
MSC35Q31, 76B03
DefinesEuler equations
UsesPartial differential equation, Vector field, Gradient, Divergence
Changed2 November 2026

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