Math 2.0

Fluid mechanics Definition

Hydrodynamic stability

Builds on Navier–Stokes equationsLyapunov stabilityNo-slip condition

Definition

Let Ω⊆ℝ3\Omega \subseteq \R^3 be a bounded open set with smooth boundary, and let ν>0\nu > 0. Let U ⁣:Ω→ℝ3U \colon \Omega \to \R^3 be a steady solution, the basic flow, of the Navier–Stokes equations with given steady boundary values and forcing. Let u0u_0 be a smooth divergence-free initial velocity with the same boundary values as UU. Let u(t)u(t) be the smooth solution with these boundary values, the same forcing and u(0)=u0u(0) = u_0, as long as it exists. The basic flow UU is stable (in the energy norm) if for every ε>0\varepsilon > 0 there is δ>0\delta > 0 such that

  1. every such u0u_0 with ∥u0−U∥L2<δ\norm{u_0 - U}_{L^2} < \delta gives a solution u(t)u(t) that exists for all t≥0t \ge 0, and
  2. ∥u(t)−U∥L2<ε\norm{u(t) - U}_{L^2} < \varepsilon for all t≥0t \ge 0.

It is asymptotically stable if, in addition, ∥u(t)−U∥L2→0\norm{u(t) - U}_{L^2} \to 0 as t→∞t \to \infty for all such u0u_0, and unstable if it is not stable.

Remarks

This is Lyapunov stability of the equilibrium UU of the Navier–Stokes evolution.

Example: the rest state U=0U = 0 with zero forcing and the no-slip condition is asymptotically stable. For a smooth solution, ddt∥u∥L22=−2ν∥∇u∥L22≤−2νλ1∥u∥L22\frac{d}{dt} \norm{u}_{L^2}^2 = -2 \nu \norm{\nabla u}_{L^2}^2 \le -2 \nu \lambda_1 \norm{u}_{L^2}^2 by the Poincaré inequality with its constant λ1>0\lambda_1 > 0, so ∥u(t)∥L2≤e−νλ1t∥u0∥L2\norm{u(t)}_{L^2} \le e^{-\nu \lambda_1 t} \norm{u_0}_{L^2}.

Non-example: take circular Couette flow with the outer cylinder at rest and the inner one rotating fast enough. Here the cylinders are infinitely long and disturbances are taken periodic in the axial direction, so the domain is a periodic gap rather than a bounded set. It is unstable: a small disturbance grows into Taylor vortices, so clause 2 fails for small ε\varepsilon.

Linear stability replaces the equations by their linearisation at UU and asks that no eigenvalue of the linearised operator has positive real part. The energy method gives sufficient conditions for nonlinear stability, and linear analysis gives sufficient conditions for instability. Both depend on the norm and on the class of disturbances, which must be stated.

Reading path

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  1. Lyapunov stability concept
  2. Navier–Stokes equations concept
  3. No-slip condition concept
  4. Hydrodynamic stability definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Hydrodynamic_stability
About this page
TagPJ4V6A4XX8
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Kinddefinition
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MSC76E99, 76E05
DefinesHydrodynamic stability
UsesNavier–Stokes equations, Lyapunov stability, No-slip condition
Changed3 November 2026

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