ExampleConstruction: AI-generatedVerification: AI-generatedPipeline-generatedprecheck passverified 2026-10-23 (gpt-6.1-sol)

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One shear mode reduced evolution

Example

Assume Countable Choice, meaning every sequence of nonempty sets admits a choice function. On $Q_{2} = R^{2} / (2 π Z)^{2}$ with normalized measure $d μ = (2 π)^{- 2} d x_{1} d x_{2}$ , take the single vector field $v_{1} (x_{1}, x_{2}) = \sqrt{2} \sin (x_{2}) e_{1}$ , where $e_{1} = (1, 0)$ . This finite real trigonometric polynomial belongs to $H^{0} (Q_{2}; R^{2})$ and has its displayed smooth periodic representative. For every $ν \geq 0$ and $a_{0} \in R$ , the one-mode reduced velocity $u (t, x) = a (t) v_{1} (x)$ has coefficient equation $a^{'} (t) + ν a (t) = 0$ and unique forward solution $a (t) = a_{0} e^{- ν t}$ for $t \geq 0$ . Its reduced energy $E (t) = \frac{1}{2} a (t)^{2}$ satisfies $E^{'} (t) = - ν \int_{Q_{2}} ∣ \nabla u (t, x) ∣^{2} d μ$ . This is an exact finite-mode calculation and asserts no convergence or approximation error for the full partial differential equation.

Example references

Skew reduced convection and global finite dynamics

Facts & Assumptions

Given: $ν \geq 0$ , $a_{0} \in R$ , $Q_{2} = (R / 2 π Z)^{2}$ with normalized measure, and the supplied $H^{0}$ class of the explicit smooth mode $v_{1} = \sqrt{2} \sin (x_{2}) e_{1}$ .

[F1]

Skew reduced convection and global finite dynamics: The specified finite coefficient system has exactly one forward-global solution for each initial coefficient vector.

[F2]

Skew reduced convection and global finite dynamics: The reduced energy obeys $∣ a (t) ∣^{2} + 2 ν \int_{0}^{t} \int_{Q_{2}} ∣ \nabla u (s, x) ∣^{2} d μ d s = ∣ a_{0} ∣^{2}$ .

Verification

1.1givenalgebra

The field is smooth and periodic, and $div v_{1} = \partial_{x_{1}} (\sqrt{2} \sin x_{2}) = 0$ . Its squared norm is $\int_{Q_{2}} 2 \sin^{2} (x_{2}) d μ = 1$ , while $A_{11} = \int_{Q_{2}} 2 \cos^{2} (x_{2}) d μ = 1$ . Also $c_{111} = \int_{Q_{2}} ((v_{1} \cdot \nabla) v_{1}) \cdot v_{1} d μ = 0$ , since $v_{1} \cdot \nabla = \sqrt{2} \sin (x_{2}) \partial_{x_{1}}$ and $v_{1}$ is independent of $x_{1}$ .

2.1givenF1F2step 1.1algebra∎

Thus the coefficient equation is $a^{'} + ν a = 0$ . The function $a (t) = a_{0} e^{- ν t}$ has $a (0) = a_{0}$ and satisfies it by differentiation; [F1] makes it the unique forward-global solution. Since $\int_{Q_{2}} ∣ \nabla u ∣^{2} d μ = a (t)^{2} A_{11} = a (t)^{2}$ , direct differentiation gives $E^{'} (t) = - ν a (t)^{2} = - ν \int_{Q_{2}} ∣ \nabla u ∣^{2} d μ$ , and [F2] gives the corresponding integrated identity.

Depends on

Skew reduced convection and global finite dynamics