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Shear mode has zero periodic pressure

Example

Assume Countable Choice and let $d \in {2, 3}$ , $A \in R$ , and $u (x) = A \sin (x_{2}) e_{1}$ on $Q_{d}$ , where $e_{1}$ is the first coordinate vector. Then $u$ is smooth and divergence-free, $u \cdot \nabla u = 0$ , and its unique mean-zero pressure recovered by $Q (u \otimes u)$ is $π = 0$ . This is a verification of the pressure recovery map; it is not asserted to solve an unforced viscous Navier–Stokes equation.

Example references

Terence Tao, 254A Notes 1: Local well-posedness of the Navier–Stokes equations (2018).

Erick Schulz, Introduction to the Navier–Stokes Equations, McGill MATH 595 notes (2014).

Facts & Assumptions

Given: The hypotheses and conventions stated in the formulation.

[F1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): Countable Choice is assumed for the periodic Fourier coefficient and pressure recovery framework.

[F2]

Periodic fluid Fourier Sobolev spaces: the torus coordinates have period $2 π$ and the real Fourier convention is fixed by conjugate-symmetric coefficients.

[F3]

Pressure recovery for periodic incompressible Sobolev velocity: for divergence-free $u \in H^{s}$ , $s > d / 2 + 1$ , $Q (u \otimes u)$ is the unique mean-zero pressure satisfying the projected convection identity.

Verification

Given: Countable Choice, $d \in {2, 3}$ , and $u = A \sin (x_{2}) e_{1}$ .

1.1givenF2algebra

The only nonzero component is $u_{1} = A \sin (x_{2})$ and it is independent of $x_{1}$ , so $div u = \partial_{1} u_{1} = 0$ . Also $u \cdot \nabla u = u_{1} \partial_{1} u = 0$ . The field is a finite Fourier sum and hence belongs to every $H^{s} (Q_{d})$ .

2.1givenF1F3step 1.1algebra∎

Choose any $s > d / 2 + 1$ . By [F3], $π = Q (u \otimes u)$ is the unique mean-zero periodic distribution satisfying the projected convection identity. By step 1.1 the convection is zero, so the zero distribution also satisfies that identity, since $P (0) = 0$ ; it is mean-zero. Uniqueness in [F3] therefore gives $π = 0$ .

Shear mode has zero periodic pressure

Example

Example references

Facts & Assumptions

Verification

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Sources