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Exact transverse plane waves, the sheared single mode, and the Taylor-Green vortex in the periodic models

Example

Work on $T^{3}$ with $d x$ the normalized measure and the operators of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection: $\partial_{j} u$ has symbol $2 π i k_{j}$ , $Δ$ has symbol $- 4 π^{2} ∣ k ∣_{2}^{2}$ , and $P$ kills gradient fields.

(a) Transverse plane waves, inviscid and viscous. Let $k \in Z^{3} ∖ {0}$ , $A \in R^{3}$ with $A \cdot k = 0$ , and put $u_{0} (x) : = A \sin (2 π k \cdot x)$ . Then $div u_{0} = 0$ , $(u_{0} \cdot \nabla) u_{0} = 0$ , $Δ u_{0} = - 4 π^{2} ∣ k ∣_{2}^{2} u_{0}$ , and for every $ν \geq 0$ the field $u (t, x) : = e^{- 4 π^{2} ν ∣ k ∣_{2}^{2} t} A \sin (2 π k \cdot x)$ is a smooth global solution of $\partial_{t} u + P ((u \cdot \nabla) u) = ν Δ u$ with $P ((u \cdot \nabla) u) = 0$ , and a solution of the primitive system with $p = 0$ . For $ν = 0$ the field $u_{0}$ is stationary; the Galerkin coefficient of the mode $k$ satisfies the scalar equation $c^{'} (t) = - 4 π^{2} ν ∣ k ∣_{2}^{2} c (t)$ .

(b) The sheared single mode. For $k = e_{2}$ , $A = e_{1}$ part (a) gives $u (t, x) = e^{- 4 π^{2} ν t} (\sin (2 π x_{2}), 0, 0)$ : the viscous decay of a shear flow, whose $P$ -image is itself because the field is divergence-free and $P$ is the orthogonal projection of $H^{0}$ onto $H_{σ}^{0}$ (Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection).

(c) The Taylor-Green vortex. Write $θ_{j} : = 2 π x_{j}$ and $u^{0} (x) : = (\sin θ_{1} \cos θ_{2}, - \cos θ_{1} \sin θ_{2}, 0), p^{0} (x) : = \frac{1}{4} (\cos (2 θ_{1}) + \cos (2 θ_{2})) .$ Then $div u^{0} = 0$ , $(u^{0} \cdot \nabla) u^{0} = - \nabla p^{0}$ , and $Δ u^{0} = - 8 π^{2} u^{0}$ ; hence for every $ν \geq 0$ the pair $u (t, x) : = e^{- 8 π^{2} ν t} u^{0} (x), p (t, x) : = e^{- 16 π^{2} ν t} p^{0} (x)$ solves the primitive system $\partial_{t} u + (u \cdot \nabla) u = - \nabla p + ν Δ u$ on $[0, \infty) \times T^{3}$ , and $u$ solves the projected equation with $P ((u \cdot \nabla) u) = 0$ because $(u^{0} \cdot \nabla) u^{0} = - \nabla p^{0}$ is a gradient field. In particular these solutions are global in time, so the finite time $T$ of Local smooth solutions of the periodic incompressible Euler and constant-viscosity equations on $T^{3}$ is not asserted to be sharp.

Verification

Given: The conventions of Periodic vector Sobolev spaces on the three-torus, the weight family, and the Leray projection: the Fourier identities $\partial_{j} e_{k} = 2 π i k_{j} e_{k}$ and $Δ e_{k} = - 4 π^{2} ∣ k ∣_{2}^{2} e_{k}$ , the coefficient formula $\hat{P u} (k) = π_{k} \hat{u} (k)$ with $π_{k} v = v - \frac{k \cdot v}{∣ k ∣_{2}^{2}} k$ for $k \neq 0$ , the identity $P \nabla q = 0$ , and the fact that $P$ is the orthogonal projection onto $H_{σ}^{s}$ , so $P u = u$ for $u \in H_{σ}^{s}$ ; in particular $P (A \sin (2 π k \cdot x)) = A \sin (2 π k \cdot x)$ whenever $A \cdot k = 0$ .

1.1givenalgebra

Part (a), nonlinear term: for $u_{0} = A \sin (2 π k \cdot x)$ one has $\partial_{j} u_{0} = 2 π k_{j} A \cos (2 π k \cdot x)$ , so $(u_{0} \cdot \nabla) u_{0} = 2 π (A \cdot k) \cos (2 π k \cdot x) A \sin (2 π k \cdot x) = 0$ because $A \cdot k = 0$ ; and $div u_{0} = 2 π (A \cdot k) \cos (2 π k \cdot x) = 0$ .

1.2givenalgebra

Part (a), norms: $\hat{u} (t, \pm k) = \pm \frac{1}{2 i} c (t) A$ are the only nonzero coefficients, so $∥ u (t) ∥_{H^{m}} = ∣ c (t) ∣ ∥ u_{0} ∥_{H^{m}} = e^{- 4 π^{2} ν ∣ k ∣_{2}^{2} t} ∥ u_{0} ∥_{H^{m}}$ for every $m$ ; in particular the solution exists for all $t \geq 0$ .

1.3givenalgebra

Part (c), divergence and Laplacian: $\partial_{1} u_{1}^{0} = 2 π \cos θ_{1} \cos θ_{2}$ , $\partial_{2} u_{2}^{0} = - 2 π \cos θ_{1} \cos θ_{2}$ , so $div u^{0} = 0$ ; and $\partial_{1}^{2} u_{1}^{0} = - (2 π)^{2} u_{1}^{0}$ , $\partial_{2}^{2} u_{1}^{0} = - (2 π)^{2} u_{1}^{0}$ , likewise for the second component, so $Δ u^{0} = - 8 π^{2} u^{0}$ .

1.4givenalgebra

Part (c), nonlinear term and pressure: $(u^{0} \cdot \nabla) u_{1}^{0} = u_{1}^{0} \partial_{1} u_{1}^{0} + u_{2}^{0} \partial_{2} u_{1}^{0} = 2 π \sin θ_{1} \cos θ_{2} \cos θ_{1} \cos θ_{2} + 2 π \cos θ_{1} \sin θ_{2} \sin θ_{1} \sin θ_{2} = 2 π \sin θ_{1} \cos θ_{1} = π \sin 2 θ_{1}$ , while $- \partial_{1} p^{0} = - \frac{1}{4} (- 2 \cdot 2 π \sin 2 θ_{1}) = π \sin 2 θ_{1}$ (because $\partial_{1} \cos 2 θ_{1} = - 2 \cdot 2 π \sin 2 θ_{1}$ ); similarly $(u^{0} \cdot \nabla) u_{2}^{0} = π \sin 2 θ_{2} = - \partial_{2} p^{0}$ . Hence $(u^{0} \cdot \nabla) u^{0} = - \nabla p^{0}$ .

2.1step 1.1given

Part (a), equation: with $c (t) : = e^{- 4 π^{2} ν ∣ k ∣_{2}^{2} t}$ one has $\partial_{t} u = \partial_{t} c A \sin (2 π k \cdot x)$ and $ν Δ u = c ν (- 4 π^{2} ∣ k ∣_{2}^{2}) A \sin (2 π k \cdot x) = \partial_{t} u$ ; since $(u (t) \cdot \nabla) u (t) = c (t)^{2} (u_{0} \cdot \nabla) u_{0} = 0$ by step 1.1 and $P$ is linear with $P 0 = 0$ , the projected equation holds; the primitive equation holds with $p = 0$ by Reconstruction of the periodic pressure from the velocity: $\nabla p = - (I - P) ((u \cdot \nabla) u)$ and $Δ p = - \sum_{i, j} \partial_{i} \partial_{j} (u_{i} u_{j})$ (whose defining formula gives $p = 0$ when $(u \cdot \nabla) u = 0$ ). The Galerkin reduction is the one-dimensional ODE $c^{'} = - 4 π^{2} ν ∣ k ∣_{2}^{2} c$ , since the mode $k$ is not mixed by the convolution.

3.1step 1.3step 1.4given∎

Part (c), the equations: with $u (t) = e^{- 8 π^{2} ν t} u^{0}$ and $p (t) = e^{- 16 π^{2} ν t} p^{0}$ , one computes $\partial_{t} u = - 8 π^{2} ν u$ , $ν Δ u = ν e^{- 8 π^{2} ν t} (- 8 π^{2} u^{0}) = - 8 π^{2} ν u$ (so $\partial_{t} u - ν Δ u = 0$ ), and $(u \cdot \nabla) u = e^{- 16 π^{2} ν t} (u^{0} \cdot \nabla) u^{0} = - e^{- 16 π^{2} ν t} \nabla p^{0} = - \nabla p$ ; therefore $\partial_{t} u + (u \cdot \nabla) u = - \nabla p + ν Δ u$ . Since $(u \cdot \nabla) u$ is a gradient, its $P$ -image vanishes, and the projected equation $\partial_{t} u = ν Δ u - P ((u \cdot \nabla) u) = ν Δ u$ holds; this also shows the solution is global.

Remarks

These exact solutions show that the model equations admit global smooth solutions with special symmetry, and they verify the sign conventions and the projection/pressure reconstruction on explicit data. They are conditional mathematical solutions of the stated model equations; no claim is made that any of them describes a physical flow, and their global existence does not address the general global regularity problem recorded in Scope of the local periodic result: what is proved and what is not.

Exact transverse plane waves, the sheared single mode, and the Taylor-Green vortex in the periodic models

Example

Verification

Remarks

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