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Continuous periodic solenoidal surrogate residual

Definition

Assume Countable Choice. Let $d \in {2, 3}$ , $T > 0$ , and $Q_{d} = R^{d} / (2 π Z)^{d}$ with normalized volume $d μ = (2 π)^{- d} d x$ . For a vector field $w$ , use the componentwise Fourier coefficients $\hat{w} (k) = \int_{Q_{d}} w (x) e^{- i k \cdot x} d μ (x)$ and norm $∥ w ∥_{2} = (\int_{Q_{d}} ∣ w (x) ∣^{2} d μ (x))^{1 / 2}$ . Define $P_{0} = I$ and, for $k \neq 0$ , $P_{k} = I - k k^{T} / ∣ k ∣^{2}$ , with $\hat{P w} (k) = P_{k} \hat{w} (k)$ ; $P$ is the Leray projector. For a smooth spatial vector field $w : Q_{d} \to R^{d}$ , define $div w (x) = \sum_{j = 1}^{d} \partial_{j} w_{j} (x)$ ; for a smooth time-dependent vector field $w : [0, T] \times Q_{d} \to R^{d}$ , define $div w (t, x) = \sum_{j = 1}^{d} \partial_{j} w_{j} (t, x)$ , with derivatives taken only in $x$ ; for a tensor field $F = (F_{i j})$ , set $(div F)_{i} = \sum_{j = 1}^{d} \partial_{j} F_{i j}$ , and set $(v \otimes v)_{i j} = v_{i} v_{j}$ . Let $Δ = \sum_{j = 1}^{d} \partial_{j}^{2}$ . A smooth spatial field $w : Q_{d} \to R^{d}$ is solenoidal, or divergence-free, iff $div w (x) = 0$ for every $x \in Q_{d}$ ; a smooth time-dependent field $w : [0, T] \times Q_{d} \to R^{d}$ is solenoidal iff $div w (t, x) = 0$ for every $(t, x) \in [0, T] \times Q_{d}$ . Given a constant $ν \geq 0$ , smooth $f : [0, T] \times Q_{d} \to R^{d}$ , a smooth solenoidal field $v : [0, T] \times Q_{d} \to R^{d}$ , and a smooth solenoidal reference initial field $u_{0} : Q_{d} \to R^{d}$ , define the continuous residual and initial defect by

$r_{v} : = \partial_{t} v + P div (v \otimes v) - ν Δ v - P f, d_{0} : = ∥ v (0, \cdot) - u_{0} ∥_{2} .$

The residual is a continuous function of time with values in $L^{2} (Q_{d}; R^{d})$ , so $\int_{0}^{T} ∥ r_{v} (t, \cdot) ∥_{2} d t$ is finite. This is a continuous-domain residual: finitely many collocation values do not by themselves bound this $L_{t}^{1} L_{x}^{2}$ quantity. All quantities here are mathematical; no physical units are assigned to $ν$ .

Definition references

Tanveer, Energy Estimate, Uniqueness and ν dependence of Smooth Solutions, Ohio State lecture notes.

Carlo Morosi and Livio Pizzocchero, On approximate solutions of the incompressible Euler and Navier-Stokes equations.

Facts & Assumptions

Given: Countable Choice; $d \in {2, 3}$ ; $T > 0$ ; the normalized torus $Q_{d}$ ; $ν \geq 0$ ; smooth $f, v, u_{0}$ as in the Definition; and $div v = div u_{0} = 0$ .

[F1]

Periodic fluid Fourier Sobolev spaces: $Q_{d}$ is the $2 π$ -periodic torus with normalized measure $(2 π)^{- d} d x$ , and its $H^{0}$ norm is the normalized $L^{2}$ norm.

[F2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every sequence of nonempty sets has a choice function; this is the explicit choice assumption of the Fourier supplier.

[F3]

Periodic Leray and pressure maps: the multiplier $P$ is a real-preserving bounded orthogonal projection on every $H^{s}$ and commutes with derivatives; its coefficient formula projects each nonzero mode onto $k^{⊥}$ .

Proof

Given: The hypotheses and conventions stated in the Definition.

1.1givenF1F2F3algebra

The smoothness assumptions make $v \otimes v$ , $\partial_{t} v$ , $Δ v$ , and $f$ well-defined periodic fields. By [F1] these fields lie in the required $L^{2}$ spaces. The adopted projector convention is conditional on the explicit Countable Choice premise [F2], and [F3] defines the multiplier on each term; hence the displayed formula defines $r_{v}$ at every $t \in [0, T]$ .

2.1givenF1F3step 1.1∎

The maps $t \mapsto \partial_{t} v$ , $t \mapsto div (v \otimes v)$ , $t \mapsto Δ v$ , and $t \mapsto f$ are continuous into $L^{2}$ ; boundedness of $P$ from [F3] preserves that continuity. Hence $r_{v} \in C ([0, T]; L^{2})$ , and compactness of $[0, T]$ gives $\int_{0}^{T} ∥ r_{v} (t) ∥_{2} d t < \infty$ . Also $v (0) - u_{0} \in L^{2}$ , so $d_{0}$ is finite.

Continuous periodic solenoidal surrogate residual

Definition

Definition references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources