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A periodic C1 velocity has a global characteristic flow on a finite horizon

Statement

Let $d \geq 1$ , $T > 0$ , $ℓ_{j} > 0$ , and let $J \subseteq R$ be an open interval containing $[0, T]$ . Suppose $b : J \times R^{d} \to R^{d}$ is $C^{1}$ and periodic in $x_{j}$ with period $ℓ_{j}$ for every $j$ . Choose a compact interval $K \subset J$ with $[0, T] \subset int K$ , and put $M_{K} : = \sup_{K \times [0, ℓ_{1}] \times \dots \times [0, ℓ_{d}]} ∥ b ∥_{2}$ . For every $s \in [0, T]$ and $x \in R^{d}$ , there is a unique solution $X (\cdot; s, x)$ of $X^{'} (t) = b (t, X (t)), X (s; s, x) = x,$ on the whole interval $[0, T]$ . The lifted flow is periodic-equivariant: $X (t; s, x + k ℓ) = X (t; s, x) + k ℓ (k \in Z^{d}),$ where $k ℓ = (k_{1} ℓ_{1}, \dots, k_{d} ℓ_{d})$ . If $M_{T} : = \sup_{[0, T] \times [0, ℓ_{1}] \times \dots \times [0, ℓ_{d}]} ∥ b ∥_{2}$ , then $∥ X (t; s, x) - x ∥_{2} \leq M_{T} ∣ t - s ∣ (t \in [0, T]) .$ Thus each characteristic stays in a compact tube $[0, T] \times \overline{B} (x, M_{T} T + 1)$ .

Facts & Assumptions

Given: $d, T, ℓ, J, K, b, s, x$ as in the Statement.

[F1]

Linear transport equations and their characteristic flow: a characteristic solves $X^{'} (t) = b (t, X (t))$ ; a single-valued flow requires existence and uniqueness for each initial value.

[F2]

Picard-Lindelöf local existence and uniqueness for first-order systems: a continuous vector field locally Lipschitz in the state has a unique local solution; on a compact cylinder with field bound $M_{0}$ and state Lipschitz constant $L_{0}$ , it applies on a symmetric interval of radius $δ$ when $δ M_{0} \leq r$ and $L_{0} δ < 1$ for the cylinder state radius $r$ .

[F3]

A first-order initial value problem is equivalent to its Volterra integral equation: a solution satisfies $X (t) = X (s) + \int_{s}^{t} b (r, X (r)) d r$ with the oriented integral.

[F4]

For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable: $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ for integrable vector-valued $f$ and ordered endpoints $a \leq b$ ; a reversed oriented integral is first negated and rewritten with ordered endpoints.

[F5]

On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ : on a convex open set, a uniform derivative bound $∥ D f (z) v ∥_{2} \leq L ∥ v ∥_{2}$ gives the corresponding $L$ -Lipschitz bound.

Proof

1.1givenF2F5choose

Periodicity and continuity on the compact cylinder $K \times [0, ℓ_{1}] \times \dots \times [0, ℓ_{d}]$ give finite bounds $M_{K} : = \sup ∥ b ∥_{2}$ and $L : = \sup ∥ D_{x} b ∥$ there. The same bounds hold on all spatial translates. By [F5] each time slice is globally $L$ -Lipschitz in $x$ . Since $[0, T] \subset int K$ , choose $δ > 0$ small enough that every radius- $δ$ time interval centered in $[0, T]$ lies in $K$ , $δ (M_{K} + 1) \leq 1$ , and $δ (L + 1) < 1$ . On each corresponding unit-radius state cylinder, [F2] therefore gives a unique local solution through its center.

2.1givenF2step 1.1construct

Partition the finite interval between $s$ and any $t \in [0, T]$ into finitely many subintervals of length at most $δ$ . Apply [F2] successively at each endpoint, using the preceding endpoint value as the next initial state. The resulting curves agree on their shared endpoints and have the same derivative $b (r, X (r))$ there, so they concatenate to a solution on $[0, T]$ . Repeated local uniqueness from [F2] proves uniqueness on the concatenated interval. This is a finite construction and selects no family of local solutions over an infinite index set.

3.1givenF1F3F4step 2.1algebra∎

For the constructed solution on $[0, T]$ , [F3] and [F4] give $∥ X (t; s, x) - x ∥_{2} \leq \int_{\min (s, t)}^{\max (s, t)} ∥ b (r, X (r)) ∥_{2} d r \leq M_{T} ∣ t - s ∣$ . Thus its graph lies in the stated compact tube. For a period vector $k ℓ$ , the curve $X (t; s, x + k ℓ) - k ℓ$ solves the same IVP as $X (t; s, x)$ by periodicity of $b$ ; uniqueness gives periodic equivariance.

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Sources

European Centre for Medium-Range Weather Forecasts, Numerical Methods, Meteorological Training Course Lecture Series, 2002 (standard reference, not scraped)
Lukas Einkemmer and Alexander Ostermann, On the error propagation of semi-Lagrange and Fourier methods for advection problems, Computers & Mathematics with Applications 69 (2015), 170–179 (standard reference, not scraped)