LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-05

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Transport characteristics depend C^1 on the initial position

Statement

Let $a$ be $C^{1}$ on an open neighborhood of the graph of a characteristic $X (\cdot; ξ_{0})$ solving

$X^{'} (t) = a (X (t), t), X (t_{0}; ξ_{0}) = ξ_{0},$

on a compact interval $I$ containing $t_{0}$ . Then, after shrinking to a neighborhood $U_{0}$ of $ξ_{0}$ , every initial point $ξ \in U_{0}$ determines a characteristic $X (\cdot; ξ)$ on the same interval $I$ , the map $(t, ξ) \mapsto X (t; ξ)$ is continuous on $I \times U_{0}$ , and for each $t \in I$ the map $ξ \mapsto X (t; ξ)$ is $C^{1}$ . Its Jacobian matrix $Y (t, ξ) = D_{ξ} X (t; ξ)$ satisfies

$Y^{'} (t, ξ) = D_{x} a (X (t, ξ), t) Y (t, ξ), Y (t_{0}, ξ) = I_{n} .$

Facts & Assumptions

Given: A $C^{1}$ transport field $a$ , a base characteristic $X (\cdot; ξ_{0})$ on a compact interval $I$ , and nearby initial points $ξ$ .

[L2]

Picard-Lindelof gives local existence and uniqueness for first-order systems (Picard-Lindelöf local existence and uniqueness for first-order systems).

[L3]

Nearby ODE solutions exist on one common compact interval and depend continuously on the initial data (Continuous dependence of ODE solutions on initial data and parameters).

[L4]

An ODE solution is equivalent to its Volterra integral equation (A first-order initial value problem is equivalent to its Volterra integral equation).

[L5]

Gronwall's inequality turns an integral inequality into an exponential bound (Gronwall's integral inequality with variable and constant coefficients).

[L6]

The norm of a vector integral is bounded by the integral of the norm (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).

[L7]

The Jacobian matrix is the matrix of first partial derivatives with respect to the initial-position variables (The Jacobian matrix of partial derivatives and the gradient in the scalar-valued case).

Proof

technique · direct

1.1

By [L2] and [L3], after shrinking to a neighborhood $U_{0}$ of $ξ_{0}$ , every $ξ \in U_{0}$ has a unique characteristic $X (\cdot; ξ)$ on the common compact interval $I$ , and $(t, ξ) \mapsto X (t; ξ)$ is continuous there.

L2L3

2.1

Fix a coordinate vector $e_{j}$ and a nonzero scalar $h$ with $ξ, ξ + h e_{j} \in U_{0}$ ; by [L4], the difference quotient $Z_{h} (t) : = \frac{X (t; ξ + h e_{j}) - X (t; ξ)}{h}$ satisfies $Z_{h} (t) = e_{j} + \int_{t_{0}}^{t} A_{h} (s, ξ) Z_{h} (s) d s$ , where $A_{h} (s, ξ) = \int_{0}^{1} D_{x} a (X (s; ξ) + θ (X (s; ξ + h e_{j}) - X (s; ξ)), s) d θ$ , and because $a$ is $C^{1}$ on a compact neighborhood of the family of graphs, the matrices $A_{h}$ are uniformly bounded.

L4step 1.1

3.1

Apply [L6] to the integral equation in step 2.1. If $t \geq t_{0}$ , then $∥ Z_{h} (t) ∥_{2} \leq 1 + M \int_{t_{0}}^{t} ∥ Z_{h} (s) ∥_{2} d s$ ; if $t \leq t_{0}$ , rewrite step 2.1 as $Z_{h} (t) = e_{j} - \int_{t}^{t_{0}} A_{h} (s, ξ) Z_{h} (s) d s,$ so $∥ Z_{h} (t) ∥_{2} \leq 1 + M \int_{t}^{t_{0}} ∥ Z_{h} (s) ∥_{2} d s$ . Gronwall therefore gives $∥ Z_{h} (t) ∥_{2} \leq e^{M ∣ t - t_{0} ∣}$ for every $t \in I$ , uniformly in $h$ . The continuity from step 1.1 together with the uniform continuity of $D_{x} a$ makes $A_{h} (\cdot, ξ) \to D_{x} a (X (\cdot; ξ), \cdot)$ uniformly as $h \to 0$ ; comparing the equations for $Z_{h}$ and $Z_{k}$ on the forward or backward interval between $t_{0}$ and $t$ and applying [L5] again shows that $(Z_{h})$ is Cauchy in $C (I; R^{n})$ .

L3L5L6step 2.1

4.1

Let $Y_{j} (\cdot, ξ)$ be the limit from step 3.1; passing to the limit in step 2.1 gives $Y_{j} (t, ξ) = e_{j} + \int_{t_{0}}^{t} D_{x} a (X (s; ξ), s) Y_{j} (s, ξ) d s$ , which [L4] rewrites as $Y_{j}^{'} (t, ξ) = D_{x} a (X (t; ξ), t) Y_{j} (t, ξ)$ with $Y_{j} (t_{0}, ξ) = e_{j}$ , and doing this for every coordinate vector while invoking [L7] identifies the matrix $Y = (Y_{1} \dots Y_{n})$ with $D_{ξ} X$ ; therefore $ξ \mapsto X (t; ξ)$ is $C^{1}$ and its Jacobian solves the displayed variational equation.

L4L7step 3.1∎

Depends on

Used by

Local linear transport has a unique solution from noncharacteristic Cauchy data Theorem

Dependency tree · two levels

52 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Andersson, First Order PDE: The Method of Characteristics (standard reference, not scraped)
Victor Ivrii, Partial Differential Equations (standard reference, not scraped)