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A $C^{2}$ vector field has a $C^{2}$ flow on a common compact tube

Statement

Let $f : U \to R^{n}$ be $C^{2}$ on an open set $U$ . Let $T_{+} > 0$ , let $W \subset U$ be open, and suppose every solution from $x \in W$ exists on an open time interval containing $[0, T_{+}]$ and stays in one compact convex set $K \subset U$ for all $t \in [0, T_{+}]$ . Write $Φ (t, x) = ϕ_{t} (x)$ . Then $Φ$ is $C^{2}$ jointly in $(t, x)$ on $(0, T_{+}) \times W$ , locally on every common-tube neighborhood in $W$ , and its derivatives extend continuously in time to $[0, T_{+}]$ . Its first state derivative $S (t, x) = D_{x} Φ (t, x)$ solves

\partial_{t} S = D f (Φ) S, S (0, x) = I_{n},

and its second state derivative is the bilinear map $H (t, x)$ determined by

\partial_{t} H [u, v] = D f (Φ) H [u, v] + D^{2} f (Φ) [S u, S v], H (0, x) [u, v] = 0.

In particular $\partial_{t} Φ = f (Φ)$ , $\partial_{t x} Φ = D f (Φ) S$ , and $\partial_{t t} Φ = D f (Φ) f (Φ)$ . The assertion is local to the stated common tube.

Facts & Assumptions

Given: The $C^{2}$ vector field, $T_{+}$ , open initial-state set $W$ , and a common compact convex tube containing all solution graphs on $[0, T_{+}]$ .

[F1]

On a common compact tube the flow is $C^{1}$ in its initial state, and $S = D_{x} Φ$ is the unique solution of the variational equation (Initial-data sensitivity of an ODE flow on a common compact tube). Its proof's difference-quotient estimate is uniform over the compact time interval.

[F2]

A classical solution of the autonomous IVP is differentiable in time and satisfies $\partial_{t} Φ = f (Φ)$ (Autonomous ordinary differential equations, First-order systems, initial value problems, and solutions on intervals).

[F3]

The $C^{2}$ convention supplies continuous first and second total derivatives; on compact $K$ , $D f$ and $D^{2} f$ are bounded and $D^{2} f$ is uniformly continuous ( $C^{k}$ Euclidean maps and diffeomorphisms, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous).

[F4]

A continuous-coefficient linear matrix IVP has a unique solution on the compact interval; adjoining a constant coordinate also solves an affine linear equation (Linear matrix ODEs have unique global solutions on a fixed interval).

[F5]

A nonnegative solution of $u (t) \leq A + L \int_{a}^{t} u (s) d s$ is bounded by $A e^{L (t - a)}$ (Gronwall's integral inequality with variable and constant coefficients).

[F6]

Integrating a differentiable curve recovers its endpoint difference, coordinatewise for vector-valued curves, and the norm of a vector integral is at most the integral of its norm (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , 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).

[F7]

Derivatives of compositions obey the chain rule (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

Proof

1.1F1F2F3F5F6given

Fix a base state $x \in W$ . By [F1], $Φ$ is $C^{1}$ in $x$ and $S$ solves $S^{'} = A S$ , where $A (t) = D f (Φ (t, x))$ . The coefficient is bounded on $[0, T_{+}]$ by [F3]; integrating the variational equation by [F6] and applying [F5] bounds $S$ uniformly in $t$ . The flow itself is jointly continuous in $(t, x)$ on the common tube by [F1] and the ODE relation [F2].

2.1F3F4F5F6step 1.1algebra

For a direction $h \in R^{n}$ , define a matrix $B_{h} (t)$ by $B_{h} (t) v = D^{2} f (Φ (t, x)) [S (t, x) h, S (t, x) v]$ . It is continuous and has norm at most $C ∥ h ∥_{2}$ for a finite constant $C$ , by [F3] and the bound on $S$ . Let $H_{h}$ solve $H_{h}^{'} = A H_{h} + B_{h}$ , $H_{h} (0) = 0$ . For each column, adjoining a constant coordinate turns its inhomogeneous linear IVP into a homogeneous linear system; applying [F4] to the augmented matrix gives a unique solution on $[0, T_{+}]$ . Its integral equation from [F6] and [F5] give $∥ H_{h} ∥_{\infty} \leq C^{'} ∥ h ∥_{2}$ for a finite constant $C^{'}$ .

2.2F1F3F6F7step 1.1algebra

For small $h$ with $x + h \in W$ , put $Δ (t) = Φ (t, x + h) - Φ (t, x)$ , $S_{h} (t) = S (t, x + h)$ , and $A_{h} (t) = D f (Φ (t, x + h))$ . The uniform remainder in [F1] gives $Δ = S h + o (∥ h ∥_{2})$ uniformly on $[0, T_{+}]$ . On each segment between $Φ (t, x)$ and $Φ (t, x + h)$ , the chain rule [F7] and [F6] applied coordinatewise to $D f$ give $A_{h} - A = \int_{0}^{1} D^{2} f (Φ + s Δ) [Δ, \cdot] d s .$ Uniform continuity of $D^{2} f$ on $K$ [F3] and $Δ = S h + o (∥ h ∥)$ therefore imply $A_{h} - A = D^{2} f (Φ) [S h, \cdot] + o (∥ h ∥_{2})$ uniformly in $t$ .

3.1F1F3F5F6step 2.1step 2.2algebra

The variational equations give $S_{h}^{'} = A_{h} S_{h}$ and $S^{'} = A S$ . Subtract these and the equation for $H_{h}$ from step 2.1, setting $R_{h} = S_{h} - S - H_{h}$ , to obtain $R_{h}^{'} = A_{h} R_{h} + (A_{h} - A) S + (A_{h} - A) H_{h} - B_{h}, R_{h} (0) = 0.$ By step 2.2, $(A_{h} - A) S - B_{h} = o (∥ h ∥)$ uniformly; also $A_{h} - A = O (∥ h ∥)$ and $H_{h} = O (∥ h ∥)$ by steps 2.2 and 2.1. Thus the forcing after $A_{h} R_{h}$ is $o (∥ h ∥)$ uniformly. The coefficients $A_{h}$ are uniformly bounded on $K$ , so [F5] applied to the integral equation for $R_{h}$ from [F6] gives $∥ R_{h} ∥_{\infty} = o (∥ h ∥_{2})$ . Therefore $D_{x} S (t, x) [h] = H_{h} (t)$ , proving existence of the second state derivative.

4.1F1F3F5F6step 2.1step 3.1

If $x_{j} \to x$ , [F1] gives uniform convergence of $Φ (\cdot, x_{j})$ and $S (\cdot, x_{j})$ on $[0, T_{+}]$ . Continuity of $D^{2} f$ on $K$ then gives uniform convergence of the coefficients and forcing in the linear equations for $H_{h}$ ; subtracting their integral equations and applying [F5] proves $H_{h} (\cdot, x_{j}) \to H_{h} (\cdot, x)$ uniformly. Thus the second state derivative is continuous in $(t, x)$ .

5.1F2F7step 1.1step 4.1∎

By [F2], $\partial_{t} Φ = f (Φ)$ . Differentiating this identity in $x$ and in $t$ using [F7] gives $\partial_{t x} Φ = D f (Φ) S$ and $\partial_{t t} Φ = D f (Φ) f (Φ)$ . These derivatives and $D_{x}^{2} Φ = H$ are continuous by steps 1.1 and 4.1, so $Φ$ is jointly $C^{2}$ on the stated tube.

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Sources

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Chapter 1, §1.5 (standard reference, not scraped)
E. J. Doedel and B. E. Oldeman, AUTO-07P full manual (2021) (standard reference, not scraped)

A C2C2 vector field has a C2C2 flow on a common compact tube