The Flow Theorem via the Contraction Principle

Abstract

This article aims to present a proof of the classical theorems for the existence, regularity, invertibility and uniqueness properties on the solutions to the Flow Ordinary Differential Equation.

Keywords

flow matching, flow models, ODE, differential equations

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Introduction

Continuous-time flow models generate data by transporting a simple source distribution along the solution of an ordinary differential equation. Continuous normalizing flows (Chen et al. 2018; Grathwohl et al. 2019) introduced this construction as a continuous-time analogue of normalizing flows (Rezende and Mohamed 2015; Papamakarios et al. 2021), with the velocity field trained by maximum likelihood. Flow matching and the closely related rectified flows and stochastic interpolants (Lipman et al. 2023; Liu, Gong, and Liu 2023; Albergo and Vanden-Eijnden 2023; Albergo, Boffi, and Vanden-Eijnden 2025; Tong et al. 2024) replaced likelihood training by a simulation-free regression objective. The probability flow ODE of score-based diffusion models (Song et al. 2021) fits the same framework. In each case the model is specified by a time-dependent velocity field \(u:[0,1]\times\mathbb{R}^d\to\mathbb{R}^d\), and samples are produced by the flow \(\psi\) of \(u\), defined by \[ \begin{aligned} \frac{d}{dt}\psi_t(x) &= u_t(\psi_t(x)) \\ \psi_0(x) &= x \end{aligned} \tag{1}\] The model distribution is the pushforward of the source distribution under the time-one map \(\psi_1\).

Every such construction presupposes that Equation 1 is well posed: the flow must exist, be uniquely determined by \(u\), and be regular enough for the pushforward and the change-of-variables formula to make sense. The result invoked for this purpose is the following, stated in this form in Lipman et al. (2024, Theorem 1) with reference to Perko (2001) and Coddington and Levinson (1955).

Theorem (flow local existence and uniqueness). Let \(u\in C^r([0,1]\times\mathbb{R}^d,\mathbb{R}^d)\) with \(r\ge1\). Then Equation 1 has a unique solution \(\psi\), defined on a relatively open set \(\Omega\subseteq[0,1]\times\mathbb{R}^d\) containing \(\{0\}\times\mathbb{R}^d\), with \(\psi\in C^r(\Omega,\mathbb{R}^d)\). For each \(t\), the map \(\psi_t\) is a \(C^r\)-diffeomorphism onto an open subset of \(\mathbb{R}^d\).

The restriction to a subset \(\Omega\) cannot be removed without growth assumptions on \(u\). For \(d=1\) and \(u_t(x)=x^2\), the solution starting from \(x>0\) is \(\psi_t(x)=x/(1-tx)\), which blows up at \(t=1/x\), so that \(\Omega=\{(t,x):tx<1\}\).

In the standard references this theorem is assembled from several distinct arguments: existence by successive approximation or the contraction principle (Picard 1890; Lindelöf 1894; Banach 1922), uniqueness and continuous dependence by Grönwall-type estimates, differentiability in the initial data through the variational equation or the implicit function theorem, and maximal solutions through an extension argument (Coddington and Levinson 1955; Hartman 2002; Amann 1990; Teschl 2012). The aim of this article is a self-contained proof in which every existence, uniqueness and regularity statement is obtained from the Banach fixed point theorem, applied to a single operator. Specifically:

1. We prove a parametrized contraction principle with \(C^r\) dependence of the fixed point on the parameter (Lemma 1), using only the Neumann series and no implicit function theorem.
2. We define the Picard operator (Definition 2) for an arbitrary continuous field and show that, in a Bielecki-weighted norm (Bielecki 1956), it is a contraction on the entire time interval for every globally Lipschitz field, uniformly in the initial point, and that it is jointly \(C^r\) when the field is. A single application of Lemma 1 then yields existence, uniqueness and joint \(C^r\) regularity of the flow (Theorem 1).
3. We obtain invertibility from a reversal identity (Lemma 4), which expresses \(\psi_T^{-1}\) as the time-\(T\) flow of the reversed field \(-u_{T-t}\), and we pass to locally Lipschitz fields by truncation (Theorem 2).
4. We patch the local flows of Theorem 2 into a single flow on the maximal domain, and demonstrate a sufficient condition for global flow existence (Theorem 5).
5. We explore an extension to fields singular at \(t=1\), such as those of flow matching (Theorem 6)

The same operator therefore serves for \(u\), for its time reversal and for its truncations.

The remainder of the article is organized as follows. We discuss related work and fix notation next. Prerequisites contains the abstract fixed-point results and the Bielecki norm. The Picard Operator verifies that the Picard operator satisfies the hypotheses of Lemma 1. Global Flow proves the globally Lipschitz case, the reversal identity and the local theorem. Local Flow constructs the flow on maximal domain. Singular Endpoints deals with x-prediction velocity fields taken from flow matching.

Related work

Classical theory. Each ingredient of our proof has a classical counterpart, and we indicate the correspondence here. The method of successive approximations is due to Picard (1890) and Lindelöf (1894), and its formulation as the fixed point of a contraction is due to Banach (1922); Definition 2 and Theorem 1 follow this line. Bielecki (1956) observed that an exponentially weighted supremum norm makes the Picard operator a contraction on the whole interval for globally Lipschitz fields; this is the norm of Definition 1. Smooth dependence of the fixed point of a uniform family of contractions on a parameter is the uniform contraction principle of Chow and Hale (1982). Our Lemma 1 is a version of it, and is equivalently a specialization of the implicit function theorem to \(F(p,z)=z-T_pz\), in which invertibility of \(D_zF\) and global existence of the fixed point map follow from the contraction hypothesis. Differentiability of superposition (Nemytskii) operators, the subject of Proposition 7, is treated systematically by Appell and Zabrejko (1990). Textbook accounts of existence, uniqueness, smooth dependence on initial data and maximal solutions include Coddington and Levinson (1955), Hartman (2002), Amann (1990), Perko (2001) and Teschl (2012). These accounts also contain the blow-up alternative, which corresponds to our Theorem 4, and global existence criteria based on growth conditions, which correspond to Theorem 5. Banach-space treatments are given by Dieudonné (1960) and Lang (1993), and the fundamental theorem on flows in the manifold setting by Lee (2013).

In these accounts, existence is typically proved by successive approximation, uniqueness by Grönwall’s inequality, differentiability through the variational equation or the implicit function theorem, and maximal solutions by an extension argument. In our proof, existence, uniqueness and \(C^r\) dependence on the initial point on a fixed interval are all consequences of Lemma 1 applied to the operator of Definition 2. The operator is used for \(u\) itself, for its time reversal (Lemma 4) and for its truncations (Theorem 2). Uniqueness in particular requires no Grönwall-type estimate, and the derivative formula Equation 5 is the integral form of the variational equation. The remaining steps use standard arguments. Joint regularity in \(t\) follows by bootstrapping from the integral equation. The maximal domain is constructed by exhaustion together with a compactness argument. Global existence (Theorem 4, Theorem 5) rests on an extension argument and a differential inequality.

Wan et al. (2025) prove existence and uniqueness on \([0,1)\) for targets with finite second moment, and the existence of \(\Psi_1\) with \((\Psi_1)_\#p_{\mathrm{prior}}=p\) under geometric assumptions on the support, and Stéphanovitch (2026) give sharp bounds \(\|\nabla u_t\|\leq C/(1-t)\) with sampling rates. Theorem 6 recovers the bounded case within our framework, adds the regularity and diffeomorphism conclusions of Theorem 3, and applies to any field of the form Equation 10 with bounded \(m\), including learned \(x_1\)-prediction fields.

Shoushtari et al. (2026) bound the Lipschitz constant of the measurement-conditional velocity field by \(1/(1-t)+tR^2/(1-t)^3\) for priors supported in \(\overline B(0,R)\), which gives well-posedness and the pushforward identity on \([0,\tau]\) for every \(\tau<1\). Theorem 6 uses no Lipschitz bound on \(m\), and its confinement bound does not degenerate as \(t\to1\).

The strategy of deriving existence, uniqueness and smooth dependence on initial data from a single abstract principle applied to an operator on a space of curves goes back to Robbin (1968), who used the implicit function theorem. Our Lemma 1 plays the same role. Because we work in the Bielecki norm, the contraction holds on the whole interval and uniformly in the initial point, so the construction is global in both time and initial data rather than local.

Low regularity. Without local Lipschitz continuity, uniqueness fails in general. DiPerna and Lions (DiPerna and Lions 1989) and Ambrosio (Ambrosio 2004) developed a theory of flows defined almost everywhere for Sobolev and \(BV\) fields, respectively. Such fields lie outside the scope of this article.

Flow-based generative models. The theorem above is often quoted as the foundation of flow matching (Lipman et al. 2024), where existence up to \(t=1\) is obtained under an additional integrability assumption. For targets with compact support, Lipman et al. (2024, sec. 3.5) work on the half-open interval \([0,1)\), since the velocity field is not defined at \(t=1\), while the theorem as stated assumes \(u\) to be \(C^r\) on the closed interval. Theorem 6 addresses this for the conditional optimal transport path (Remark 1). Well-posedness of particular constructions has also been studied directly: Gao, Huang, and Jiao (2024) establish Lipschitz regularity of the velocity field of Gaussian interpolation flows for several classes of target distributions, and deduce existence and uniqueness of the flow together with Lipschitz continuity of the flow map and its inverse. Error analyses of deterministic flow matching samplers, such as Benton, Deligiannidis, and Doucet (2024), likewise work under regularity assumptions on the velocity field.

Our position. Our results are classical, and textbook accounts cover more ground, for instance flows on manifolds (Lee 2013). The contribution is expository: a single parametrized contraction principle and a single operator yield every conclusion of the theorem. We take the field \(u\) as given and ask only for the qualitative conclusions of the theorem: existence, uniqueness, joint regularity and the diffeomorphism property. The flow matching papers discussed above (Gao, Huang, and Jiao 2024; Wan et al. 2025; Shoushtari et al. 2026; Stéphanovitch 2026) start instead from the probability path, and use the posterior structure of its marginal field. This structure yields results we do not obtain: explicit Lipschitz constants (Gao, Huang, and Jiao 2024; Shoushtari et al. 2026; Stéphanovitch 2026), sampling rates (Stéphanovitch 2026), pushforward identities (Wan et al. 2025; Shoushtari et al. 2026), existence for targets with only a finite second moment, and the limit \(t\to1\) (Wan et al. 2025). Our arguments never use this structure, and so cannot reach these conclusions. In exchange, Theorem 6 depends only on the form Equation 10 and a bound on \(m\), so it applies unchanged to learned fields of that form with bounded output, for which no posterior structure is available.

Notation and conventions

The following conventions are used throughout; notation specific to a single result is introduced where it is used.

• Time is written as a subscript, \(u_t(x)=u(t,x)\) and \(\psi_t(x)=\psi(t,x)\), following
  1. When the dependence on the field matters, we write \(\psi[u]\) for the flow of \(u\), and \(\psi[u]_\cdot(x)\) for the trajectory \(t\mapsto\psi[u]_t(x)\), regarded as an element of \(C([0,T],\mathbb{R}^d)\).
• \(|\cdot|\) is the Euclidean norm on \(\mathbb{R}^d\), \(||\cdot||_{op}\) an operator norm, and \(B(E,E)\) the bounded linear operators on a Banach space \(E\). \(B(x,R)\) and \(\overline B(x,R)\) are the open and closed balls.
• For a curve \(z\), \(||z||_\infty\) is the supremum of \(|z|\) over its interval of definition. For a field \(v\), \(||v||_\infty\) is the supremum of \(|v_t(x)|\) over \([0,1]\times\mathbb{R}^d\); it may be infinite, in which case any estimate involving it is vacuous.
• A field \(v\) is Lipschitz with constant \(L\) if \(|v_t(x)-v_t(y)|\le L|x-y|\) for all \(t,x,y\), that is, Lipschitz in \(x\) uniformly in \(t\). \(\mathrm{Lip}_x(v)\) is the least such \(L\).
• \(C^r\) regularity on \([0,T]\times\mathbb{R}^d\) is understood with one-sided derivatives in \(t\) at \(t\in\{0,T\}\).
• Multi-indices are elements of \(\mathbb{N}^d\), where \(0\in\mathbb{N}\). For \(\alpha\in\mathbb{N}^d\), \(|\alpha|=\alpha_1+\dots+\alpha_d\) and \(\partial_x^\alpha=\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}\).
• Landau symbols \(o(\cdot)\) and \(O(\cdot)\) refer to the limit in which their argument tends to zero. \(a\leftarrow b\) denotes substitution of \(b\) for \(a\) in an expression already written.
• For \(s>0\) we fix cutoffs \(b_s\in C^\infty_c(\mathbb{R}^d;[0,1])\) with \(b_s=1\) on \(B(0,s)\) and \(b_s=0\) outside \(\overline B(0,2s)\), for instance \(b_s=b_1(\cdot/s)\).

Several letters are reused with different meanings in different results. These are collected in Table 1; in each case the meaning is fixed by the arguments the symbol carries.

Table 1: Symbols with more than one meaning.

Symbol	Meanings	Distinguished by
\(T\)	Contraction family \(T_p\); Picard operator \(T[v]_y\); time horizon \(T\in(0,1]\); local existence time \(T_{x_0}\)	Operators carry a parameter or a bracketed field; scalars occur only as time bounds
\(I\)	Identity operator; integration operator \(I\varphi(t)=\int_0^t\varphi(\tau)\,d\tau\)	\(I-A\) versus \(I\varphi\)
\(B\)	Balls \(B(x,R)\); operator space \(B(E,E)\); \(D_pT_p\) in Lemma 1; \(J(A+H)-J(A)\) in the proof of Proposition 3	Arguments; the last two are local to their proofs
\(A\), \(F\), \(M\)	Generic operators, maps and spaces, local to each result; \(M\) is also \(\max\lvert u\rvert\) in the proof of Theorem 2	Local to each result
\(P\)	Parameter space; integrand \(P_\alpha\) in Equation 7	Multi-index subscript
\(d\)	Dimension; metric in \((E,d)\); differential	Position
\(\lvert\cdot\rvert\)	Euclidean norm; order of a multi-index	Vector or multi-index argument
Subscripts	Time (\(u_t\), \(\psi_t\)); parameter (\(T_p\)); initial point (\(T[v]_y\), and \(g_x\) in Lemma 4); a point of \(\mathbb{R}^d\) (\(x_0\)); an index (\(S_n\), \(e_i\), \(P_\alpha\))	In particular, \(\psi_0\) is the time-zero map while \(x_0\in\mathbb{R}^d\)
Superscripts	Power (\(A^n\)); regularity (\(C^r\)); derivative order (\(D^k\), \(\partial^\alpha\)); truncation radius (\(u^s\)); horizon (\(\tilde u^{(T)}\)); fixed point (\(z^*\))	A parenthesized superscript is a horizon

Prerequisites

This section collects the abstract results used throughout; none of them refers to ODEs. Proposition 2 and Lemma 1 establish, respectively, continuous and \(C^r\) dependence of the fixed point of a parametrized family of contractions on its parameter. Proposition 1 and Proposition 3 supply the two ingredients of the latter. Definition 1 and Proposition 4 introduce the weighted norm in which the Picard operator will be shown to be a contraction.

Proposition 1 (\(C^1\) functions are Locally Lipschitz) Let \(E, F\) be normed spaces, \(U\) be an open convex subset of \(E\), and \(f \in C^1(U, F)\), then:

\[ \sup_{x \in U}||Df(x)||_{op} \leq L \iff f \text{ is } L\text{-Lipschitz on } U \]

Moreover, if \(U\) lies in the interior of a compact set \(K\), \(L = \sup_{x \in K}{||Df(x)||_{op}} < \infty\).

Remark. Implication in the forward direction follows by applying Jensen’s inequality and the operator norm upper bound on the line segment \((1-t)x + ty\) of integration for any \(x, y \in U\). The line segment is contained within \(U\), by convexity. The reverse direction follows by simply applying the definition of Lipschitz to the difference quotient of directional derivatives.

Proposition 2 (Continuously Parametrized Contraction Principle) Let \((E, d)\) be a complete metric space, and \(P\) be a topological space on which a parametrized family of self-maps \((T_p)_{p \in P}, T_p: E \to E\) are defined such that \((p \mapsto T_p z)\) is continuous for every \(z \in E\). If the family is a locally uniform family of contractions, i.e, every \(p \in P\) has a neighbourhood \(U_p\) and a constant \(\kappa \in [0, 1)\) (depending on \(p\)) such that for all \(p' \in U_p\) and all \(z, w \in E\):

\[ d(T_{p'}z, T_{p'}w) \leq \kappa d(z, w) \tag{2}\] Then there exists a unique fixed point \(z^{*}\) for each \(p \in P\) which satisfies \(T_pz^{*} = z^{*}\). Moreover, the map taking the contraction parameter \(p\) to its unique fixed point \(z^{*}(p)\), i.e, the map \(p \mapsto z^{*}(p)\) is continuous.

Proof. The existence of the map \(z^{*}(p)\) is given by classical BFP, and the completeness of \(E\) which ensures the sequence of \(z_{(n)} = T_pz_{(n-1)}\), which is Cauchy by Equation 2, converges to a unique stationary point. To prove continuity of \(z^{*}(p)\), fix \(p\) and let \(p' \in U_p\), so that \(T_p\) and \(T_{p'}\) are both \(\kappa\)-contractions with the same \(\kappa\). We must examine the distance:

\[ \begin{aligned} d(z^{*}(p'), z^{*}(p)) &= d(T_{p'}z^{*}(p'), T_{p}z^{*}(p)) \\ & \leq d(T_{p'}z^{*}(p'), T_{p'}z^{*}(p)) + d(T_{p'}z^{*}(p), T_{p}z^{*}(p)) \\ & \leq \kappa d(z^{*}(p'), z^{*}(p)) + d(T_{p'}z^{*}(p), T_pz^{*}(p)) \\ \end{aligned} \]

Inserting \(T_pz^{*}(p')\) in place of \(T_{p'}z^{*}(p)\) and using the contraction bound of \(T_p\) instead gives the same estimate with \(z^{*}(p')\) in the last term. Thus, for either choice of \(w\), \[ d(z^{*}(p'), z^{*}(p)) \leq \frac{1}{1 - \kappa}d(T_{p'}w, T_pw) \text{ where } w \in \{z^{*}(p), z^{*}(p')\} \tag{3}\] Take \(w = z^{*}(p)\), which does not move with \(p'\). By continuity of the family on \(p\) at this fixed \(w\), we have: as \(p' \to p, d(T_{p'}w, T_pw) = o(1)\). Consequently, \(d(z^{*}(p'), z^{*}(p)) = (1 - \kappa)^{-1} o(1)\), hence \(z^{*}: P \to E\) is continuous over \(P\). The factor \(\frac{1}{1 - \kappa}\) does not change due to local uniformity.

Proposition 3 (Neumann Series) Let \(E\) be Banach, and \(B(E,E)\) be bounded linear operators \(E \to E\) under the operator norm. For \(||A||_{op} < 1, I - A\) is invertible with \(J(A) = (I - A)^{-1} = \sum_{n \geq 0}A^n\) is \(C^{\infty}\) in \(A\). Moreover, \(DJ(A)H = J(A)HJ(A)\)

Proof. Let us consider the partial sum \(S_n = \sum_{k=0}^{n}{A^k}\). Clearly \((I - A)S_n = I - A^{n+1}\), thus \((I-A)S_n\) converges in norm to \(I\). By submultiplicativity, the operator \(I - A\) is continuous with respect to the operator norm, and by completeness \(S_n\) converges (\(||A^n||_{op} \leq ||A||^n_{op}\)). So now we have \((I - A)S = I\). Likewise, the same arguments establish \(S(I-A) = I\). Now let us examine the difference \(B = J(A + H) - J(A)\). Multiplying on the left first by \(I - A - H\) and \(I - A\), we get \(H = (I - A - H)B(I - A)\). We get the identity \[ J(A+H) - J(A) = J(A+H)HJ(A) \tag{4}\] We can first obtain continuity of \(J\) by bounding the RHS of Equation 4 and choosing \(||H||_{op} \leq \frac{1}{2}(1 - ||A||_{op})\): \[ \begin{aligned} ||J(A+H) - J(A)||_{op} = ||J(A+H)HJ(A)||_{op} &\leq \frac{1}{1 - ||A||_{op} - ||H||_{op}}\frac{1}{1 - ||A||_{op}}||H||_{op} \\ & \leq \frac{2}{(1 - ||A||_{op})^2}||H||_{op} \end{aligned} \] Subtracting \(J(A)HJ(A)\) from the RHS of Equation 4 creates the difference term again, and applying the previous inequality again changes the bound to \(O(||H||^2_{op})\), so we then have: \[ J(A + H) = J(A) + J(A)HJ(A) + o(||H||_{op}), \text{ as } ||H||_{op} \to 0 \]

Since \(DJ\) is expressed solely in terms of \(J\), this process can be repeated ad-infinitum with each derivative being a smooth combination of continuous maps.

Remark. The Neumann series essentially tells us that the expression \(\frac{1}{I - A}\) is as meaningful as the function \(\frac{1}{1-x}\) provided we work only with \(||A||_{op} < 1\). In particular, that it is well-defined and smooth.

Lemma 1 (Regularly Parametrized Contraction Principle) Let \((E, ||.|||)\) be Banach, and \(P\) be an open subset of a normed space on which a parametrized family of contractions (Proposition 2) \((T_p)_{p \in P}, T_p: E \to E\) such that \((p,z \mapsto T_p z) \in C^r\), then the map taking the contraction parameter \(p\) to its unique fixed point \(z^{*}(p)\), i.e, the map \(p \mapsto z^{*}(p)\) is \(C^r(P, E)\) with \(r \geq 1\).

Proof. Existence, uniqueness and continuity follow from Proposition 2. We need to argue regularity, by exploiting the identity \(z^{*}(p) = T_pz^{*}(p)\). Contraction family is at least \(C^1\), so there exists a Taylor expansion of the form: \[ T_{p + q}(z + \Delta) = T_pz + A\Delta + Bq + o(||\Delta|| + ||q||) \] If we can choose \(||\Delta|| = O(||q||)\), then the remainder is \(o(||q||)\). Take \(\Delta \coloneqq z^{*}(p+q) - z^{*}(p)\), applying the fixed point property and simplifying via Equation 3: \[ \begin{aligned} ||z^{*}(p+q) - z^{*}(p)|| &= ||T_{p+q}z^{*}(p+q) - T_pz^{*}(p)|| \\ & \leq \frac{1}{1-\kappa}||T_{p+q}z^{*}(p+q) - T_pz^{*}(p+q)|| \\ \end{aligned} \] \(C^1\) regularity of \(T_pz\) and arguments similar to Proposition 1 provide an upper-bound \(\frac{1}{1-\kappa}\Upsilon||q||\) for some constant \(\Upsilon\). So we have \(\Delta = O(||q||)\). We can now simplify the Taylor expansion at \((p, z^{*}(p))\) as: \[ (I - A)\Delta = Bq + o(||q||) \\ \implies \Delta = (I - A)^{-1}Bq + o(||q||) \] The last equality follows from Proposition 3 applied on \(A\) which has \(||A||_{op} \leq \kappa < 1\) (from Proposition 1 and Equation 2). Writing in common notation, we get: \[ \frac{dz^{*}}{dp} = \left(\frac{1}{I - D_zT_pz}D_pT_pz\right)\bigg|_{p, z^{*}(p)} \tag{5}\] It is clear from Proposition 3 that the reciprocal is in fact smooth, so the expression, discounting the evaluation at \(p, z^{*}(p)\) is \(C^{r-1}\) in \(p, z\). The evaluation is tantamount to composition, and \(\frac{dz^{*}}{dp}\) has the same regularity as \(z^{*}\) upto \(C^{r-1}\), allowing us to induct. Hence \(z^{*} \in C^r\).

Definition 1 (Bielecki-style Norm) Let \(\beta \in \mathbb{R}^{+}_{0}\), then define a norm on \(C([0,1], \mathbb{R}^d)\), using the Euclidean norm \(|a|\) on \(\mathbb{R}^d\) as: \[ z \mapsto ||z||_{\beta} = \sup_{t \in [0, 1]}{e^{-\beta t} |z(t)|} \] This map is indeed topologically equivalent to sup-norm, as we show in Proposition 4. It is trivial to verify that it is indeed a norm, since it inherits its properties from the Euclidean norm.

Proposition 4 (Equivalence of Bielecki and sup-norm) Let \(z \in C([0,1], \mathbb{R}^d)\). We have:

1. \(||z||_{\beta}\) is monotonically decreasing in \(\beta\).
2. \(e^{\beta}||z||_{\beta}\) is monotonically increasing in \(\beta\).

Combining for \(0 \leq \beta_1 < \beta_2\) we get: \[ e^{-(\beta_2 - \beta_1)}||z||_{\beta_1} \leq ||z||_{\beta_2} \leq ||z||_{\beta_1} \leq e^{(\beta_2 - \beta_1)}||z||_{\beta_2} \] Thus, the norms are equivalent for any distinct \(\beta_1, \beta_2\). In particular, the norms are equivalent to the standard sup norm taking \(\beta_1 \coloneq 0\).

Proof. First we have: \[ \sup_{t \in [0,1]}{e^{-\beta_2 t} |z(t)|} = \sup_{t \in [0,1]}{e^{-(\beta_2 - \beta_1)t} e^{-\beta_1 t} |z(t)|} \] Since \(e^{-(\beta_2 - \beta_1)} \leq e^{-(\beta_2 - \beta_1)t} \leq 1\), both (1) and (2) follow.

Remark. The demonstrated property allows for proofs of qualitative properties such as completeness, for instance, to be transferred since the original sup-norm can be bounded in either direction incurring only a constant which can be absorbed by suitable choice of \(\epsilon\). Hereafter, we restrict \(\beta \neq 0\), and adopt the standard notation \(||z||_{\infty}\) for the sup-norm.

Remark. Additionally, note that our development of the Bielecki-style norm does not depend on \([0,1]\). We could use the same arguments to demonstrate the same properties on an arbitrary compact interval say, \([0, a]\), with the only change being a scaling factor: \((\beta_2 - \beta_1) \leftarrow (\beta_2 - \beta_1)a\).

Proposition 5 (Max on compact index is continuous) Let \(S\) be a compact metric space, \(P\) a metric space, and \(g:S\times P\to\mathbb{R}\) continuous for the product topology. Then \(F(p)\coloneq\max_{s\in S}g(s,p)\) exists for every \(p\in P\), and \(F:P\to\mathbb{R}\) is continuous.

Proof. For fixed \(p\), the map \(s\mapsto g(s,p)\) is continuous on the compact space \(S\), so it attains its maximum. Fix \(p\in P\) and \(\varepsilon>0\). If we first find a single \(\delta>0\) such that: \[ d_P(p',p)<\delta\implies|g(s,p')-g(s,p)|<\varepsilon\quad\text{for all }s\in S. \] Since \(|F(p') - F(p)| \leq |g(s,p')-g(s,p)|\). For each \(s\in S\), continuity of \(g\) at \((s,p)\) gives \(\delta_s>0\) such that \(d_S(s',s)<\delta_s\) and \(d_P(p',p)<\delta_s\) imply \(|g(s',p')-g(s,p)|<\varepsilon/2\). By compactness, finitely many balls \(B(s_i, \delta_{s_i})\) suffice to cover \(S\). Put \(\delta\coloneq\min_i\delta_{s_i}\). Given \(s\in S\), choose \(i\) with \(d_S(s,s_i)<\delta_{s_i}\). If \(d_P(p',p)<\delta\), the choice of \(\delta_{s_i}\) applies both at \((s,p')\) and at \((s,p)\), giving \[ |g(s,p')-g(s,p)|\le|g(s,p')-g(s_i,p)|+|g(s_i,p)-g(s,p)|<\varepsilon. \]

The Picard Operator

We define the Picard operator \(T[v]_y\) (Definition 2) for an arbitrary continuous field \(v\), and verify the two hypotheses of Lemma 1 with the initial point \(y\) as the parameter: uniform contraction (Lemma 2) and joint \(C^r\) regularity (Lemma 3). The latter rests on the regularity of the superposition map \(z\mapsto v\circ z\) (Proposition 7).

Definition 2 (Picard Operator) For \(v \in C([0, 1] \times \mathbb{R}^d, \mathbb{R}^d)\), \(y \in \mathbb{R}^d\) and \(z \in C([0,1], \mathbb{R}^d)\), define \(T[v]_y: C([0,1], \mathbb{R}^d) \to C([0,1], \mathbb{R}^d)\) as: \[ (T[v]_yz)(t) = y + \int_{0}^{t}{v_{\tau}(z(\tau)) d\tau} \tag{6}\] The operator is well-defined since the integrand is continuous, and the result is \(C^1\).

Remark. All solutions of Equation 1 are fixed points of \(T[u]_xz\), as can be seen by integrating both sides.

Lemma 2 (Joint Lipschitz Bound) Let \(v, v' \in C([0, 1] \times \mathbb{R}^d, \mathbb{R}^d)\) be Lipschitz with constants \(L_v, L_{v'}\). Put \(L \coloneq \max(L_v, L_{v'}), \beta \coloneq 2L + 1, \kappa \coloneq L/\beta < \frac{1}{2}\). Then, for every \(y, y' \in \mathbb{R}^d, z, z' \in C([0,1],\mathbb{R}^d)\), and every restriction of \(v, v'\) to a subinterval \([0, a] \subseteq [0, 1]\), we have (using suitable definitions of the norms): \[ ||T[v']_{y'}z' - T[v]_{y}z||_{\beta} \leq |y' - y| + ||v' - v||_{\infty} + \kappa ||z' - z||_{\beta} \]

Proof. Examine the difference: \[ \begin{aligned} T[v']_{y'}z' - T[v]_{y}z &= (y' - y) + \int_{0}^{t}{v'_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) d\tau} \\ &= (y' - y) + \int_{0}^{t}{v'_{\tau}(z'(\tau)) - v_{\tau}(z'(\tau)) + v_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) d\tau} \end{aligned} \] We can now simply apply the triangle inequality to get three terms of Bielecki norm. However, the Bielecki norm of a constant function is just the Euclidean norm. So we get: \[ ||T[v']_{y'}z' - T[v]_{y}z||_{\beta} \leq |y' - y| + \left\lVert\int_{0}^{t}{v'_{\tau}(z'(\tau)) - v_{\tau}(z'(\tau)) d\tau}\right\rVert_{\beta} + \left\lVert\int_{0}^{t}{v_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) d\tau}\right\rVert_{\beta} \] The second integrand can be upper bound by \(||v'_t(z'(t)) - v_t(z'(t))||_{\infty} \leq ||v' - v||_{\infty}\). For the third term we have from Definition 1: \[ \begin{aligned} \left\lVert\int_{0}^{t}{v_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) d\tau}\right\rVert_{\beta} &= \sup_{t \in [0,1]}{e^{-\beta t}\left|\int_{0}^{t}{v_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) d\tau}\right|} \\ & \leq \sup_{t \in [0,1]}{\int_{0}^{t}{e^{-\beta (t - \tau)} e^{-\beta \tau}|v_{\tau}(z'(\tau)) - v_{\tau}(z(\tau)) | d\tau}} \\ & \leq L \sup_{t \in [0,1]}{\int_{0}^{t}{e^{-\beta (t - \tau)} e^{-\beta \tau}|z'(\tau)) - z(\tau))| d\tau}} \\ & \leq L ||z' - z||_{\beta} \sup_{t \in [0, 1]}{\int_{0}^{t}{e^{-\beta\tau} d\tau}} \\ &= \frac{L(1 - e^{-\beta})}{\beta}||z' - z||_{\beta} \\ &\leq \frac{L}{\beta}||z' - z||_{\beta} \end{aligned} \] The simplification follows by the Lipschitz property, creating the norm term inside the integrand and the substitution \(\tau \leftarrow t - \tau\). The proof can easily be transferred to the case of \([0, a]\).

Proposition 6 Let \(F\) be \(C^r\) at \(z\) (into a normed space \(Y\)), \(r\geq1\), and let \(M:Y\to Z\) be linear and bounded. Then \(M\circ F\) is \(C^r\) at \(z\), with \(D(M\circ F)(z)h = M(DF(z)h)\).

Proof. We examine the difference and apply linearity of \(M\): \[ \begin{aligned} & M(F(z+h)) - M(F(z)) - M(DF(z)h) \\ &= M\big(F(z+h)-F(z)-DF(z)h\big) \\ &= o(||h||) \end{aligned} \] Last equality follows by boundedness of \(M\). So \(M\circ F\) is differentiable at \(z\) with derivative \(M\circ DF(z)\). Since \(M\) does not depend on \(z\), this same computation applies to each derivative of \(F\) in turn: \(D^k(M\circ F) = M\circ D^kF\) for every \(k\leq r\), so \(M\circ F\) is \(C^r\) whenever \(F\) is.

Remark. Integration \(I : C([0,1];\mathbb{R}^d)\to C([0,1];\mathbb{R}^d)\), \(I\varphi(t):=\int_0^t\varphi(\tau)\,d\tau\), is linear, and bounded since we have for every \(t\in[0,1]\), \[|I\varphi(t)| = \Big|\int_0^t\varphi(\tau)\,d\tau\Big| \leq \int_0^t|\varphi(\tau)|\,d\tau \leq ||\varphi||_\infty,\] so \(||I\varphi||_\infty\leq||\varphi||_\infty\). So Proposition 6 applies to \(I\).

Proposition 7 (Regularity of the composition map) Let \(V\) be a finite-dimensional normed space, \(r \geq 1\), and let \(g: [0,1] \times \mathbb{R}^d \to V\) be such that the partial derivatives \(\partial_x^k g\) exist and are jointly continuous in \((t,x)\) for \(0 \leq k \leq r\). Write \((g\circ z)(t) := g_t(z(t))\) for \(z \in C([0,1];\mathbb{R}^d)\). Then \(z \mapsto g\circ z\) is \(C^r(C([0,1],\mathbb{R}^d),C([0,1],V))\), with derivative \(h \mapsto (\partial_xg\circ z)h\), where \(((\partial_xg\circ z)h)(t) := \partial_xg_t(z(t))h(t)\).

Proof. Write \(Ah := (\partial_xg\circ z)h\). As usual, we begin by examining the difference: \[ \begin{aligned} (g\circ(z+h))(t) - (g\circ z)(t) - Ah(t) &= g_t(z(t)+h(t)) - g_t(z(t)) - \partial_xg_t(z(t))h(t) \\ &= \int_{0}^{1}{[\partial_xg_t(z(t) + \xi h(t)) - \partial_xg_t(z(t))]\, h(t)\, d\xi} \end{aligned} \] The RHS would admit the required simplification to \(o(||h||_{\infty})\) if we can obtain a uniform control over \(||\partial_xg_t(z(t) + \xi h(t)) - \partial_xg_t(z(t))||_{op}\). Since at the least \(\partial_xg_t(z)\) is jointly continuous, we can obtain uniform control by restricting to a compact set \(K := [0,1]\times\overline{B}(0,||z||_\infty+1)\), and choosing \(||h||_\infty<1\) ensures \(z+h\) never leaves the ball. So, there exists \(\delta\in(0,1)\) such that \(||h||_\infty<\delta \implies ||\partial_xg_t(z+\xi h) - \partial_xg_t(z)||_{op}<\varepsilon\). Taking the sup-norm on both sides: \[ \begin{aligned} ||g\circ(z+h) - g\circ z - Ah||_{\infty} & \leq \sup_{t \in [0,1]}{\int_{0}^{1}{||\partial_xg_t(z(t)+\xi h(t)) - \partial_xg_t(z(t))||_{op}\, |h(t)|\, d\xi}} \\ & < \varepsilon\, ||h||_{\infty}. \end{aligned} \] So \(g\circ z\) is differentiable at \(z\) with derivative \(A\). Moreover, \(A\) is continuous with respect to the operator norm, as: \[ \begin{aligned} ||z' - z||_{\infty} < \delta \implies & ||A(z)-A(z')||_{op} \\ &= \sup_{||h||_\infty\leq1}||(\partial_xg_t(z(t))-\partial_xg_t(z'(t)))h(t)|| \\ & \leq \sup_t||\partial_xg_t(z(t))-\partial_xg_t(z'(t))||_{op} < \varepsilon \end{aligned} \]

The operator \((\partial_x g \circ z)\) so found satisfies the conditions of Proposition 7 after switching to operator norm, so we induct to find all \(C^r\) derivatives.

Lemma 3 (Regularity of Picard Operator) Let \(v \in C^r([0,1]\times\mathbb{R}^d,\mathbb{R}^d)\), \(r \geq 1\). Then the map \((z,y)\to T[v]_yz\) is jointly \(C^r\), and \(DT[v]z\,h(t) = \int_0^t{\partial_xv_\tau(z(\tau))h(\tau)\,d\tau}\) is independent of \(y\) (\(h\) is the operand).

Proof. Write \(T[v]_yz = y + I(v\circ z)\). Since \(v\in C^r\), Proposition 7 gives \(z\mapsto v\circ z\) is \(C^r\); by Proposition 6, \(I(v\circ z)\) is \(C^r\) too, with \[D(I(v\circ z))h = I\big(D(v\circ z)h\big), \quad\text{i.e.}\quad DT[v]z\,h(t) = \int_0^t{(\partial_xv\circ z)(\tau)h(\tau)\,d\tau} = \int_0^t{\partial_xv_\tau(z(\tau))h(\tau)\,d\tau}.\] Differentiability in \(y\) is immediate.

Global Flow

We first treat globally Lipschitz fields (Theorem 1), then obtain invertibility of the time-\(T\) map from the reversal identity (Lemma 4). Finally, truncating \(u\) with the cutoffs \(b_s\) yields globally Lipschitz fields that agree with \(u\) near a given point, from which we obtain the local theorem (Theorem 2).

Theorem 1 (Global Flow) Let \(T\in(0,1]\) and \(u\in C^r([0,T]\times\mathbb{R}^d;\mathbb{R}^d)\), \(r\ge1\), globally \(L\)-Lipschitz in \(x\). Then there is a map \(\psi[u]:[0,T]\times\mathbb{R}^d\to\mathbb{R}^d\) with:

1. Global existence. For every \(x\), \(t\mapsto\psi[u]_t(x)\) solves Equation 1 on all of \([0,T]\).
2. Uniqueness. If \(z\) solves Equation 1 on \([0,T]\) with \(z(0)=x\), then \(z=\psi[u]_\cdot(x)\).
3. Joint regularity. \(\psi[u]\in C^r([0,T]\times\mathbb{R}^d;\mathbb{R}^d)\).

Proof. Clearly, fixed points of the Picard operator \(T[u]_x\) (Definition 2) give the solutions to Equation 1. By Lemma 2, taking \(v'\coloneq v\coloneq u\) and \(y'\coloneq y\), \(T[u]_x\) is a contraction in \(||\cdot||_\beta\), \(\beta=2L+1\), with the same \(\kappa=L/\beta\) for every \(x\), and the space is complete by Proposition 4. So each \(T[u]_x\) has a unique fixed point \(z^*(x)\), and we put \(\psi[u]_t(x)\coloneq z^*(x)(t)\). This gives (1) and (2).

The family is uniformly (in particular, locally uniformly) contracting in the sense of Proposition 2. With Lemma 3, transferred to \(||\cdot||_\beta\) by Proposition 4, it satisfies the conditions of Lemma 1, so \(x\mapsto z^*(x)\) is \(C^r\) into \(C([0,T],\mathbb{R}^d)\). For a multi-index \(\alpha\in\mathbb{N}^d\) we write \(|\alpha|\coloneq\alpha_1+\dots+\alpha_d\) and \(\partial_x^\alpha\coloneq\partial_{x_1}^{\alpha_1}\cdots\partial_{x_d}^{\alpha_d}\). It follows that the partials \(\partial_x^\alpha\psi[u]_t(x)=(\partial^\alpha z^*(x))(t)\) (evaluation in compact domain, Proposition 6) for \(|\alpha|\le r\) are jointly continuous in \((t,x)\), since \(\partial^\alpha z^*\) is continuous with respect to \(||\cdot||_{\infty}\) and each \(\partial^\alpha z^*(x)\) is continuous.

For joint regularity, differentiate the fixed point identity \(z^*(x)=x+I(u\circ z^*(x))\) in \(x\). By Proposition 7 and the chain rule, and since integration and evaluation are bounded linear (Proposition 6), for \(|\alpha|\le r\): \[ \partial_x^\alpha\psi[u]_t(x)=\partial^\alpha x+\int_0^tP_\alpha(\tau,x)\,d\tau,\qquad P_\alpha(\tau,x)\coloneq\partial_x^\alpha\big[u_\tau(\psi[u]_\tau(x))\big] \tag{7}\] where \(\partial^\alpha x\) is \(x\), \(e_i\) or \(0\) for \(\alpha=0\), \(\alpha=e_i\) and \(|\alpha|\ge2\) respectively.

By the chain rule, \(P_\alpha\) is a polynomial in the \((\partial_x^\eta u)(\tau,\psi[u]_\tau(x))\) and the \(\partial_x^\gamma\psi[u]_\tau(x)\) with \(|\eta|,|\gamma|\le|\alpha|\). Hence \(P_\alpha\) is continuous, and \(\partial_t\partial_x^\alpha\psi[u]=P_\alpha\), one-sided at the end points.

Effectively, all \(x\)-partials up to order \(r\), and their \(t\)-partials, are jointly continuous. We will inductively increase regularity of \(\partial_x^\alpha\psi \in C^{r-k}\) by lowering the bound \(|\alpha| \leq k\), starting from \(k = r\). Fix \(\alpha\) such that \(|\alpha| \leq k - 1\). Any \(x\)-partial is jointly \(C^{r-k}\). It remains to show that \(t\)-partials too are jointly \(C^{r-k}\), after which we may upgrade regularity. The \(t\)-derivatives are given by \(P_\alpha\). Since we have \(|\eta|, |\gamma| \leq |\alpha| < k\), \(\partial_x^{\eta}u \in C^{r-k}, \partial_x^{\gamma}\psi \in C^{r-k}, \psi \in C^{r-k}\), so \(P_{\alpha}\) is polynomial in terms of \(C^{r-k}\) maps, so \(t\)-partials are jointly \(C^{r-k}\). Therefore \(\partial_x^\alpha\psi\in C^{r-(k-1)}\). Inducting to \(k = 0\) gives (3).

Remark. Continuity of \(u\mapsto\psi[u]\) holds on any class of fields sharing a Lipschitz bound. Fix \(L\) and let \(\mathcal U_L\) be the set of fields satisfying the hypotheses of Theorem 1 with \(\mathrm{Lip}_x\le L\), under the sup-norm \(||u-v||_\infty\). On \(\mathcal U_L\times\mathbb{R}^d\), every \(T[v]_y\) is a contraction in \(||\cdot||_\beta\), \(\beta=2L+1\), with the same \(\kappa=L/\beta\), and by Lemma 2 (taking \(z'\coloneq z\)), \((v,y)\mapsto T[v]_yz\) is continuous. So Proposition 2 applies directly: \((u,x)\mapsto\psi[u]_\cdot(x)\) is continuous on \(\mathcal U_L\times\mathbb{R}^d\), in \(||\cdot||_\beta\) and hence in \(||\cdot||_\infty\) by Proposition 4. An alternative route to this would be to bake in control over the Lipschitz constant into the metric itself, and using the resulting topology. Either case, the informal idea is that small perturbations that don’t drastically change the Lipschitz bound only result in small changes in the solutions.

Proposition 8 (Stability in Initial Point and Field) Let \(u, v \in C([0, T]\times \mathbb{R}^d, \mathbb{R}^d)\) and \(v\) be \(L\)-Lipschitz. Consider solutions \(w(t), z(t)\) of Equation 1 for \(u, v\) on an interval \([0, T]\) such that \(y = z(0) \neq w(0) = y'\) satisfy the inequality: \[ |w(t) - z(t)| \leq e^{Lt}|y' - y| + \int_{0}^{t}{e^{L\tau}\alpha(t - \tau) d\tau} \] where \(\alpha(t) = |u_t(w(t)) - v_t(w(t))|\)

Proof. Apply fixed point identity to \(z, w\) and subtract two equations after expanding via Definition 2 to get: \[ \begin{aligned} |w(t) - z(t)| &\leq |y' - y| + \int_{0}^{t}{\alpha(\tau) + |v_\tau(w(\tau)) - v_\tau(z(\tau))| d\tau} \\ & \leq |y' - y| + \int_{0}^{t}{\alpha(\tau) + L|w(\tau) - z(\tau)| d\tau} \\ & = T[\alpha + Lx]_{|y'-y|}|w(t) - z(t)| \end{aligned} \] Let \(\beta\) be the fixed point of \(T[\alpha + Lx]_{y' - y}\), then it satisfies the following differential equation: \[ \dot{\beta} = \alpha + L \beta, \beta(0) = |y' - y| \] This ODE can be solved via an integrating factor of \(e^{-Lt}\), which gives: \[ \beta(t) = e^{Lt}|y' - y| + \int_{0}^{t}{e^{L(t - \tau)} \alpha(\tau) d\tau} \] Clearly, \(|w(t) - z(t)|\) is bounded above by the fixed point.

Lemma 4 (Reversal Identity) Define the reverse-flow velocity field at time \(T\), denoted \(\tilde{u}^{(T)}\), as \(\tilde{u}^{(T)}_t(x) \coloneq -u_{T-t}(x)\). We then have, for any \(t \in [0, T]\): \[ \psi[u]_t(x) = \psi[\tilde{u}^{(T)}]_{T - t}(\psi[u]_T(x)) \tag{8}\] In particular, \(\psi[u]_T\) is a \(C^r\)-diffeomorphism with inverse \(\psi[\tilde{u}^{(T)}]_T\).

Proof. Fix \(x\). Let \(g_x(t) = \psi[\tilde{u}^{(T)}]_t(x)\). By Equation 1: \[ \begin{aligned} \frac{d}{dt}g_x(T-t) &= -\frac{dg_x}{dt}\bigg|_{(T - t)} \\ &= - \tilde{u}^{(T)}_{t}(g_x(t))|_{T - t} \\ &= - \tilde{u}^{(T)}_{T - t}(g_x(T-t)) \\ &= u_t(g_x(T-t)) \end{aligned} \] Thus \(g_x(T-t)\) also satisfies the ODE from Equation 1, but not the initial value condition, since \(g_x(T-t)|_{t = 0} = \psi[\tilde{u}^{(T)}]_T(x)\). However, for fixed \(x\), the initial point of a flow \(\dot{\varphi} = u_t(\varphi(t))\) can be recovered by \(t \coloneq 0\), so, \(g_x(T-t)\) is indistinguishable from \(\psi[u]_{t}(\psi[\tilde{u}^{(T)}]_T(x))\) by Theorem 1. Since both \(t \to T - t\) and \(u \to \tilde{u}^{(T)}\) both involutions, swapping them leads to Equation 8. Take \(t \coloneq 0\), so Equation 8 gives: \(\psi[\tilde{u}^{(T)}]_T \circ \psi[u]_T = \text{id}\). Diffeomorphism follows from regularity of flow and sending \(u \to \tilde{u}^{(T)}\).

Theorem 2 (Local Flow around a Point) Let \(u\in C^r([0,1]\times\mathbb{R}^d;\mathbb{R}^d)\), \(r\ge1\), and fix \(x_0\in\mathbb{R}^d\) and \(R>0\). Then there are \(T_{x_0}\in(0,1]\), depending on \(x_0\) and \(R\), and a map \(\psi^{\mathrm{loc}}[u]:[0,T_{x_0})\times B(x_0,R)\to\mathbb{R}^d\) with:

1. Existence. For every \(x\in B(x_0,R)\), \(t\mapsto\psi^{\mathrm{loc}}[u]_t(x)\) solves Equation 1 on \([0,T_{x_0})\) and stays in \(B(x,R)\).
2. Uniqueness. If \(a<T_{x_0}\) and \(z\) solves Equation 1 on \([0,a]\) with \(z(0)=x\in B(x_0,R)\), then \(z=\psi^{\mathrm{loc}}[u]_\cdot(x)\) on \([0,a]\).
3. Regularity. \([0,T_{x_0})\times B(x_0,R)\) is open in \([0,1]\times\mathbb{R}^d\), and \(\psi^{\mathrm{loc}}[u]\in C^r([0,T_{x_0})\times B(x_0,R);\mathbb{R}^d)\).
4. Diffeomorphism. For every \(t<T_{x_0}\), \(\psi^{\mathrm{loc}}[u]_t\) is a \(C^r\)-diffeomorphism of \(B(x_0,R)\) onto an open subset of \(\mathbb{R}^d\).

Proof. For \(s>0\) put \(u^s_t(x)\coloneq b_s(x-x_0)\,u_t(x)\). Each \(u^s\) is \(C^r\) and agrees with \(u\) on \([0,1]\times B(x_0,s)\). Its derivative \(\partial_xu^s\) is continuous with compact support, hence bounded, so \(u^s\) is globally Lipschitz in \(x\) by Proposition 1. Let \(M\coloneq\max_{[0,1]\times\overline B(x_0,4R)}|u|\), so that \(|u^{2R}|\le M\) everywhere, put \(T_{x_0}\coloneq\min(1,R/M)\) (\(T_{x_0}\coloneq1\) if \(M=0\)), and let \(\psi^{\mathrm{loc}}[u]\) be the restriction of \(\psi[u^{2R}]\) (Theorem 1) to \([0,T_{x_0})\times B(x_0,R)\).

For \(t<T_{x_0}\), the fixed point identity gives \[ |\psi[u^{2R}]_t(x)-x|\le\int_0^t|u^{2R}_\tau(\psi[u^{2R}]_\tau(x))|\,d\tau\le Mt<R, \] so the trajectory stays in \(B(x,R)\subseteq B(x_0,2R)\), where \(u^{2R}=u\), and solves the true ODE. For any true solution \(z(t)\) starting from \(x\), compactness of \([0, a]\) ensures that we can choose \(R'\) such that \(z([0, a]) \subseteq B(x_0, R') \supseteq B(x_0, 2R)\). We know that our candidate \(\psi[u^{2R}]\) also solves Equation 1 with \(u^{R'}\) since \(R' \geq 2R\), remaining within \(B(x_0, 2R)\) on the interval \([0, a]\). But, applying Theorem 1 on \(u^{R'}\)-equation means that our two solutions must agree. Finally, \(\psi^{\mathrm{loc}}[u]\) inherits regularity from \(\psi[u^{2R}]\), and \([0,T_{x_0})\times B(x_0,R)\) is open in \([0,1]\times\mathbb{R}^d\). Diffeomorphism just follows from Lemma 4 and restriction.

Local Flow

Set \(x_0 \leftarrow 0\), and translate the apparatus of Theorem 2 to ensure equivalence.

Definition 3 (Maximal Flow) For \(R>0\) put: \[ \Omega_R\coloneq\bigl\{(t,x)\in[0,1]\times\mathbb{R}^d:\ |\psi[u^R]_\sigma(x)|<R\ \forall\sigma\in[0,t]\bigr\}, \qquad \Omega\coloneq\bigcup_{R>0}\Omega_R, \] and define \(\Psi[u]\coloneq\psi[u^R]\) on \(\Omega_R\). This is a well-defined construction, since the local flows are consistent \(\Omega_R \subseteq \Omega_{R'}, \psi[u^{R'}]|_{\Omega_R} = \psi[u^R]|_{\Omega_R}\) (by Theorem 2 uniqueness argument)

Proposition 9 (Maximal Domain is Open) \(\Omega\), as constructed in Definition 3, is open in \([0,1]\times \mathbb{R}^d\). Moreover, \(\{0\} \times \mathbb{R}^d \subseteq \Omega\).

Proof. Clearly, \((0, x)\) lies in some \(\Omega_R\) for each \(x\). If we show each \(\Omega_R\) to be open in \([0,1] \times \mathbb{R}^d\) then \(\Omega\) is open as a union of open sets. Substituting \(\sigma=st\) with \(s\in[0,1]\), \[ \Omega_R=\{(t,x):F(t,x)<R\},\qquad F(t,x)\coloneq\max_{s\in[0,1]}g(s,(t,x)),\qquad g(s,(t,x))\coloneq|\psi[u^R]_{st}(x)-x_0|. \] Take \(S \coloneq [0,1]\) and \(P\coloneq[0,1]\times\mathbb{R}^d\) with their subspace topologies, and apply Proposition 5 since \(g\) inherits continuity from \(\psi[u^R]\). Hence \(\Omega_R=F^{-1}\bigl((-\infty,R)\bigr)\) is open in \(P\).

Theorem 3 (Flow Local Existence) Let \(u\in C^r([0,1]\times\mathbb{R}^d,\mathbb{R}^d)\) with \(r\ge1\). Then Equation 1 has a unique solution \(\Psi\), defined on \(\Omega \supseteq \{0\}\times\mathbb{R}^d\) open in \([0,1]\times\mathbb{R}^d\) with \(\Psi\in C^r(\Omega,\mathbb{R}^d)\) such that:

1. Domain. Each slice \(\Omega_x\coloneq\{t:(t,x)\in\Omega\}\) is an interval containing \(0\).
2. Uniqueness and maximality. If \(z\) solves Equation 1 on \([0,a]\subseteq[0,1]\) with \(z(0)=x\), then \(a\in\Omega_x\) and \(z(t)=\Psi[u]_t(x)\) on \([0,a]\).
3. Diffeomorphism. For every \(t\in[0,1]\), \(\Psi[u]_t\) is a \(C^r\)-diffeomorphism of the open set \(U_t\coloneq\{x:(t,x)\in\Omega\}\) onto an open subset of \(\mathbb{R}^d\).

Proof. We can conclude that the slice \(\Omega_x\) is an interval containing 0, since \(\forall 0 \leq t' \leq t, (t, x) \in \Omega_R \implies (t', x) \in \Omega_R\). Uniqueness and maximality follow by construction and the fact that we may by compactness of \([0, a]\) choose \(R\) larger than the furthest distance so that \(z\) agrees with \(\psi[u^{R}]\), in turn agreeing with \(\Psi[u]\). \(U_t\) is open by lowering the arity of \(g\) in Proposition 9. Regularity of \(\Psi\) is inherited from local flows, since it is a local property. Diffeomorphism follows by observing that the proof of Lemma 4 can be transferred.

Remark. The hypothesis \(u\in C^r([0,1]\times\mathbb{R}^d;\mathbb{R}^d)\) is stronger than the proofs require. Lemma 2 uses only continuity of the field and the Lipschitz bound. Existence, uniqueness, continuous dependence and the construction of the maximal domain therefore hold for \(u\) continuous and Lipschitz in \(x\) on \([0,1]\times\overline B(0,s)\) for every \(s>0\). Moreover, \(x\mapsto\psi_t(x)\) is \(C^r\) as soon as \(\partial_x^\alpha u\) is jointly continuous for \(|\alpha|\le r\), and in that case \(\partial_x^\alpha\psi\) and \(\partial_t\partial_x^\alpha\psi\) are jointly continuous, while further regularity of \(\psi\) in \(t\) is inherited from that of \(u\).

Remark. Continuity in \(t\) can be weakened further. Suppose \(v\) is measurable in \(t\) and satisfies \(|v_t(x)-v_t(y)|\le L(t)|x-y|\) with \(L\) integrable on \([0,1]\). Replacing the weight \(e^{-\beta t}\) of Definition 1 by \(e^{-\beta\Lambda(t)}\), where \(\Lambda(t)\coloneq\int_0^tL(\tau)\,d\tau\), the argument of Lemma 2 gives a contraction with constant \(\kappa=1/\beta\), with solutions understood as continuous solutions of Equation 6. Existence and uniqueness then extend to fields with integrable singularities in \(t\), such as \(L(t)\sim(1-t)^{-1/2}\).

We shall now look at necessary and sufficient conditions for solutions upto to time \(t = 1\).

Theorem 4 (A priori bounds imply full-time solutions) Fix \(x \in \mathbb{R}^d\), \(\Psi[u]\) be a flow on velocity field \(u \in C^r\), then the following are equivalent:

1. \(\Omega_x = [0, 1]\)
2. There exists a constant \(K\) such that every solution \(z\) of Equation 1 on an interval \([0, a]\), with \(z(0) = x\) travels at most \(K\) distance from origin, i.e: \[ \max_{t \in [0, a]}{|z(t)|} \leq K \]
3. \(\Psi[u]_{\cdot}(x)\) is bounded on \(\Omega_x\).

Proof. For \((1)\implies(2)\), choose \(K\) as max over \([0,1]\) of \(\Psi[u](x)\) and apply uniqueness from Theorem 3. \((2)\implies(3)\) is straightforward, \(t \in \Omega_x \implies \forall t' \in [0, t] \max_{s \in [0, t']}{|z(s)|} \leq K\) In particular, \(|z(t)| \leq K\). From \((3)\) we have that \(\exists K \in \mathbb{R}^{+}, \Psi[u]_{\Omega_x}(x) \subseteq \overline{B}(0, K), a \coloneq \sup{\Omega_x}\). Since \([0,1] \times \overline{B}(0, K)\) is compact, there is a maximum speed \(u_{max}\), and \(\Psi[u](x)\) is Lipschitz (points \(\delta t\) apart in time can’t get farther than \(u_{max} \delta t\) away) on \([0, a)\), and the one-sided limit \(\lim_{t \to a^{-}}\Psi[u]_t(x)\) exists . Therefore, we can now extend the solution to \([0, a]\) and have \(|\Psi[u]_a(x)| \leq K\). At the same time, \((a, x) \in \Omega_{K+1} \subseteq \Omega \implies a \in \Omega_x\). But, \(\Omega_x\) is open in \([0, 1]\), so \(\Omega_x = [0, 1]\).

Theorem 5 (Global existence under one-sided growth) Let \(u \in C^r([0,1]\times\mathbb{R}^d;\mathbb{R}^d)\), \(r \geq 1\). Suppose there are constants \(R_0 \geq 0\) and \(C_0 \geq 0\) such that \[ y \cdot u_t(y) \leq C_0 |y|^2 \qquad \text{for all } t \in [0,1] \text{ and all } |y| \geq R_0. \tag{9}\] Then \(\Omega = [0,1]\times\mathbb{R}^d\). Moreover, with \(M_0 := \max_{[0,1]\times\overline{B}(0,R_0)} |u|\) and \(C := \max(R_0 M_0, C_0)\), \[ 1 + |\Psi[u]_t(x)|^2 \leq (1 + |x|^2)\, e^{2Ct} \qquad \text{for all } (t,x) \in [0,1]\times\mathbb{R}^d. \]

Proof. First, a one-sided bound on all of \(\mathbb{R}^d\). \(M_0\) is finite because \(u\) is continuous on a compact set. For \(|y| < R_0\) we have \(y \cdot u_t(y) \leq |y|\,|u_t(y)| \leq R_0 M_0\), and for \(|y| \geq R_0\) we use Equation 9. In both cases \[ y \cdot u_t(y) \leq C\,(1 + |y|^2) \qquad \text{for all } (t,y) \in [0,1]\times\mathbb{R}^d. \]

Now let \(z\) solve Equation 1 on \([0,a] \subseteq [0,1]\) with \(z(0) = x\). Then \(z\) is \(C^1\), so \(g := 1 + |z|^2\) is differentiable with \[ g' = 2\, z \cdot u_t(z) \leq 2C\,(1 + |z|^2) = 2C\, g. \] Hence \(\big(e^{-2Ct} g\big)' = e^{-2Ct}(g' - 2Cg) \leq 0\), and by the mean value theorem \(e^{-2Ct} g(t) \leq g(0)\), i.e. \[ 1 + |z(t)|^2 \leq (1 + |x|^2)\, e^{2Ct} \leq (1 + |x|^2)\, e^{2C} \qquad (t \in [0,a]). \] This is condition (2) of Theorem 4 with \(K := \sqrt{(1+|x|^2)\,e^{2C} -1}\), for every \(x\). So \(\Omega_x = [0,1]\) for every \(x\), i.e. \(\Omega = [0,1]\times\mathbb{R}^d\).

Corollary 1 (Linear growth) If \(|u_t(y)| \leq A + B|y|\) for all \((t,y)\), then \(\Omega = [0,1]\times\mathbb{R}^d\). In particular, this holds if \(u\) is globally \(L\)-Lipschitz in \(x\), with \(A := \max_{t} |u_t(0)|\) and \(B := L\).

Proof. For \(|y| \geq 1\), \(y \cdot u_t(y) \leq A|y| + B|y|^2 \leq (A+B)|y|^2\), so Equation 9 holds with \(R_0 = 1\) and \(C_0 = A + B\). In the Lipschitz case, \(|u_t(y)| \leq |u_t(0)| + L|y|\).

Remark. In this way, we recover the global case Theorem 1 from Theorem 3

Singular Endpoints

The results so far need \(u\) on the closed interval \([0,1]\), or at worst with an integrable singularity in \(t\). The marginal velocity fields of flow matching typically have neither. They have the form \[ u_t(y)=\frac{m_t(y)-y}{1-t},\qquad (t,y)\in[0,1)\times\mathbb{R}^d \tag{10}\] with \(|m_t(y)|\leq R\) (Remark 1). Since \(|u_t(y)-u_t(y')|\geq(|y-y'|-2R)/(1-t)\), the Lipschitz constant of \(u_t\) is at least \(1/(1-t)\), which is not integrable. Outside a fixed ball, however, such a field points inward, uniformly in \(t\). We show that this alone gives a flow on \([0,1)\) whose trajectories stay in a fixed ball. Lemma 5 specializes \(C_0=0\) in Equation 9, the bound of Theorem 5, whose replacing the rate by one that involves neither \(t\) nor \(u\).

Lemma 5 (Confinement) Let \(J\subseteq[0,1]\) be an interval containing \(0\), let \(R\geq0\), and let \(u:J\times\mathbb{R}^d\to\mathbb{R}^d\) satisfy \[ y\cdot u_t(y)\leq0\qquad\text{for all } t\in J\text{ and all } |y|\geq R. \tag{11}\] If \(z\) solves Equation 1 on \(J\) with \(z(0)=x\), then \(|z(t)|\leq\max(|x|,R)\) for all \(t\in J\).

Proof. First, observe that Equation 11 tells us \(|z|^2\) is decreasing whenever \(|z| \geq R\). We can show this by applying Equation 1: \[ \frac{d}{dt}|z(t)|^2 = 2z \cdot \dot{z} = 2z \cdot u_t(z) \leq 0 \] Put \(R' \coloneq max(|x|, R)\). Suppose \(z(t)\) is a solution for which \(|z(t_1)| > R'\) at some time \(t_1\). The set \(\{t \in [0, t_1]| |z(t)| \leq R'\}\) is closed and contains \(0\). Let \(t_0\) be its max. Since \(|z|^2\) is increasing at \(\xi \in (t_0, t_1)\) by mean value theorem with \(|z(\xi)| > R'\), Equation 11 doesn’t hold.

Theorem 6 (Flow on Field singular at endpoint) Let \(R\geq0\), \(r\geq1\), and let \(m:[0,1)\times\mathbb{R}^d\to\mathbb{R}^d\) such that \(|m_t(y)|\leq R\) for all \((t,y)\in[0,1)\times\mathbb{R}^d\), and \(m|_{[0, \tau]\times \mathbb{R}^d}\) is \(C^r\) for every \(\tau\in(0,1)\). Let \(u\) be given by Equation 10. Then there is a map \(\psi\in C^r([0,1)\times\mathbb{R}^d;\mathbb{R}^d)\) with:

1. Existence. For every \(x\), \(t\mapsto\psi_t(x)\) solves Equation 1 on \([0,1)\).
2. Uniqueness. If \(z\) solves Equation 1 on \([0,a]\) with \(a<1\) and \(z(0)=x\), then \(z=\psi_\cdot(x)\) on \([0,a]\).
3. Confinement. \(|\psi_t(x)|\leq\max(|x|,R)\) for all \((t,x)\in[0,1)\times\mathbb{R}^d\).
4. Diffeomorphism. For every \(t\in[0,1)\), \(\psi_t\) is a \(C^r\)-diffeomorphism of \(\mathbb{R}^d\) onto an open subset of \(\mathbb{R}^d\).

Proof. For \(|y|\geq R\), the we have the following bound on \(m\) via Cauchy-Schwarz: \[ y\cdot u_t(y)=\frac{y\cdot m_t(y)-|y|^2}{1-t}\leq\frac{|y|\,(R-|y|)}{1-t}\leq0, \] By Lemma 5 every solution \(z\) of Equation 1 on \([0,a]\subseteq[0,1)\) satisfies \(|z|\leq\max(|z(0)|,R)\). So all solutions satisfy \((3)\). Moreover, this bound satisfies Theorem 4, for any restriction of \(u\) to fixed \([0, \tau] \subset [0, 1]\). On \([0,\tau]\) the field \(u\) is \(C^r\), and the results of the previous section hold. So we have \(\Psi^\tau\) of Theorem 3 for \(u\) on \([0,\tau]\) defined on all of \([0,\tau]\times\mathbb{R}^d\). By its uniqueness, \(\Psi^{\tau'}\) extends \(\Psi^\tau\) when \(\tau<\tau'\), so the \(\Psi^\tau\) patch to a single map \(\psi\) on \([0,1)\times\mathbb{R}^d\). Every \([0,a]\) with \(a<1\) lies in \([0,\tau)\) for some \(\tau<1\), and \(\psi=\Psi^\tau\) there. So (1), (2) and the regularity of \(\psi\) are inherited from Theorem 3, and (4) holds because \(\psi_t=\Psi^\tau_t\) is defined on all of \(\mathbb{R}^d\).

Remark 1 (Flow matching fields). Let \(q\) be a probability density supported in \(\overline B(0,R)\), and let \(x_1\sim q\) and \(x_0\sim\mathcal N(0,I)\) be independent. For the path \(x_t=tx_1+(1-t)x_0\), the marginal velocity field is Equation 10 with \(m_t(y)=\mathbb{E}[x_1\mid x_t=y]\), by the \(x_1\)-prediction formula of 1 [§4.8]. This path satisfies Assumption 1 of 1, under which the marginal field is \(C^1\) on \([0,1)\times\mathbb{R}^d\) (Lipman et al. 2024, proof of Theorem 3), and hence so is \(m_t(y)=y+(1-t)\,u_t(y)\). Moreover, \(|m_t(y)|\leq R\), since \(m_t(y)\) is a conditional mean of \(x_1\). So Theorem 6 applies with \(r=1\).

Conclusion

We have given a self-contained proof of the flow local existence and uniqueness theorem, in the form in which it is cited in the flow matching literature (Lipman et al. 2024). The theorem and its ingredients are classical. The contribution of this article is expository: it organizes the proof around a single principle. For flow-based generative models, the practical question is whether the time-\(T\) map \(\Psi_T\), and hence the pushforward of the source distribution, is defined on all of \(\mathbb{R}^d\). The escape alternative (Theorem 4) reduces this question to an a priori bound on trajectories, and Theorem 5 provides such a bound under a one-sided growth condition that can be checked directly on the velocity field. The condition holds for fields of linear growth, and in particular for globally Lipschitz fields (Corollary 1).

Three directions extend naturally from this work. First, Theorem 6 stops short of \(t=1\). The time-one map of a flow matching model is the limit of \(\psi_t\) as \(t\to1\), and its existence, together with the pushforward identity at \(t=1\), requires a separate argument. Extending Theorem 6 to targets with unbounded support under moment assumptions would complement Gao, Huang, and Jiao (2024). Second, Lemma 1 applies equally with the field itself as the parameter. With explicit constants, this would relate the approximation error of a learned velocity field to the error in the generated distribution, in the spirit of Benton, Deligiannidis, and Doucet (2024). Third, the proof relies on a single abstract principle and few estimates, which makes it well suited to formal verification. Below local Lipschitz regularity uniqueness fails, and the fixed point approach no longer applies. That regime is addressed by the theory of DiPerna and Lions (1989) and Ambrosio (2004).

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