5 The superposition principle: weak \(\Rightarrow \) Lagrangian
Uniqueness in Theorem 1.3 is uniqueness among Lagrangian solutions. Ruling out phantom weak solutions — distributional solutions not transported by any flow — is the superposition principle, proved here on a short window at the cost of one degree of regularity (Assumption 2). The argument freezes the field at the solution’s own marginal, establishes the \(C^1\) dependence of the flow on its initial point via the variational equation, enlarges the test class from \(C^\infty _c\) to \(C^1_c\) to admit transported tests, and closes with a diagonal chain rule: differentiating \(\int \psi _s\, \mathrm{d}f_s\) through both moving arguments, the transport identity and the weak PDE cancel exactly.
In addition to Assumption 1, let \(\nabla W \in C^1\) (equivalently \(W \in C^2\)). One degree of regularity beyond Assumption 1, spent to make the phase-space field \(C^1\) in space — exactly what the variational equation for the flow’s derivative in its initial point requires. Only the superposition principle carries this assumption.
The canonical Dyson-series solution \(M(t)\) of the matrix variational equation \(\dot M = A(s)\, M\), \(M(0) = I\), continuous jointly in time and the parameter. It provides the candidate derivative of the flow with respect to its initial point — constructed explicitly rather than chosen, so that its regularity in the parameter is provable.
The flow’s Fréchet derivative in its initial point is the Dyson-series fundamental matrix \(M(t)\) evaluated along the trajectory.
Both the trajectory difference \(u_h(s)=\Phi _s(z+h)-\Phi _s(z)\) and its linearization \(M(s)h\) solve the linear ODE \(\dot w=A(s)w\) up to the first-order Taylor remainder of the field, which is \(o(\| h\| )\) uniformly over the compact flow image; a Gronwall estimate on approximate trajectories bounds their gap, giving \(\Phi _t(z+h)-\Phi _t(z)-M(t)h=o(\| h\| )\).
The map \(z \mapsto \Phi _t(z)\) is Fréchet differentiable in the initial point, with derivative solving the linearized (variational) equation \(\dot M = (D_z b)\, M\), \(M(0)=I\), and depending continuously on \(z\). The research-grade core of the bridge.
Construct the fundamental matrix explicitly as a Dyson series and prove its joint \((t,z)\)-continuity by a parameter Weierstrass \(M\)-test; the difference-quotient Gronwall bound then identifies \(M(t)\) as the Fréchet derivative of \(z\mapsto \Phi _t(z)\).
The two-time flow \((s,z) \mapsto \Phi _{s \to t}(z)\) and its inverse are jointly \(C^1\) on the window — the jointly-smooth change of variables used to transport test functions along characteristics.
A lower Gronwall bound makes \(\Phi _t\) antilipschitz, hence injective with closed range; the inverse function theorem gives open range, so by connectedness \(\Phi _t\) is a \(C^1\) diffeomorphism. Applying the inverse function theorem to the space-time chart \((s,z)\mapsto (s,\Phi _s z)\) — block-triangular, invertible derivative — makes the inverse jointly \(C^1\), hence so is \(\Phi _{s\to t}=\Phi _t\circ \Phi _s^{-1}\).
The weak-solution test class is enlarged from \(C_c^\infty \) to \(C^1_c\). Needed because the transported test \(\psi _s = \varphi \circ \Phi _{s \to t}\) is only \(C^1_c\) — the flow is once, not infinitely, differentiable in space.
Mollify the \(C^1_c\) test \(\chi \) by a shrinking smooth bump, \(\chi _n=\rho _n\star \chi \), and apply the \(C^\infty _c\) weak equation to each. Pass to the limit using \(\chi _n\to \chi \) and the uniform convergence \(\nabla \chi _n\to \nabla \chi \) (the analytic core of the step), the field bound providing a uniform dominating function.
The dual transport identity: the transported test \(\psi _s = \varphi \circ \Phi _{s \to t}\) satisfies \(\partial _s \psi _s + D\psi _s \cdot b_s = 0\) along the flow — the dual of the pushforward chain rule.
Set \(z_0=\Psi _s w\), so \(w=\Phi _s z_0\). Then \(\psi _{s'}(\Phi _{s'}z_0)=\varphi (\Phi _t z_0)\) is constant in \(s'\); differentiating this composite at \(s'=s\) and splitting by the chain rule gives \(\partial _s\psi _s(w)+(D\psi _s(w))\, b_s(w)=0\).
\(\dfrac {\mathrm{d}}{\mathrm{d}s} \int \psi _s\, \mathrm{d}f_s = 0\) for \(s \in (0,T)\): differentiating through both the moving test function and the moving measure, the transport identity and the weak PDE cancel exactly.
Split \(I(\sigma )=\int \psi _\sigma \, \mathrm{d}f_\sigma =B(\sigma )+q(\sigma )\), with \(B\) the fixed-integrand part — its derivative \(V_b\) supplied by the weak equation on the enlarged \(C^1_c\) test class — and \(q(\sigma )=\int (\psi _\sigma -\psi _s)\, \mathrm{d}f_\sigma \). A little-\(o\) argument gives \(q'(s)=-V_b\): the remainder splits into a uniform-differentiability term (Heine–Cantor over the fixed compact carrying the moving supports) and a narrow-continuity term. The transport identity forces \(V_b=-\int \partial _s\psi _s\, \mathrm{d}f_s\), so the two contributions cancel and \(I'(s)=0\).
\(\int \varphi \, \mathrm{d}f_T = \int \varphi \, \mathrm{d}g_T\) for every \(C_c^\infty \) test function \(\varphi \), where \(g = (\Phi )_\# f_0\) is the pushforward of the initial datum along the frozen-field flow. Hence \(f_T = g_T\) by measure extensionality.
By the diagonal chain rule the map \(s\mapsto \int \psi _s\, \mathrm{d}f_s\) has zero derivative on \((0,T)\), hence is constant. Evaluating at the endpoints — \(\psi _T=\varphi \) and \(\psi _0=\varphi \circ \Phi _T\) — gives \(\int \varphi \, \mathrm{d}f_T=\int \varphi \circ \Phi _T\, \mathrm{d}f_0 =\int \varphi \, \mathrm{d}g_T\). Ranging over \(\varphi \in C_c^\infty \) and measure extensionality give \(f_T=g_T\).
Let \(W\) satisfy Assumption 2 and let \(T {\gt} 0\) satisfy \(L\, T^2 {\lt} 1\). Every weak solution on \([0,T]\) whose first moments are uniformly bounded on the window and whose force field is regular — \((\nabla W * \rho _t)(x)\) continuous in \(t\) for each \(x\), with jointly continuous spatial derivative — is Lagrangian on \([0,T]\): it coincides with the pushforward of its initial datum along the characteristic flow it generates. Weak solutions cannot branch away from the Lagrangian one on the window.
Freeze the field at \(\rho ^f\) and build its characteristic flow \(\Phi \); the pushforward \(g=(\Phi _t)_\# f_0\) is Lagrangian by construction and solves the same frozen linear equation. The dual core shows \(f_T=g_T\) for every window endpoint, so \(f\) coincides with its own characteristic pushforward and is therefore Lagrangian.