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For $s\ge 0$ and $\xi \in \mathbb{R}$, define the characteristic curve $X(t;s,\xi )$ byThe bounded derivative $F_x$ makes $F(t,\cdot )$ globally Lipschitz, uniformly in $t$. On each finite time interval, $|F(t,x)|\le |F(t,0)|+L|x|$, so Gronwall inequality prevents finite-time escape. The Picard-Lindelof theorem therefore gives a unique trajectory for every finite $t\ge 0$. Differentiation in $\xi$ givesso the characteristic flow map is a $C^1$ increasing diffeomorphism.

Along a characteristic, the chain rule changes the equation intoTracing $(t,x)$ backward by the flow therefore givesThe regularity of the flow makes this a $C^1$ classical solution. Conversely, every classical solution obeys the same ordinary differential equation along every characteristic, so the formula also proves uniqueness.

On the finite-dimensional space $H_m$, the Galerkin equation is an autonomous system of ordinary differential equations whose right-hand side is a quadratic polynomial in the coefficients of $u_m$. It is locally Lipschitz, so the Picard-Lindelof theorem gives a unique maximal local solution.

Taking the $H$ inner product with $u_m$ givesThe advecting field $P_N{u}_m$ is divergence free. Periodicity and the skew-symmetry of incompressible transport therefore make the nonlinear term zero. HenceA finite-dimensional solution can cease to exist only if its norm diverges. This uniform bound prevents such blow-up, so the solution extends uniquely through every interval $[0,T]$.

The first equation determines $u_m$ linearly from ${\theta }_m$:Substitution into the second equation gives an ordinary differential equation on the finite-dimensional space ${\tilde{H}}_m$. Its right-hand side is polynomial, and therefore locally Lipschitz continuous. The Picard-Lindelof theorem supplies a unique local solution. The energy estimate in part (iii) bounds ${\theta }_m$ on every finite time interval, so the finite-dimensional continuation criterion rules out finite-time escape. The solution is consequently unique on every interval $[0,T]$.