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We investigate the Lax equation in the context of infinite-dimensional Liealgebras. Explicit solutions are discussed in the sequentially completeasymptotic estimate context, and an integral expansion (sums of iteratedRiemann integrals over nested commutators with correction term) is derived forthe situation that the Lie algebra is inherited by an infinite-dimensional Liegroup in Milnor's sense. In the context of Banach Lie groups (and Lie groupswith suitable regularity properties), we generalize theBaker-Campbell-Dynkin-Hausdorff formula to the product integral (withadditional nilpotency assumption in the non-Banach case). We combine thisformula with the results obtained for the Lax equation to derive an explicitrepresentation of the product integral in terms of the exponential map. Animportant ingredient in the non-Banach case is an integral transformation thatwe introduce. This transformation maps continuous Lie algebra-valued curves tosmooth ones and leaves the product integral invariant. This transformation isalso used to prove a regularity statement in the asymptotic estimate context.

The Lax Equation and Weak Regularity of Asymptotic Estimate Lie Groups