Smooth flows with all parameter derivatives

Complete selected AN-03 programme proof, §§17.1–17.5 of Geometric and microlocal calculus. CC0 1.0. Original ownership is retained. See licence, rights, and selection history.

The scalar calculus and topology foundation, §§12–13, supplies finite-dimensional completeness, compactness, all finite-coordinate derivative rules, the oriented Riemann integral, exponential and uniform limits. The finite algebra foundation, §§10.1–10.6, supplies matrix operations, finite norm bounds and the injective/surjective equivalence. All arguments below retain the original norms and two-sided time interval.

Receiving detail: operator completeness. If is the complete continuous-path space proved in §17.1 and is Cauchy in operator norm, then is Cauchy for each . Define . Linearity passes to the limit. Boundedness of the sequence gives , and its uniform Cauchy bound gives

\|T-T_m\|\le\sup_{l\ge m}\|T_l-T_m\|\to0.

This proves the operator completeness used for the series in §17.2. This added detail is an AN-01 CC0 contribution to the selected AN-03 component.

17.1. The original integral equation and its complete local solution

Let be the original real finite-dimensional normed spaces, with their given norms. Let be open and let be , . Fix the original point . The equation and datum are

\partial_tu(t)=F(t,u(t),p),\qquad u(t_0)=x. \tag{NF1}

Choose positive such that the compact rectangle lies in . An open neighborhood contains a sufficiently small product of balls; their closures are compact in the original norms and can still be chosen inside that neighborhood. Let the actual bounds be

M=\sup_K\|F\|_E,\qquad L=\sup_K\|D_xF\|_{E\to E},\qquad Q=\sup_K\|D_pF\|_{P\to E}. \tag{NF2}

They are finite by continuity and compactness. Choose and with and . Such an exists also when either bound is zero; no division by a zero bound is used. Keep the original interval , the state ball, and , . On continuous paths in the original closed state ball set

(\Phi_{x,p}w)(t)=x+\int_{t_0}^{t}F(s,w(s),p)\,ds . \tag{NF3}

All integrals retain their original oriented endpoints. A finite-coordinate continuous integral exists coordinatewise by the proved Riemann integral. Its norm is at most the integral of the norm: the finite tagged sums have this bound by the triangle inequality; their vector limits and the scalar integral limit preserve it. The continuous-path space is complete in . Indeed a uniform Cauchy sequence converges pointwise by the original completeness of , its Cauchy estimates pass uniformly to the limit, and the triangle inequality with one continuous approximant proves continuity. The paths with values in the closed ball form a closed subset of that complete space.

NF2--NF3 show that stays within distance of . The full state-segment fundamental theorem gives inside the original convex ball. Thus . Starting with the actual constant path , define . For every and , retain

\begin{aligned} \|u_{m+1}-u_m\|_\infty &\leq\theta^m\|u_1-u_0\|_\infty,\qquad \|u_1-u_0\|_\infty\leq hM,\\ \|u_{m+k}-u_m\|_\infty &\leq\|u_1-u_0\|_\infty \sum_{\nu=m}^{m+k-1}\theta^\nu,\\ \|u-u_m\|_\infty &\leq\frac{\theta^m}{1-\theta}\|u_1-u_0\|_\infty . \end{aligned} \tag{NF4}

The finite sum proves the Cauchy property, completeness supplies , and continuity of gives NF3 with . The last line follows by taking the limit in the full finite-tail estimate. The fundamental theorem then proves NF1, including at . Two fixed points have distance at most times that distance and therefore coincide.

For another solution with the same datum, restrict to an interval on which both solutions lie in a common compact rectangle. On a sufficiently short subinterval beginning at any time at which they agree, the same contraction estimate forces agreement. Their equality set is closed by continuity and open by this two-sided local argument. On their connected common time interval containing the datum it is therefore the whole interval. This proves local uniqueness even for a solution not initially confined to the particular closed ball used to construct .

The original parameter differences satisfy

\|u(\,\cdot\,;x',p')-u(\,\cdot\,;x,p)\|_\infty \leq\frac{\|x'-x\|_E+hQ\|p'-p\|_P}{1-\theta}. \tag{NF5}

To prove it, subtract the two full equations NF3, use the state and parameter segment formulas with their original derivative bounds, and bring the term to the left. Convexity of the original parameter ball keeps every intermediate point in . Continuity in time and this uniform estimate give joint continuity in . The retained buffer to the boundary of the larger state ball permits the full parameter Taylor comparisons below.

17.2. Ordered linear transport on the original two-sided interval

Let be continuous and be continuous. For the same original , define

(\mathcal V_Az)(t)=\int_{t_0}^t A(s)z(s)\,ds,\qquad z=B+\mathcal V_Az. \tag{NF6}

For , nested integration proves . In detail the -fold application has the full integral of . Taking absolute values reverses the bounds when ; the resulting scalar nested integral of one is , by induction from the scalar power integral. Every oriented sign remains in the original operator integral. Norm completeness therefore supplies

\begin{aligned} \mathcal R_A&=\sum_{m=0}^{\infty}\mathcal V_A^m,& (I-\mathcal V_A)\mathcal R_A &=I=\mathcal R_A(I-\mathcal V_A),\\ z&=\mathcal R_AB,& \|\mathcal R_A\|&\leq \sum_{m=0}^{\infty}\frac{(hL_A)^m}{m!} =\exp(hL_A). \end{aligned} \tag{NF7}

The inverse products follow from both finite identities $(I-\mathcal V_A)\sum_{m=0}^N\mathcal V_A^m =I-\mathcal V_A^{N+1} =(\sum_{m=0}^N\mathcal V_A^m)(I-\mathcal V_A)$ and the factorial bound. This proves uniqueness as well as existence. When , the additional geometric bounds and hold with their complete factors. Neither estimate replaces the original operator or the factorial estimate.

For a matrix coefficient acting on a finite-dimensional original fiber, the fundamental matrix is

Y(t)=I_E+\sum_{m=1}^{\infty} \int_{t_0}^{t}ds_1\int_{t_0}^{s_1}ds_2 \cdots\int_{t_0}^{s_{m-1}}ds_m\, A(s_1)A(s_2)\cdots A(s_m). \tag{NF8}

The order is exactly the displayed one. Its factorial bound proves uniform convergence. Its integral equation and the fundamental theorem give , . Construct , , by the same integral argument, now on the original matrix space with right multiplication. Differentiating gives zero with the two full terms , so . Finite-dimensional injectivity and surjectivity imply as well. Thus this constructed is the actual inverse at every original time.

For , , both full maps are

z(t)=Y(t)\left(c+\int_{t_0}^{t}Y(s)^{-1}b(s)\,ds\right), \qquad Y(t)^{-1}z(t)=c+\int_{t_0}^{t}Y(s)^{-1}b(s)\,ds . \tag{NF9}

The product rule and the proved inverse equation verify both identities, including their initial values and multiplication order. Uniqueness follows from NF7. Arbitrary complex matrices are handled on their original real and imaginary coordinates, with the same complex matrix products; no diagonalization, self-adjointness or commutation is used.

17.3. All parameter derivatives of the linear inverse

Let range in an open finite-dimensional original normed parameter space. Suppose is into continuous matrix or operator paths in the original supremum norm. Integration in NF6 is a bounded linear map of , with norm at most , so is and every derivative retains that integral. Locally bounded gives locally bounded inverses by NF7. Their exact difference identity is

\mathcal R_{\eta'}-\mathcal R_\eta =\mathcal R_{\eta'}(\mathcal V_{\eta'}-\mathcal V_\eta) \mathcal R_\eta. \tag{NF10}

Multiplication on the left by and on the right by verifies the identity directly. It proves norm continuity. Insert the full differentiability remainder of into NF10; continuity and the local inverse bound make its remaining error . Consequently . This proof uses no Banach inverse-function theorem. Bounded operator multiplication has the required product rule: expand the actual two-factor increment, retaining the bilinear increment product whose norm is bounded by the product of its two increment norms. The remainder is therefore of second order. Induction gives its full higher product rule.

For every integer allowed by the parameter differentiability, the entire inverse derivative is

D^N\mathcal R[v_1,\ldots,v_N] =\sum_{k=1}^N \sum_{\substack{(I_1,\ldots,I_k)\ {\rm ordered}\\ I_j\ne\varnothing,\ I_1\sqcup\cdots\sqcup I_k=\{1,\ldots,N\}}} \mathcal R D_{I_1}\mathcal V\,\mathcal R\cdots D_{I_k}\mathcal V\,\mathcal R . \tag{NF11}

Here retains precisely the labeled directions in . A new differentiation either joins an existing derivative block or differentiates one inverse factor and inserts its new singleton block there. Every ordered partition of the enlarged label set has exactly one predecessor, determined by the new label's block. This proves NF11 and regularity by induction, with all repeated-direction multiplicities and every noncommuting factor retained. For the value is the full inverse .

If is in the continuous-path norm, NF7 and the full product rule therefore make in that same norm. For coefficient paths of the form , whose finite-coordinate derivatives are jointly continuous, their path-valued derivatives exist uniformly on compact parameter neighborhoods. The segment Taylor remainder is bounded by the uniform oscillation of the next continuous derivative on the compact time/parameter product, which tends to zero. This proves the asserted path-valued regularity, rather than merely pointwise differentiability.

17.4. The first actual nonlinear parameter derivative

Write , retaining its given product-space norm and both coordinate projections. For a direction , the candidate derivative solves

\begin{aligned} A_\eta(t)&=D_xF(t,u_\eta(t),p),\\ B_{\eta,v}(t)&=v_x+ \int_{t_0}^tD_pF(s,u_\eta(s),p)v_p\,ds,\\ z_v&=B_{\eta,v}+\mathcal V_{A_\eta}z_v =\mathcal R_{A_\eta}B_{\eta,v}. \end{aligned} \tag{NF12}

NF7 supplies this entire linear solution and uniqueness. It is linear in the original direction, and continuous as an operator from the original parameter norm to the path norm: keep the original projection norms , so . No original norm is replaced by a coordinate norm in this estimate.

To prove that it is the derivative, let and . The exact segment expansion of the original field is

\begin{aligned} &F(t,u_\eta+\Delta u,p+\Delta p)-F(t,u_\eta,p)\\ &\quad=A_\eta(t)\Delta u+ D_pF(t,u_\eta,p)\Delta p+\epsilon_\eta(t),\\ &\|\epsilon_\eta\|_\infty \leq\omega_\eta\bigl(\|\Delta u\|_\infty+\|\Delta p\|_P\bigr) \bigl(\|\Delta u\|_\infty+\|\Delta p\|_P\bigr), \qquad \omega_\eta(q)\longrightarrow0\quad(q\downarrow0). \end{aligned} \tag{NF13}

For the bound, subtract the derivatives at the start of each full state/parameter segment from their values along it and integrate. Uniform continuity on a compact rectangle containing these segments gives the displayed modulus. NF5, with both original projection factors, bounds the argument of the modulus by . Subtract NF12 for from the exact difference of NF3. Its residual is . NF7 bounds it by . This proves the full path-valued Fréchet derivative. Joint continuity of the field derivatives, NF5 and NF10 give continuity of the derivative operator. Thus is into continuous paths before any higher dependence is used.

17.5. Every nonlinear parameter derivative, including its original blocks

Assume is in the path norm, with . The path-valued maps and in NF12 are , because is and . To justify the composition statement in the path norm, apply the full finite-coordinate chain and product rules at each time. The derivatives are finite sums of continuous products of the original derivatives of and those of . All their Taylor remainder bounds are uniform on the compact rectangle, by the same segment and uniform-continuity argument as NF13. The bound remains uniform for in the unit ball of its original finite-dimensional direction space. NF10--NF11 then show that is as an operator-valued map. Hence is . Induction proves the full parameter theorem, and every order for .

Here is its exact higher equation. Retain labeled directions in the original parameter space, and put . For a nonempty block of labels use

D_B\zeta_\eta(t)= \begin{cases} (D u_\eta(t)[v_b],(v_b)_p),&B=\{b\},\\ (D^{|B|}u_\eta(t)[v_b:b\in B],0),&|B|\geq2. \end{cases} \tag{NF14}

For , the whole -th derivative equation is

\begin{aligned} D^Nu_\eta(t)[v_1,\ldots,v_N] &=\int_{t_0}^t A_\eta(s) D^Nu_\eta(s)[v_1,\ldots,v_N]\,ds\\ &\quad+\int_{t_0}^t \sum_{\substack{\Pi\in\mathfrak P(\{1,\ldots,N\})\\|\Pi|\geq2}} D_{(x,p)}^{|\Pi|}F(s,u_\eta(s),p) [D_B\zeta_\eta(s):B\in\Pi]\,ds . \end{aligned} \tag{NF15}

The initial -th derivative is zero because the original datum is linear in . The one-block term of the full chain rule is , since the parameter component of NF14 is zero for that block. Every other partition remains explicitly in NF15. Differentiating a block appends the new direction to that block; differentiating the outer derivative creates its singleton block. Every partition is obtained once, proving the full chain formula with no omitted multiplicity. The outer derivatives are symmetric multilinear maps on the original state/parameter product; the order of their arguments may be fixed by the least label in each block. Their values do not authorize commuting any linear transport factors. NF7 applied to the entire displayed inhomogeneous integral gives the actual unique derivative, including all pure-state, pure-parameter and mixed derivatives.

All derivatives just constructed are jointly continuous in time and parameters. NF1 gives the time derivative. More explicitly, the first time derivative of each parameter derivative of total order at most is its full chain-rule derivative of . Its right side is a finite continuous sum of the just proved parameter derivatives and the original field derivatives. Differentiating these identities in time gives all further mixed time/parameter derivatives of total order at most , inductively using the complete product and chain rules. At a given total order only field derivatives and solution derivatives of lower total order occur on the right of the time equation. Their continuity proves the next derivatives and permits equality of the mixed derivatives by the proved finite-coordinate calculus. The case of no time derivative and parameter derivatives was already constructed in the path norm. Thus is jointly , not only separately differentiable.