Duhamel formula for a real-linear propagator

PDF p. 20: paragraph after (4.6), real-linearity qualification and Duhamel formulas

Assume the Axiom of Choice and hypotheses (H) and (W). In particular, $\mathrm{\Omega }$ is the prescribed smooth connected exterior domain, $T>0$, $\lambda \in \mathbb{R}\setminus \{0\}$, and the reference trajectory $w$ belongs to $C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$. Let $\theta \in [-2,2]$, put $E=H_D^{\theta }(\mathrm{\Omega })$, and let $S(t,r)$ be the real-linear evolution for $i{z}_t+{\mathrm{\Delta }}_{D}z+V_{1}z+V_2\bar{z}=0$, where $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ and $V_2=\lambda w^2$. Real-linear means linear over real scalars; complex linearity is not assumed. Choose $R>0$ with $\mathrm{\Theta }\subset B_R(0)$ and a smooth real cutoff $\phi :{\mathbb{R}}^n\to [0,1]$ which is zero for $\mathrm{\mid }x\mathrm{\mid }\le R$, positive for $\mathrm{\mid }x\mathrm{\mid }>R$, and one for $\mathrm{\mid }x\mathrm{\mid }\ge R+1$.

Use the following precise convention for a locally integrable spatial function $q$. At $\theta \ge 0$, its regular scale realization ${\mathcal{R}}_{\theta }q$ exists when its almost-everywhere class belongs to $H_D^{\theta }\subset L^2$, and is that class. At $\theta =-s<0$, it exists when $q\bar{a}\in L^1(\mathrm{\Omega })$ for every $a\in H_D^s$ and there is a finite constant $C_q$ such that $\mathrm{\mid }{\int }_{\mathrm{\Omega }}q\bar{a}dx\mathrm{\mid }\le C_q\mathrm{\parallel }a{\mathrm{\parallel }}_{H_D^s}$ for all such $a$; it is then the element of $H_D^{-s}$ represented by this bounded conjugate-linear functional. This convention specifies the entire pivot functional at negative orders.

Let $z_0\in E$ and $z\in C([0,T];E)$. Put $Q=(0,T)\times \mathrm{\Omega }$ and fix a jointly measurable representative $\tilde{w}$ of the reference trajectory. For the nonlinear assertion, suppose there are jointly measurable complex-valued functions $Z,h$ on $Q$ with $Z(t,\cdot )\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^3(\mathrm{\Omega })$ and $h(t,\cdot )\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^1(\mathrm{\Omega })$ for almost every $t$, such that ${\mathcal{R}}_{\theta }Z(t,\cdot )=z(t)$ almost everywhere in time. Form the pointwise functions

$q_N=\lambda (2\tilde{w}\mathrm{\mid }Z{\mathrm{\mid }}^2+\overline{\stackrel{~}{w}}{Z}^2+\mathrm{\mid }Z{\mathrm{\mid }}^{2}Z),q_g=\phi h\mathrm{.}$

Require their regular scale realizations $N(t)={\mathcal{R}}_{\theta }{q}_N(t,\cdot )$ and $g(t)={\mathcal{R}}_{\theta }{q}_g(t,\cdot )$ to exist almost everywhere and to be strongly measurable $E$-valued functions with ${\int }_0^T(\mathrm{\parallel }N(t){\mathrm{\parallel }}_E+\mathrm{\parallel }g(t){\mathrm{\parallel }}_E)dt<\mathrm{\infty }$. In the mild equation and formula below write $F(w,z)=N$ and $\phi h=g$. Thus the nonlinear expression is constructed on the stated representative class, rather than on arbitrary negative-order elements of $E$.

Then $z$ satisfies the free-flow mild formulation of $i{z}_t+{\mathrm{\Delta }}_{D}z+V_{1}z+V_2\bar{z}+F(w,z)=\phi h$, with initial datum $z_0$, if and only if, for every $t\in [0,T]$,

$z(t)=S(t,0)z_0+{\int }_0^{t}S(t,r)(iF(w,z)(r))dr+{\int }_0^{t}S(t,r)(-i\phi h(r))dr\mathrm{.}$

The integrals are Bochner integrals in the realification of $E$. For the linearized controlled equation the same formula holds with the $F$ integral omitted. More generally this linear assertion holds for every $z_0\in E$ and every strongly measurable $g\in L^1(0,T;E)$ in place of $\phi h$, without a function-representative assumption on the state or forcing. A real-linear propagator need not commute with multiplication by $i$, so these factors are retained inside $S(t,r)$.

Exterior and truncated obstacle domains

Reference trajectory hypothesis W

Dirichlet spectral Sobolev scale and duality

Linearized equation and its real-duality adjoint

Linear and nonlinear internally controlled exterior systems

Control region and nested real cutoffs

Quadratic-cubic remainder after subtracting a trajectory

Forward and backward real-linear propagators

The Axiom of Choice

Strongly measurable Banach-valued function

Bochner-integrable function

The original article, frozen v2 PDF page 20 immediately after (4.6), supplies the attribution and the nonlinear and linearized formulas. Its image and corresponding extracted text were read for this attribution. The explicit regular-realization convention repairs the generated proposition's nonlinear domain; it is not an assertion that the authors defined a cubic operation on every negative-order distribution. The proof uses the named evolution, spectral-scale and Bochner suppliers, not the article as a proof premise. The conditions specify when the pointwise nonlinear forcing and the integrals are defined; no independent nonlinear existence or trace result is asserted here.