Abstract nonlinear control-space hypotheses

PDF p. 14: Theorem 3.7, assumptions (i)–(iii)

Assume the Axiom of Choice. Let $T>0$, $s\in [1,2]$ and $m\in \{3,5\}$. Let $\mathrm{\Omega }$ and $H_D^s(\mathrm{\Omega })$ be the exterior domain and Dirichlet spectral space, let $W(t)=e^{it{\mathrm{\Delta }}_D}$ be the free evolution, and let $g(z)=\lambda \mathrm{\mid }z{\mathrm{\mid }}^{m-1}z$ with $\lambda \in \mathbb{R}\setminus \{0\}$. Write $H=H_D^s(\mathrm{\Omega })$.

A control space satisfying the abstract perturbation hypotheses is a real or complex Banach space $X_T$ of $H$-valued continuous paths on $[0,T]$ whose inclusion into $C([0,T];H)$ is linear and satisfies $\mathrm{\parallel }u{\mathrm{\parallel }}_{C([0,T];H)}\le \mathrm{\parallel }u{\mathrm{\parallel }}_{X_T}$, and for which there is one constant $C_0\ge 1$ such that all of the following hold:

(i) For every $u_0\in H$, the path $t\mapsto W(t)u_0$ belongs to $X_T$ and $\mathrm{\parallel }W(\cdot )u_0{\mathrm{\parallel }}_{X_T}\le C_0\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_H$.

(ii) For every $f\in L^1(0,T;H)$, the retarded path $Df(t)={\int }_0^{t}W(t-r)f(r)dr$, with the integral understood in the Bochner sense in $H$, is defined for each $t\in [0,T]$, belongs to $X_T$, and satisfies $\mathrm{\parallel }Df{\mathrm{\parallel }}_{X_T}\le C_0\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(0,T;H)}$.

(iii) For every $v\in X_T$, the pointwise power expression $g(v)$ represents an element of $L^1(0,T;H)$, and for all $u,v\in X_T$,

$\mathrm{\parallel }g(v){\mathrm{\parallel }}_{L^1(0,T;H)}\le C_0\mathrm{\parallel }v{\mathrm{\parallel }}_{X_T}^m,\mathrm{\parallel }g(v)-g(u){\mathrm{\parallel }}_{L^1(0,T;H)}\le C_0(\mathrm{\parallel }v{\mathrm{\parallel }}_{X_T}^{m-1}+\mathrm{\parallel }u{\mathrm{\parallel }}_{X_T}^{m-1})\mathrm{\parallel }v-u{\mathrm{\parallel }}_{X_T}\mathrm{.}$

The same constant is required in all three conditions, although it may depend on the fixed domain, $T$, $s$, $m$ and $\lambda$. These are hypotheses on $X_T$; they do not assert that such a space exists or that any controlled equation is solvable.

Linear and nonlinear internally controlled exterior systems

Dirichlet spectral Sobolev scale and duality

Scale isometries and free Schrödinger evolution

Continuous and mixed time-space function spaces

Banach space

The Axiom of Choice

The domain, nonlinearity and parameter $\lambda$ are fixed by Linear and nonlinear internally controlled exterior systems, and the spectral space and its norm by Dirichlet spectral Sobolev scale and duality. The Axiom of Choice is retained from those inputs and the evolution supplier The Axiom of Choice.

A Banach space is a complete normed space over its stated scalar field Banach space. The inclusion condition therefore specifies the path represented by each element of $X_T$ and bounds every evaluation by its $X_T$ norm. The time-space conventions in Continuous and mixed time-space function spaces define norm-continuous paths, strongly measurable $L^1$ classes and Bochner integration. Condition (ii) explicitly requires the displayed integral and its $X_T$ membership; condition (iii) explicitly requires the nonlinear expression to have the stated strongly measurable integrable representative. Neither is inferred merely from pointwise notation.

The group and its strong continuity on $H$ are supplied by Scale isometries and free Schrödinger evolution, so the homogeneous path in (i) is meaningful. The three conditions prescribe bounds and membership rather than deriving them from the definitions. In particular, this definition alone does not claim that multiplication by $i$ preserves an arbitrary real $X_T$; an argument using that operation must supply the necessary property.

The article is cited for the formulation of assumptions (i)–(iii) in Theorem 3.7, PDF page 14. The cited local and library items provide the independent terminology; the article attribution is not used as a proof of a prerequisite or of the estimates.