Linear and nonlinear internally controlled exterior systems

PDF p. 4: (1.8)–(1.9); PDF p. 13: (3.16)–(3.17)

Assume the Axiom of Choice. Let $n\ge 2$, let $\mathrm{\Theta }\subset {\mathbb{R}}^n$ be nonempty and compact with smooth boundary, and let its exterior $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ be connected. Choose $R>0$ with $\mathrm{\Theta }\subset B_R(0)$ and a smooth real cutoff $\phi$ which is zero on $\mathrm{\mid }x\mathrm{\mid }\le R$, positive on $\mathrm{\mid }x\mathrm{\mid }>R$, and one on $\mathrm{\mid }x\mathrm{\mid }\ge R+1$. Put $\omega =\{x\in \mathrm{\Omega }:\mathrm{\mid }x\mathrm{\mid }>R\}$ and let $T>0$.

For a control $h$, write $Kh=\phi h$. For $m\in \{3,5\}$ and $\lambda \in \mathbb{R}\setminus \{0\}$, write $g(v)=\lambda \mathrm{\mid }v{\mathrm{\mid }}^{m-1}v$. The nonlinear internally controlled system with initial state $v_0$ is

$i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+g(v)=Kh\text{in }\mathrm{\Omega }\times (0,T),v{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0,v(0)=v_0\mathrm{.}$

The linear internally controlled system with initial state $z_0$ is

$i{\mathrm{\partial }}_{t}z+\mathrm{\Delta }z=\phi h\text{in }\mathrm{\Omega }\times (0,T),z{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0,z(0)=z_0\mathrm{.}$

Here $\mathrm{\Delta }$ is the spatial Laplacian. The case $m=3$ is the cubic system and $m=5$ is its quintic counterpart. The forcing is localized: whenever $h$ is locally integrable, $Kh$ vanishes almost everywhere outside $\omega$ and its space-time support relative to $\mathrm{\Omega }\times [0,T]$ lies in $\bar{\omega }\times [0,T]$. The control $h$ itself need not be localized.

These equations specify systems, not an existence or uniqueness assertion. In a distributional solution class the interior equation means equality on compactly supported smooth space-time test functions, provided every term, including $g(v)$, is defined as a distribution. Initial and boundary conditions must be interpreted in the trace or evolution sense appropriate to the chosen solution class; an interior distributional identity alone supplies neither trace. When states are specified in the Dirichlet scale, $H_D^{\sigma }(\mathrm{\Omega })$ denotes the spectral scale associated with $I-{\mathrm{\Delta }}_D$; in particular $H_D^1(\mathrm{\Omega })=H_0^1(\mathrm{\Omega })$ and $H_D^2(\mathrm{\Omega })=H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$.

Control region and nested real cutoffs

Dirichlet spectral Sobolev scale and duality

Distributional equation convention for complex Schrödinger solutions

The Axiom of Choice

The geometry, cutoff and region are supplied by Control region and nested real cutoffs. Since $\phi$ vanishes outside $\omega$, its product with a locally integrable control vanishes there; the closure in the support statement allows contact with $\mathrm{\mid }x\mathrm{\mid }=R$. The formulas for $K$ and $g$ specify pointwise multiplication wherever the stated functions are defined, without asserting that a nonlinear expression exists for an arbitrary distribution.

The distributional convention and its explicit limitation concerning initial and boundary traces are those of Distributional equation convention for complex Schrödinger solutions. The scale and the displayed identifications are those of Dirichlet spectral Sobolev scale and duality, under the Axiom of Choice recorded in The Axiom of Choice. The equations impose the indicated conditions once a solution class makes them meaningful; the definition does not assert that such a solution exists.

The article is cited for the controlled equations and notation in (1.8), (1.9), (3.16), and (3.17). The independent suppliers above fix the cutoff, scale, and distributional terminology; the article citation is attribution, not a proof of those prerequisites.