Linearized propagator bounds in the nonlinear control spaces

PDF p. 20: Duhamel formula for S and properties (i)–(ii)

Assume the Axiom of Choice and hypothesis (H): $n\in \{2,3\}$, $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ is connected, $\mathrm{\Theta }$ is a nonempty compact obstacle with smooth boundary, and its generalized reflected and glancing rays leave every compact set in finite forward time. Let $T>0$, $\lambda \in \mathbb{R}\setminus \{0\}$, and let $w$ satisfy (W): $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ and $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ distributionally. Here $H_D^s=D((I-{\mathrm{\Delta }}_D{)}^{s\mathrm{/}2})$ for $s\ge 0$, where ${\mathrm{\Delta }}_D$ is the Dirichlet Laplacian, and $W^{2,\mathrm{\infty }}$ consists of classes whose weak derivatives through order two are essentially bounded.

Put $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$, $V_2=\lambda w^2$ and $W(t)=e^{it{\mathrm{\Delta }}_D}$. Let $S(t,r)$ be the real-linear propagator for $i{z}_t+{\mathrm{\Delta }}_{D}z+V_{1}z+V_2\bar{z}=0$. For $I=[r,b]\subseteq [0,T]$, let $X_I$ consist of continuous $H_0^1(\mathrm{\Omega })$-valued paths admitting a compatible jointly measurable representative with finite scalar $L^4(I;L^{\mathrm{\infty }}(\mathrm{\Omega }))$ mixed norm. Equip it with the sum of that norm and the supremum inherited $H_0^1$ norm; $H_0^1$ is the $H^1$-closure of compactly supported smooth functions.

There is a constant $C$, depending on the fixed domain, $T$, $\lambda$ and $w$, such that for $s=1,2$, every $h\in H_D^s$ and every strongly measurable $f\in L^1(I;H_D^s)$, $\mathrm{\parallel }S(\cdot ,r)h{\mathrm{\parallel }}_{X_I^s}+{\parallel {\int }_r^{\cdot }S(\cdot ,\rho )f(\rho )d\rho \parallel }_{X_I^s}\le C(\mathrm{\parallel }h{\mathrm{\parallel }}_{H_D^s}+\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(I;H_D^s)}),$ where $X_I^1=X_I$ and $X_I^2=C(I;H_D^2)$ with its supremum spectral norm. The integrals are Bochner integrals in the corresponding $H_D^s$ space. The mixed norm is sectionwise and does not assert strong measurability into $L^{\mathrm{\infty }}(\mathrm{\Omega })$.

The homogeneous propagator also satisfies $S(t,r)h=W(t-r)h+i{\int }_r^{t}W(t-\rho )(V_1(\rho )S(\rho ,r)h+V_2(\rho )\overline{S(\rho ,r)h})d\rho \mathrm{.}$ In a forced equation, a factor $i$ belonging to the forcing remains inside $S(t,\rho )$, since $S$ is only real-linear.

The frozen article is cited for attribution of the transfer in the proof of Theorem 1.5, PDF page 20. The deduction above uses the named mathematical suppliers and does not use that passage as a premise proving itself.