Resolvent-regularized adjoint and its forcing

PDF p. 17: notation after Lemma 4.1; PDF p. 18: (4.2)–(4.3)

Assume the Axiom of Choice and hypotheses (H) and (W). Thus $\mathrm{\Omega }={\mathbb{R}}^n\setminus \mathrm{\Theta }$ is the connected exterior of a nonempty compact smooth nontrapping obstacle, $T>0$, $\lambda \in \mathbb{R}\setminus \{0\}$, and $w\in C([0,T];H_D^2(\mathrm{\Omega })\cap W^{2,\mathrm{\infty }}(\mathrm{\Omega }))$ is the reference solution of $i{\mathrm{\partial }}_{t}w+\mathrm{\Delta }w+\lambda \mathrm{\mid }w{\mathrm{\mid }}^{2}w=0$ in distributions. Let ${\mathrm{\Delta }}_D$ be the Dirichlet Laplacian, $A=I-{\mathrm{\Delta }}_D$, and let $H_D^s(\mathrm{\Omega })$ be the spectral scale with norm $\mathrm{\parallel }{A}^{s\mathrm{/}2}\cdot {\mathrm{\parallel }}_{L^2}$ on its defining core. Put $R=A^{-1}$ and $W(t)=e^{it{\mathrm{\Delta }}_D}$.

Set $V_1=2\lambda \mathrm{\mid }w{\mathrm{\mid }}^2$ and $V_2=\lambda w^2$. Let $U(t)$ be the real-linear solution map at time $t$ for $i{\mathrm{\partial }}_{t}v+{\mathrm{\Delta }}_{D}v+V_{1}v-V_2\bar{v}=0$, with initial time zero. At negative orders the products and conjugation are their bounded scale extensions and the equation is understood through its mild formulation.

For $\sigma \in \{-1,-2\}$ put $\tau =\sigma +2$. Given $v_0\in H_D^{\sigma }(\mathrm{\Omega })$, define the regularized adjoint $y$ and its initial value by $v(t)=U(t)v_0$, $y(t)=Rv(t)$ and $y_0=R{v}_0$. Define its forcing by $f(t)=R(V_1(t)v(t)-V_2(t)\overline{v(t)})$. Then $y_0\in H_D^{\tau }$ and $\mathrm{\parallel }{y}_0{\mathrm{\parallel }}_{H_D^{\tau }}=\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^{\sigma }}$.

More generally, for $\theta \in [0,1]$ and $v_0\in H_D^{\theta -2}(\mathrm{\Omega })$, the same construction at order $\theta -2$ gives $y,f\in C([0,T];H_D^{\theta }(\mathrm{\Omega }))$ and

$y(t)=W(t)y_0+i{\int }_0^{t}W(t-r)f(r)dr\mathrm{.}$

The integral is a Bochner integral in $H_D^{\theta }$. There is a constant $C$ depending on the fixed data and $\theta$, but not on $v_0$ or $r\in [0,T]$, such that

$\mathrm{\parallel }f(r){\mathrm{\parallel }}_{H_D^{\theta }}\le C\mathrm{\parallel }v(r){\mathrm{\parallel }}_{H_D^{\theta -2}}\le C\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^{\theta -2}}=C\mathrm{\parallel }{y}_0{\mathrm{\parallel }}_{H_D^{\theta }},\mathrm{\parallel }y(r){\mathrm{\parallel }}_{H_D^{\theta }}\le C\mathrm{\parallel }{y}_0{\mathrm{\parallel }}_{H_D^{\theta }}\mathrm{.}$

The last equality uses the exact spectral norms. If the datum is also specified at an endpoint order $\sigma$, the constructions agree in the common lower-order space.

Exterior and truncated obstacle domains

Continuous and mixed time-space function spaces

Bochner-integrable function

Reference trajectory hypothesis W

Dirichlet spectral Sobolev scale and duality

Linearized equation and its real-duality adjoint

Forward and backward real-linear propagators

Scale isometries and free Schrödinger evolution

Time-dependent potential and conjugation bounds on the scale

Positive spectral Hilbert scale: duality and interpolation

The Axiom of Choice

Hypothesis (H), including the dimension and generalized-ray nontrapping convention, is fixed by Exterior and truncated obstacle domains. The reference trajectory and distributional meaning of (W) are fixed by Reference trajectory hypothesis W. The norm-continuous path and Lebesgue time-measure conventions are those of Continuous and mixed time-space function spaces, and Bochner integration means the norm limit of integrals of simple approximants in Bochner-integrable function. The scale, its norms and $R$ are fixed by Dirichlet spectral Sobolev scale and duality, and the potential formulas and adjoint-system signs are those of Linearized equation and its real-duality adjoint. The assumption recorded in The Axiom of Choice carries the choice hypotheses of the stated suppliers.

By Forward and backward real-linear propagators, for each $s\in [-2,2]$ the adjoint system has a unique mild solution $v\in C([0,T];H_D^s)$ and ${\sup }_t\mathrm{\parallel }v(t){\mathrm{\parallel }}_{H_D^s}\le C_s\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{H_D^s}$. By Time-dependent potential and conjugation bounds on the scale, $V_1$, $V_2$ and conjugation act boundedly on that scale, and the potential multipliers are norm-continuous in time. Consequently $V_{1}v-V_2\bar{v}$ is a continuous $H_D^s$-valued path. At negative orders these are the pivot-dual multiplier actions, so no pointwise product of an arbitrary negative-order element is implied.

Take $s=\theta -2$. The isometry $R:H_D^s\to H_D^{s+2}$ and the commutation $RW(t)=W(t)R$ follow from Scale isometries and free Schrödinger evolution. Put $g(r)=V_1(r)v(r)-V_2(r)\overline{v(r)}$, so $f=Rg$ is continuous in $H_D^{\theta }$. For fixed $t\in (0,T]$, unitarity and strong continuity of $W$, together with continuity of $g$ and $f$, make $r\mapsto W(t-r)g(r)$ and $r\mapsto W(t-r)f(r)$ continuous on $[0,t]$ in $H_D^s$ and $H_D^{\theta }$, respectively.

For either path $h$, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous gives uniform continuity on this compact interval. On the partition into $2^m$ equal subintervals take the left endpoint values, and take $h(t)$ at the final endpoint, obtaining measurable finite-range simple paths $h_m$ with ${\sup }_{r\in [0,t]}\mathrm{\parallel }{h}_m(r)-h(r)\mathrm{\parallel }\to 0$. This proves strong measurability in the convention of Continuous and mixed time-space function spaces. That supplier also gives boundedness of a continuous path on the interval, so ${\int }_0^t\mathrm{\parallel }h(r)\mathrm{\parallel }dr\le t{\sup }_{r\in [0,t]}\mathrm{\parallel }h(r)\mathrm{\parallel }<\mathrm{\infty }$. The criterion Bochner integrability criterion therefore makes both integrands Bochner integrable in their respective Banach scale spaces. At $t=0$ the integral over the zero-measure interval is zero.

The bounded complex-linear isometry $R$ can now be moved through the mild-equation integral by Bounded linear maps commute with Bochner integration; its commutation with $W$ gives the displayed Duhamel identity. The isometry and multiplier bounds give $\mathrm{\parallel }f(r){\mathrm{\parallel }}_{H_D^{\theta }}\le C\mathrm{\parallel }v(r){\mathrm{\parallel }}_{H_D^{\theta -2}}$; the uniform propagator bound and the same isometry give the remaining inequalities and the equality of initial norms. Taking $s=\sigma$ also gives the endpoint identity with $\tau =\sigma +2$.

The compatible scale embeddings supplied by Positive spectral Hilbert scale: duality and interpolation place a common datum and its higher-order mild solution in the lower-order scale. The consistent free flow and multiplier extensions give the same lower-order mild equation there; uniqueness in Forward and backward real-linear propagators identifies the two solutions. Thus the endpoint and intermediate-order constructions agree whenever both are applicable.

The original article supplies the notation and assertions on PDF pages 17–18. It is cited for attribution; the justification uses the mathematical suppliers listed above.