Lemma 4.4: compactness for regularized perturbed trajectories

Put $a_j=v_{j,0}-v_0$ and $d_j(t)=RU(t)a_j$. The data $a_j$ are weakly null and norm bounded in $H_D^{\sigma }$ by [F5]. If $\sigma =-1$, the continuous inclusion $H_D^{-1}\hookrightarrow H_D^{-2}$ also makes them weakly null and bounded in $H_D^{-2}$, by composition of bounded functionals with the inclusion; the assertion is immediate when $\sigma =-2$. Set $M={\sup }_j\mathrm{\parallel }{a}_j{\mathrm{\parallel }}_{H_D^{-2}}<\mathrm{\infty }$. The real-linearity of $U(t)$ gives $d_j=y_j-y$.

Put $Y=(H_0^2(G){)}^{\mathrm{\prime }}$ and denote the continuous pivot from [F3] by $J:L^2(G)\to Y$. For $z\in L^2(\mathrm{\Omega })$ and $\zeta \in C_c^{\mathrm{\infty }}(G)$ let $b_{\zeta }$ be the zero extension of $\psi \zeta$ to $\mathrm{\Omega }$ and define $(D_{\psi }z)(\zeta )=i⟨z,\mathrm{\Delta }{b}_{\zeta }{⟩}_2$. The product rule and bounded derivatives of $\psi$ give $\mathrm{\parallel }\mathrm{\Delta }{b}_{\zeta }{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}\le C_{\psi }\mathrm{\parallel }\zeta {\mathrm{\parallel }}_{H^2(G)}$; the zero extension introduces no boundary term since $\zeta$ is compactly supported in $G$. Cauchy–Schwarz gives $\mathrm{\mid }(D_{\psi }z)(\zeta )\mathrm{\mid }\le C_{\psi }\mathrm{\parallel }z{\mathrm{\parallel }}_2\mathrm{\parallel }\zeta {\mathrm{\parallel }}_{H^2(G)}$. Density of $C_c^{\mathrm{\infty }}(G)$ in $H_0^2(G)$ therefore extends this conjugate-linear functional uniquely and defines a bounded complex-linear map $D_{\psi }:L^2(\mathrm{\Omega })\to Y$.

Applying [F3] with $\theta =0$ and restricting to $G$ gives a uniform bound on $q_j=(\psi d_j){\mathrm{\mid }}_G$ in $L^2(0,T;H^{1\mathrm{/}2}(G))$. The regularized mild equation gives $d_j(t)=W(t)R{a}_j+i{p}_j(t)$, where $p_j(t)={\int }_0^{t}W(t-r)f_j(r)dr$ and $d_j,f_j\in C([0,T];L^2(\mathrm{\Omega }))$ satisfy ${\sup }_{j,t}(\mathrm{\parallel }{d}_j(t){\mathrm{\parallel }}_2+\mathrm{\parallel }{f}_j(t){\mathrm{\parallel }}_2)\le CM$. Here $W(t)=e^{it{\mathrm{\Delta }}_D}$.

For arbitrary $z\in L^2(\mathrm{\Omega })$ and $\zeta \in C_c^{\mathrm{\infty }}(G)$, $b_{\zeta }\in C_c^{\mathrm{\infty }}(\mathrm{\Omega })\subset D({\mathrm{\Delta }}_D)$. Unitarity gives $⟨W(t)z,b_{\zeta }{⟩}_2=⟨z,W(-t)b_{\zeta }{⟩}_2$. The right side is differentiable by the domain-valued flow derivative in [F4]. Since the second slot is conjugate-linear, its derivative is $i⟨W(t)z,\mathrm{\Delta }{b}_{\zeta }{⟩}_2=(D_{\psi }W(t)z)(\zeta )$. Both $J((\psi W(t)z){\mathrm{\mid }}_G)$ and $D_{\psi }W(t)z$ are norm-continuous $Y$-valued maps, hence their products with a smooth compactly supported scalar time test or its derivative are Bochner integrable. Scalar integration by parts and commutation of evaluation $F\mapsto F(\zeta )$ with these integrals yield the $Y$-valued identity ${\int }_0^{T}J((\psi W(t)z){\mathrm{\mid }}_G){\phi }^{\mathrm{\prime }}(t)dt=-{\int }_0^T{D}_{\psi }W(t)z\phi (t)dt$: the two sides agree on all smooth $\zeta$ and hence on $H_0^2(G)$ by density. This proves the free identity for every $L^2$ datum without assuming it has a boundary trace or a strong time derivative in $L^2$.

Fix $\zeta \in C_c^{\mathrm{\infty }}(G)$ and set $K_j(t,r)=⟨W(t-r)f_j(r),b_{\zeta }{⟩}_2$. By the same adjoint calculation, ${\mathrm{\partial }}_t{K}_j(t,r)=i⟨W(t-r)f_j(r),\mathrm{\Delta }{b}_{\zeta }{⟩}_2$. Both $K_j$ and this derivative are continuous in $(t,r)$ on the compact time square, and $\mathrm{\mid }{\mathrm{\partial }}_t{K}_j(t,r)\mathrm{\mid }\le {\sup }_r\mathrm{\parallel }{f}_j(r){\mathrm{\parallel }}_2\mathrm{\parallel }\mathrm{\Delta }{b}_{\zeta }{\mathrm{\parallel }}_2$. Moving the bounded scalar pairing through the Bochner integral gives $⟨p_j(t),b_{\zeta }{⟩}_2={\int }_0^t{K}_j(t,r)dr$. Taking a difference quotient, the added or removed endpoint interval tends to $K_j(t,t)=⟨f_j(t),b_{\zeta }{⟩}_2$ by continuity, while the quotient on the common interval tends to ${\int }_0^t{\mathrm{\partial }}_t{K}_j(t,r)dr$ by dominated convergence and the displayed derivative bound. Thus its derivative is $⟨f_j(t),b_{\zeta }{⟩}_2+i{\int }_0^t⟨W(t-r)f_j(r),\mathrm{\Delta }{b}_{\zeta }{⟩}_{2}dr$. Combining this with the homogeneous calculation and $d_j=W(t)R{a}_j+i{p}_j$ gives $\frac{d}{dt}⟨d_j(t),b_{\zeta }{⟩}_2=i⟨d_j(t),\mathrm{\Delta }{b}_{\zeta }{⟩}_2+i⟨f_j(t),b_{\zeta }{⟩}_2$.

For each fixed $t$, the map $a\mapsto \psi RU(t)a$ is bounded and real-linear from $H_D^{-2}$ to $L^2(\mathrm{\Omega })$. Weak convergence in the complex space implies weak convergence of its realification: any bounded real-linear functional is the real part of a bounded complex-linear functional. Thus $\psi d_j(t)$ converges weakly to zero in the realification of $L^2(\mathrm{\Omega })$, and in particular its complex pairing with any fixed $L^2$ vector tends to zero. The uniform $L^2$ bound in step 2.1 gives $\mathrm{\mid }⟨\psi d_j(t),g(t)⟩\mathrm{\mid }\le CM\mathrm{\parallel }g(t){\mathrm{\parallel }}_2$ for $g\in L^2(0,T;L^2(\mathrm{\Omega }))$. The scalar pairing is measurable, and the right side is integrable on the finite interval by Cauchy–Schwarz. Dominated convergence therefore gives ${\int }_0^T⟨\psi d_j(t),g(t)⟩dt\to 0$.

Define $g_j(t)=D_{\psi }{d}_j(t)+iJ((\psi f_j(t)){\mathrm{\mid }}_G)$. Boundedness of $D_{\psi }$, $J$ and cutoff restriction makes $J{q}_j$ and $g_j$ norm-continuous $Y$-valued maps; step 2.1 gives ${\sup }_{j,t}\mathrm{\parallel }{g}_j(t){\mathrm{\parallel }}_Y\le CM$. Multiplying the scalar derivative identity in step 3.1 by any $\phi \in C_c^{\mathrm{\infty }}((0,T);\mathbb{C})$ and integrating by parts gives, after evaluation on $\zeta$, ${\int }_0^{T}J{q}_j(t){\phi }^{\mathrm{\prime }}(t)dt=-{\int }_0^T{g}_j(t)\phi (t)dt$. All integrals exist by [F6], evaluations commute with them, and density of the smooth spatial tests proves equality in $Y$. This is precisely the derivative convention in [F6], with no conjugation of $\phi$. Consequently ${\mathrm{\partial }}_t(J{q}_j)=g_j$ is bounded in $L^{\mathrm{\infty }}(0,T;Y)$ and in $L^1(0,T;Y)$, with the latter norm at most $TCM$.

The bounds in steps 2.1 and 4.1 and the compact triple in [F3] meet all hypotheses of [F5], so $(q_j)$ is relatively compact in $L^2(0,T;L^2(G))$. Since $\psi$ is supported in $G$ relative to $\mathrm{\Omega }$, zero extension is an isometry on these localized functions; consequently $(\psi d_j)$ is relatively compact in $L^2(0,T;L^2(\mathrm{\Omega }))$.

Every subsequence has a further subsequence converging strongly in $L^2(0,T;L^2(\mathrm{\Omega }))$ by step 5.1. If its limit is $h$, taking $g=h$ in step 3.2 and passing to the strong limit yields $\mathrm{\parallel }h{\mathrm{\parallel }}_{L_t^2{L}_x^2}^2=0$. Hence every such limit is zero. If the full sequence failed to converge strongly to zero, a subsequence would remain at distance at least some positive constant from zero, contradicting the preceding conclusion for a strongly convergent further subsequence. Squaring the norm gives the displayed integral limit.