Three-dimensional intermediate Sobolev control

PDF p. 21: §5.2(B), three-dimensional intermediate range

Assume the Axiom of Choice. Let $\mathrm{\Omega }={\mathbb{R}}^3\setminus \mathrm{\Theta }$ be the connected exterior of a nonempty compact obstacle with smooth boundary, and suppose that $\mathrm{\Theta }$ is non-trapping: each admissible generalized ray leaves each compact set in finite forward time. Let $T>0$ and $1<s\le 3\mathrm{/}2$. Let $\phi$ be the cutoff in Linear and nonlinear internally controlled exterior systems, zero on $\mathrm{\mid }x\mathrm{\mid }\le R$ and one for $\mathrm{\mid }x\mathrm{\mid }\ge R+1$, where $\mathrm{\Theta }\subset B_R(0)$.

For $m\in \{3,5\}$ and $\lambda \in \mathbb{R}\setminus \{0\}$, consider the Dirichlet system $i{\mathrm{\partial }}_{t}v+\mathrm{\Delta }v+\lambda \mathrm{\mid }v{\mathrm{\mid }}^{m-1}v=\phi h$ on $(0,T)\times \mathrm{\Omega }$. Is it locally exactly controllable near zero in $H_D^s(\mathrm{\Omega })$? Explicitly, is there $\delta >0$ such that every $v_0,v_T\in H_D^s(\mathrm{\Omega })$ with norms less than $\delta$ can be joined by a solution $v\in C([0,T];H_D^s(\mathrm{\Omega }))$ with $v(0)=v_0$, $v(T)=v_T$ and a control $h\in C([0,T];H_D^s(\mathrm{\Omega }))$?

Here $H_D^s(\mathrm{\Omega })$ is the spectral domain of $(I-{\mathrm{\Delta }}_D{)}^{s\mathrm{/}2}$, with its graph norm, and $C([0,T];H_D^s(\mathrm{\Omega }))$ means norm-continuous paths. The boundary condition is homogeneous Dirichlet, interpreted in the solution class; the interior equation and endpoint conditions have the meanings specified in Linear and nonlinear internally controlled exterior systems.

The article proposes combining an $L_t^4{L}_x^{\mathrm{\infty }}$ bound with suitable Moser-type estimates in the Dirichlet scale. In this question, such estimates mean bounds controlling the nonlinearity and its differences in $L^1(0,T;H_D^s)$ by appropriate powers of a solution-space norm, as required in Abstract nonlinear control-space hypotheses. The mixed norm is the scalar sectionwise norm of Continuous and mixed time-space function spaces; it does not by itself require strong measurability into $L^{\mathrm{\infty }}(\mathrm{\Omega })$. These estimates and the existence of an appropriate solution space are part of the proposed route, not assumptions that resolve the question.

The frozen article v2, PDF page 21, §5.2(B), records this intermediate range as a question and suggests the above route. Its Remark 5, PDF page 16, reports a separate algebra-based extension for $s>3\mathrm{/}2$ in dimension three; the endpoint $s=1$ is treated in its Proposition 3.9. The present item makes no claim that the proposed fractional estimates or the requested controllability have been established. This is an article-status record as of the frozen v2; no independent survey establishing the current worldwide status has been performed.