# Upgrading weak-convergence in $\mathbb{R}$ to strong-convergence in compact sets

Let $$s,s'\in\mathbb{R}$$ satisfying $$s'. Consider $$u=u(t,x)\in C(\mathbb{R},H^s(\mathbb{R}))$$ and suppose that, as $$t$$ goes to infinity, we have the following weak convergence: $$u(t,\cdot)\rightharpoonup u^*(x) \quad \hbox{in} \quad H^{s'}(\mathbb{R}),$$ where $$s' and "$$\rightharpoonup$$" denotes the weak convergence in $$H^{s'}$$. If we additionally assume that $$u(t,\cdot)$$ is uniformly bounded in $$H^s$$, say $$u\in L^{\infty}(\mathbb{R},H^s(\mathbb{R})),$$ does this imply that $$u(t,\cdot)$$ strongly converge to $$u^*$$ in $$H^{s'}(K)$$ for any $$K\subset\mathbb{R}$$ compact? Moreover, if we additionally assume that $$u^*$$ belongs to $$H^s$$ (recall that $$s'), does the previous hypotheses (weak convergence in $$H^{s'}(\mathbb{R})$$ and uniform boundedness in $$H^s(\mathbb{R})$$) imply strong convergence in $$H^{r}(K)$$ for all $$s'\leq r and all $$K\subset\mathbb{R}$$ compact? In other words, if we have a function is converging in a very weak topology, but this function is also uniformly bounded in a stronger topology, does that implies that the function is locally-strongly converging in the topologies "in between" them (whenever "in between" has sense, like in this case)?