Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$ 
*

*Can we find an explicit example of a sequence of functions $f_k \in H^1({\mathbf R}^3)$ such that, $f_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f_k \to f$ strongly converges in $L^6(R^3)$, but $f_k$ does not strongly converge to  $f$ in $H^1({\mathbf R}^3)$?

*What happens if one changes ${\mathbf R}^3$ into ${\mathbf T}^3$? 

*I know that we can use the orthogonal basis decomposition to get a abstract consturction of a sequence $\{f_i\}_{i=1}^{\infty}$ such that $\|f_i\|_{H^1({\mathbf R}^3)}\leq M$ that does not have a limit in $H^1({\mathbf R}^3)$, similar in $H^1({\mathbf R}^3)$, and use the diagnoal take subsequence method we can find a sequence $\{f_i\}_{i=1}^{\infty}$, coverges weakly in $H^1(R^3)$, strongly in $L^6(R^3)$, but $f_i$ is not strong converges to $f$ in $H^1(R^3)$, if $f_i$ is not a cauchy sequence in $H^1(R^3)$

*The question I can not understand is, is the a mechanism for PDE to make the sequence weak solution $\{u_k\}_{k=1}^{\infty}$ come from a sequences of approximation to the original PDE, and a weak solution $u$ in $H^1(R^3)$) which is a weak limit of a subsequences of $\{u_k\}_{k=1}^{\infty}$ satisfied weak converage in $H^1(R^3)$ and strong converage in $L^{q}, \forall q<q^*$, but by some reason (after take subsequence) it is finally strong converage in $H^1(R^3)$? or do there exists such an counterexample come from a particular system differential equations, i.e. if we can construct a sequence of systems partial differential equations $S=\{P_i(u,\nabla u,\nabla^2 u,...,\nabla^ku, f_1,...,f_{s_i})=0, 1\leq i\leq t\}$ and a sequence of approximation of it, said $S_1,.S_2,...,S_n,...$, for every $S_k$ we have a weak solution $u_k$ of $S_k$, and $u_k$ are uniformly bounded in some suitable space $H$, and by take subsequencese of $u_k$ we gain a weak solution $u$ of $S$ and $u$ is a strong limit of $u_k$ in $M $ for all $H \hookrightarrow M$, but we can not prove $u$ is in any higher regularity space than some $M,$ such that $H \hookrightarrow M$ by Rellich–Kondrachov theorem, and in fact $u_k$ is not strong converage to $u$ in $H$?

Let me give some example, from PDE

*

*Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}, a_{i j}$ be bounded functions in $\Omega,$ with $a_{j i}=a_{ji},$ and $c$ be an integrable function in $\Omega .$ consider a differential operator $L$ of the second order defined by
\begin{equation*}
L u=-\sum_{i, j=1}^{n} \partial_{j}\left(a_{i j}(x) \partial_{i} u\right)+c(x) u
\end{equation*}
we can consider weak solution of this equation, at least in $H^1(\Omega)$
For some $f \in L^{2}(\Omega),$ a function $u \in H_{\mathrm{loc}}^{1}(\Omega)$ is called a weak solution if $L u =f$ holds weakly in $\Omega,$ i.e.,
\begin{equation*}
\int_{\Omega}\left(a_{i j} \partial_{i} u \partial_{j} \varphi+c u \varphi\right) d x = \int_{\Omega} f \varphi d x
\end{equation*}
for any $\varphi \in D(\Omega)$ the test function space with $\varphi \geq 0$, assume we can find weak solution $u_k \in H^2(\Omega)$ solve a relative equation $L_{k}u=f_k$ where $L_k \to L, f_k\to f$ as $k\to \infty$, than by take subsequence and use then by Rellich–Kondrachov campact embedding theorem we can only gain weak solution $u$ of $Lu=f$ is strong converage limit of $u_k$ in $L^{q}(\Omega), q<\frac{n^2}{(n-1)^2}$.
But in this case, beacuse of the equation ifself, we have a priori estimate, said, let $\Omega=B_{R}$ be a ball in $\mathbb{R}^{n}$, for $a_{i j} \in L^{\infty}\left(B_{R}\right)$ uniformly elliptic and $c \in L^{q}\left(B_{R}\right),$ for some $q>n / 2 .$ Assume $u \in H^{1}\left(B_{R}\right)$ satisfies $L u = f$ weakly in $B_{R}$. If $f \in L^{q}\left(B_{R}\right),$ then $u^{+} \in L_{\text {loc }}^{\infty}\left(B_{R}\right)$ and
\begin{equation*}
\sup _{B_{R / 2}} u^{+} \leq C\left\{R^{-\frac{n}{2}}\left\|u^{+}\right\|_{L^{2}\left(B_{R}\right)}+R^{2-\frac{n}{\sigma}}\|f\|_{L^{0}\left(B_{R}\right)}\right\}
\end{equation*}
where $C$ is a positive constant depending only on $n, \lambda, \Lambda, q$ and $R^{2-n / q}\|c\|_{L^{q}\left(B_{R}\right)}$
This said, morally, in a smaller set of $\Omega$, we have $L^{\infty}$ estimate for the weak solution $u$, which is out of the scope of Rellich–Kondrachov campact embedding theorem, and the reason to make it hold is come from the structure of the equation. And the prove , roughly speaking, just use the structure of the equation to gain something can not hold if $u$ do not have enough regularity, i.e.
\begin{equation*}
\|u\|_{L^{p_{1}}\left(B_{r_{1}}\right)} \leq C\|u\|_{L^{p_{2}}\left(B_{r_{2}}\right)}
\end{equation*}
for $p_{1}>p_{2}$ and $r_{1}<r_{2} .$ This is a reversed Hölder inequality. And interaction with this inequality to get a $L^{\infty}$ bound of $u$.

*

*In parobalic revolution equation, We intend to build a weak solution of the parabolic problem
\begin{equation*}
\left\{\begin{aligned}
u_{t}+L u=f & \text { in } U_{T} \\
u=0 & \text { on } \partial U \times[0, T] \\
u=g & \text { on } U \times\{t=0\}
\end{aligned}\right.
\end{equation*}
This can be done by construct weak solution of a sequence of finitespace approximation problem to the original parabolic problem, and proof these solution a uniformly bounded in a suitable space, then take a sub sequences to gain a weak solution of the original equation, said the solution is $u$, then by Rellich–Kondrachov campact embedding theorem, we can(by taken subsequence in the limit process to get $u$) assume $u$ in $L^{q}$, for $q<q^*, q^*$ is the critical exponent of compact embedding, but we have a refine priori estimate(which can not gain from Rellich), Assume $g \in H_{0}^{1}(U), \mathrm{f} \in L^{2}\left(0, T ; L^{2}(U)\right)$
Suppose also $\mathbf{u} \in L^{2}\left(0, T ; H_{0}^{1}(U)\right),$ with $\mathbf{u}^{\prime} \in L^{2}\left(0, T ; H^{-1}(U)\right),$ is the weak
solution of
\begin{equation*}
\left\{\begin{aligned}
u_{t}+L u=f & \text { in } U_{T} \\
u=0 & \text { on } \partial U \times[0, T] \\
u=g & \text { on } U \times\{t=0\}
\end{aligned}\right.
\end{equation*}
Then in fact
\begin{equation*}
\mathbf{u} \in L^{2}\left(0, T ; H^{2}(U)\right) \cap L^{\infty}\left(0, T ; H_{0}^{1}(U)\right), \mathbf{u}^{\prime} \in L^{2}\left(0, T ; L^{2}(U)\right)
\end{equation*}
and we have the estimate
\begin{equation*}
\operatorname{ess} \sup _{0 \leq t \leq T}\|\mathbf{u}(t)\|_{H_{0}^{1}(U)}+\|\mathbf{u}\|_{L^{2}\left(0, T ; H^{2}(U)\right)}+\left\|\mathbf{u}^{\prime}\right\|_{L^{2}\left(0, T ; L^{2}(U)\right)}
\end{equation*}
\begin{equation*}
\leq C\left(\|\mathbf{f}\|_{L^{2}\left(0, T ; L^{2}(U)\right)}+\|g\|_{H_{0}^{1}(U)}\right)
\end{equation*}
constant $C$ depending only on $U, T$ and the coefficients of $L$.
$U\subset R^n$, in particuler $\mathbf{u} \in L^{\infty}\left(0, T ; H_{0}^{1}(U)\right)$ , this is already beyond the scope of Rellich–Kondrachov campact embedding theorem
A similar situation happen in,
$\left\{\begin{aligned} u_{tt}+L u &=f \text { in } U_{T} \\ u &=0 \text { on } \partial U \times[0, T] \\ u &=g \text { on } U \times\{t=0\} \end{aligned}\right.$

*

*considering the following initial-value problem (IVP) for the
gKdV equation on the real line $\mathbb{R}$ :
\begin{equation*}
\left\{\begin{array}{l}
\partial_{t} u+\partial_{x}^{3} u+\partial_{x} F(u)=0 \\
u(0, x)=g(x)
\end{array}\right.
\end{equation*}
where the solution $u(t, x)$ is a real-valued function of two real variables and the given function $g(x)$ is its initial profile. then  with $F(u)=u^{5}$. Then there exists $\varepsilon_{0}>0$ such that if $g$ satisfies
\begin{equation*}
\|g\|_{1}+\|g\|_{H^{1}}<\varepsilon_{0}
\end{equation*}
the solution to the corresponding gKdVequation is dispersive; more precisely, it satisfies
\begin{equation*}
\sup _{t \in \mathbb{R}}\langle t\rangle^{1 / 3}\|u(t)\|_{\infty}<\infty
\end{equation*}
This is also out of the scope of Rellich–Kondrachov campact embedding theorem
Finally, I state the question I wondering a answer

*

*if we can construct a sequence of systems partial differential equations $S=\{P_i(u,\nabla u,\nabla^2 u,...,\nabla^ku, f_1,...,f_{s_i})=0, 1\leq i\leq t\}$ and a sequence of approximation of it, said $S_1,.S_2,...,S_n,...$, for every $S_k$ we have a weak solution $u_k$ of $S_k$, and $u_k$ are uniformly bounded in some suitable space $H$, and by take subsequencese of $u_k$ we gain a weak solution $u$ of $S$ and $u$ is a strong limit of $u_k$ in $M $ for all $H \hookrightarrow M$, but we can not prove $u$ is in any higher regularity space than some $M,$ such that $H \hookrightarrow M$ by Rellich–Kondrachov theorem?

*if we can construct a sequence of systems partial differential equations $S=\{P_i(u,\nabla u,\nabla^2 u,...,\nabla^ku, f_1,...,f_{s_i})=0, 1\leq i\leq t\}$ and a sequence of approximation of it, said $S_1,.S_2,...,S_n,...$, for every $S_k$ we have a weak solution $u_k$ of $S_k$, and $u_k$ are uniformly bounded in some suitable space $H$, and by take subsequencese of $u_k$ we gain a weak solution $u$ of $S$ and $u$ is a strong limit of $u_k$ in $M $ for all $H \hookrightarrow M$, but in fact $u_k$ is not strong converage to $u$ in $H$?

 A: First, note that the embedding $H^1(\mathbf{R}^3)\hookrightarrow L^p_{\text{loc}}(\mathbf{R}^3)$ is compact only for $p<6$, and the "loc" is mandatory for this compactness to hold. I know that you did not wrote anything in contradiction with this, but your assumptions surprised me a bit.
Now, consider $f_n:(x_1,x_2,x_3)\mapsto \frac{1}{n}\cos(n x_1)$.
We have $\|f_n\|_\infty\leq \frac{1}{n}\rightarrow_n 0$ and $\|\nabla f_n\|_\infty \leq 1$. In particular, for $\varphi\in\mathscr{D}(\mathbf{R}^3)$ a test function equalling $1$ on the cube $C:=[0,2\pi]^3$, the sequence defined by $g_n:=\varphi f_n$ is bounded in $H^1(\mathbf{R}^3)$ and converges to $0$ in all $L^p(\mathbf{R}^3)$.
On the other hand $(g_n)_n$ weakly converges to $0$ in $H^1(\mathbf{R}^3)$ (for the gradient part, this is due to the Riemann-Lebesgue Lemma) but cannot converge strongly in this space, neither do any of its subsequences. Indeed, one has, on the cube $C$, $\partial_1 g_n = \cos(nx_1)$. For $n\neq p$, by Fubini's Theorem :
\begin{align*}
\|\partial_1 g_n -\partial_1 g_p\|_{L^2(C)}^2 = 4\pi^2 \int_0^{2\pi}|\cos(nz)-\cos(pz)|^2\mathrm{d}z.
\end{align*}
Using the orthogonality of the family $(z\mapsto \cos(kz))_k$ in $L^2(0,2\pi)$ we get
\begin{align*}
\|\partial_1 g_n -\partial_1 g_p\|_{L^2(C)}^2 &= 4\pi^2\left( \int_0^{2\pi}\cos^2(nz)\,\mathrm{d}z+\int_0^{2\pi}\cos^2(pz)\,\mathrm{d}z\right)\\
&=4\pi^3.
\end{align*}
Bottom line : $(\partial_1 g_n)_n$ does not have any (strongly) converging subsequence in $L^2(\mathbf{R}^3)$, so neither does $(g_n)_n$ in $H^1(\mathbf{R}^3)$.
For the torus the previous analysis applies verbatim, replacing $\cos(n x_1)$ by $\cos(2\pi n x_1)$ depending on the normalization you choose. In this case of course you can stick to $(f_n)_n$ (no need to multiply by a test function).
