# 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 $$L^{q}, \forall q, but by some reason (after take subsequence) it is finally strong converage in $$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 $$r_{1} This is a reversed Hölder inequality. And interaction with this inequality to get a $$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 $$q is the critical exponent of compact embedding, but we have a refine priori estimate(which can not gain from Rellich), Assume $$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 $$H$$?
• Uh, $H^1$ is weakly closed, no? Nov 18, 2020 at 13:26
• sorry, the original intention $f_k$ is not strong converage to $f$ in $H_1(R^3)$, already corrected. Nov 18, 2020 at 16:45
• The last two points are somehow obscure. Could you make them more clear? Nov 18, 2020 at 18:34
• @Pietro Majer , I already make it more clear. thanks for point out. Nov 19, 2020 at 10:28

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).

• thanks for this answer, this is a very explict construnction to a sequence of functions $f_k\in H^1(R^3)$ such that, $f_k\to f$ weakly converges in $H^1(R^3)$ and $f_k\to f$ strongly converges in $L^6(R^3)$, and $f_k$ does not strongly converge to $f$ in $H^1(R^3)$. Could there still have a counterexample when $u_k$ is a weak solution of k-th approximation of a PDE(or system of PDE)? Nov 19, 2020 at 10:52