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I started recently studying the theory of Partial Differential Equations, so sorry if my question is maybe trivial.

I learned about the Sobolev spaces $W(n,p)$, and the spaces $H^n = W(n,2)$. I learned about their structure of Banach spaces, and the structure of Hilbert space of $H^2$. But frequently, when I hear people talking about PDEs, they mention spaces like $H^{2}L^{2}$, etc ... i.e. spaces where you have both Sobolev space and an $L^p$ space. I asked some friends about these spaces and they told me that in these spaces, you can define the regularity with respect to space and time at once, that it allows to have absolutely continuous energies etc ... But I did not understood at all why, and even what are really these spaces.

Thank you in advance for your answers

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2 Answers 2

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I basically repeat what lolo said. If people talk about $H^2L^2$ spaces in the area of pde, they most likely are talking about Bochner Spaces.

To give an example why we are interested in these spaces: Let $\Omega \subset \mathbb{R}^n$ be an open and bounded set, and $T > 0$. We look for a solution $u \colon \Omega \times (0, T) \to \mathbb{R}$ of the heat equation \begin{equation} \partial_t u - \Delta u = 0 \text{ in } \Omega \quad \text{ and } \quad u = 0 \text{ on } \partial \Omega\, . \quad (1) \end{equation}

One way of proving that $(1)$ has a classical solution $u$ is to first prove it has a weak solution and then conclude using regularity theory. One framework for proving existence (and uniqueness) of weak solutions of time-dependent pdes are Bochner spaces. There are some very helpful identities that make it easier to understand what kind of functions we are dealing with in certain Bochner spaces, e.g., \begin{align} L^2(0, T; L^2(\Omega)) &= L^2(\Omega \times (0, T))\, ,\\ L^2(0, T; H^1(\Omega)) &= \Big\lbrace u \in L^2(\Omega \times (0, T))\, \Big\lvert \, \nabla_x u \in L^2(\Omega \times (0, T)) \Big\rbrace\, ,\\ L^2(0, T; H^1_0(\Omega)) & = \Big\lbrace u \in L^2(0, T; H^1(\Omega))\, \Big\lvert \, u(t) \in H^1_0(\Omega) \text{ for almost all } t \in (0, T)\Big\rbrace\, . \end{align}

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  • $\begingroup$ Thank you very much for your answer ! Now i can look to references on these Bochner spaces, especially to understand why having solutions in these spaces imply absolute continuity of the energy $\endgroup$
    – Galois
    Jul 15, 2022 at 13:16
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Maybe $H^2L^2$ is the notation for the spaces of functions $u:\Omega\subset\mathbb{R}^n\times(0,T)\rightarrow\mathbb{R}$ verifying $u_t\in H^2(\Omega)$ and $u_x\in L^2(0,T)$.

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