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Fix $d\geq 1$ and $1<p,q<+\infty$. I am searching for a reference concerning the following result.

There exists a constant $C=C(d,p,q)$ such that, for any $u\in\mathscr{C}^\infty(\mathbb{R}_+\times\mathbb{T}^d)$ where $\mathbb{T}^d$ is the flat torus, we have the following inequality : \begin{align*} \|\partial_t u\|_{L^p(\mathbb{R}_+;L^q(\mathbb{T}^d))} & + \|\Delta u\|_{L^p(\mathbb{R}_+;L^q(\mathbb{T}^d))} \\ & \leq C\Big[ \|\partial_t u -\Delta u\|_{L^p(\mathbb{R}_+;L^q(\mathbb{T}^d))} + \|u(0,\cdot)\|_{L^q(\mathbb{T}^d)}\Big]. \end{align*}

Precisions:

  • my original interest is a similar maximal estimate for the (unsteady) Stokes equation, but the Leray projector being continuous from $L^q(\mathbb{T}^d)$ to itself, the above inequality would satisfy me.
  • I found some references (without the initial data though...) replacing $\mathbb{T}^d$ by $\mathbb{R}^d$ but I don't know if a transference argument exists for maximal regularity issues.

I am sure that the above result is true and it is quite oftenly used in the literature. I would be relieved to find a definitive reference for it !

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