To fix the ideas, let's work on the flat periodic torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$.

My question is the following.

Does there exist an infinite dimensional Banach (sub-)algebra $A \subset \mathscr{C}^\infty(\mathbf{T}^d)$ stable by derivation ?

I am almost certain that the answer is no, but I do not manage to prove it.

My original interrogation concerns nonlinear evolution equations of the form $\partial_t u = \Delta_x F(u)$, where $F$ is some (non degenerative) nonlinearity. The only method I know to prove that starting from $u_0\in\mathscr{C}^\infty(\mathbf{T}^d)$ such equations have at least a local smooth solution is by bootstraping Sobolev-like estimates (differentiating the equation itself). I wonder how much one can use the (Banach-valued) Picard-Lindelöf Theorem, starting from $u_0$ in some algebra as above to produce directly a non-trivial smooth (local) solution. It is the Fréchet structure of $\mathscr{C}^\infty(\mathbf{T}^d)$ which forbids to use this type of Picard-based theorem.

But I have a strong feeling that if such a proof is possible, it would have been written somewhere !

**EDIT** : several comments below made me realize that my question was not precise enough, sorry for that. I wanted to simplify as much as possible but I think it will be more clear if I explicit what I precisely need. Consider the following non linear system of PDE (named " cross-diffusion system " in the literature) on $\mathbf{R}_{\geq 0}\times \mathbf{T}^d$

\begin{align*} \partial_t u &= \Delta_x((d_1+a_{12}v)u),\\ \partial_t v &= \Delta_x((d_2+a_{21}u)v). \end{align*}

The parameters $d_i$ and $a_{ij}$ are all positive numbers, the unkowns being $u$ and $v$.

My goal is to have the most simple proof of existence of one local (non constant) solution, starting from a smooth initial data (I want it to be smooth all the way down to $t=0$). Existence of such smooth solution can be achieved differentiating the equation and iterating a big hammer (Theorem 17.1) but this is precisely what I would like to avoid. This type of results give more information than I need : existence is obtained in Sobolev spaces, together with a criterion of explosion in these spaces. To be completely honest I also prefer to avoid the use of a long article that I obviously will never read in detail in my whole academic life.

Let me now precise my question in this very setting. I am wondering if there exists an infinite dimensional algebra $A\subset \mathscr{C}^\infty(\mathbf{T}^d)$ equipped with a norm $\|\cdot\|_A$ such that $(A,\|\cdot\|_A)$ is Banach, with $\Delta_x(A)\subset A$ (and thus, in particular, $\Delta_x : A\rightarrow A$ is continuous for $\|\cdot\|_A$). If such a structure were to exist, then the previous system could be written as $U'(t) = F(U(t))$, where $U:=(u,v)\in A\times A$ and $F:A\times A\rightarrow A\times A$ is locally lipschitz : we would get in this way existence of local smooth solutions by the Picard-Lindelöf Theorem.

In line with suggestions below : note that such a strategy will produce a solution in a neighboorhood $(-\varepsilon,\varepsilon)$ of $t=0$. The simple case of the heat equation ($d_2=a_{ij}=0$) suggests such an algebra $A$ is necessarily contained in $\mathscr{C}^\omega(\mathbf{T}^d)$ (analytic regularity), for the heat equation cannot be solved backward otherwise.

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