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 a banach algebra $A \subset \mathscr{C}^\infty(\mathbf{T}^d)$ stable by derivation and pointwise multiplication and for which those operations are continuous ? 

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 has at least a  local smooth solution is by bootstraping Sobolev-like estimates (differentiating the equation itself). I wonder how much one can use a 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 !