The Picard-Lindelöf theorem for smooth singular PDE

Paolo Giordano (Università di Vienna)

In this seminar, we first present the Banach fixed point theorem for a contraction $P$ with loss of derivatives $L\in\mathbb N$, i.e.~satisfying $\left\Vert P^{n+1}\left(y_{0}\right)-P^{n}\left(y_{0}\right)\right\Vert _{i}\leq\alpha_{i,n}\left\Vert P\left(y_{0}\right)-y_{0}\right\Vert _{i+nL}$. This result allow us to prove a Picard-Lindelöf theorem for normal smooth PDE, i.e.~of the form \[ \begin{cases}\partial_{t}^{k}y(t,x)=F\left[t,x,\left(\partial_{x}^{a}y\right)_{|a|\leq L},\left(\partial_{t}^{b}y\right)_{|b|<k}\right],\\ \partial_{t}^{j}y(t_{0},x)=y_{0j}(x)\ j=0,\ldots,k-1.\end{cases}\] We compare these results with the classical counter-examples of Lewy and Mizohata. In the second part of the seminar, I review recent results of generalized smooth functions (GSF) theory. This is a minimal extension of Colombeau theory where generalized functions are directly defined as suitable set-theoretical maps between (non-Archimedean) Colombeau generalized numbers, so as to share a lot of properties with ordinary smooth functions. Essentially, all classical theorems of multidimensional differential and integral calculus hold for GSF and there are already non-trivial applications to calculus of variations, optimal control, harmonic analysis, mathematical physics. In contrast to Colombeau generalized functions, GSF are freely closed with respect to composition (e.g.~we can even consider $\delta\circ\delta$) and can also be defined on infinite numbers or on purely infinitesimal domains. Finally, we see that for GSF (in particular, for Colombeau generalized functions and hence for Schwartz distributions), the above mentioned Picard-Lindelöf theorem implies that every Cauchy problem with a normal generalized PDE is well-posed in the Hadamard sense, but only if we allow for solutions defined in infinitesimal intervals in $t$. The aforementioned classical counter-examples (for existence) and the works of de Giorgi et al.~(for uniqueness), show that a better general result is not possible.