Thursday, March 22, 2012

1203.4700 (Jochen Schmid)

A note on the well-posedness of non-autonomous linear evolution

Jochen Schmid
In this note we investigate a commonly used theorem by Yosida (Theorem XIV.4.1 of Yosida's book on functional analyis or Theorem X.70 of Reed and Simon's book on mathematical physics) on the well-posedness of the initial value problems corresponding to a family $A$ of linear operators $A(t)$ with common dense domain $D$ in some Banach space $X$. We prove that the rather long regularity conditions of this theorem can be replaced -- without changing the content of the theorem -- by the single (and simple) condition that $t \mapsto A(t)x$ is continuously differentiable for every $x \in D$. We also extend this simplification to the case of sequentially complete locally convex spaces $X$. With these observations we finally clarify the relation between Yosida's theorem and other classical theorems on well-posedness.
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