Non-Autonomous Maximal Regularity in Hilbert Spaces
Abstract
We consider non-autonomous evolutionary problems of the form $u'(t)+A(t)u(t)=f(t)$, $u(0)=u_0,$ on $L^2([0,T];H)$, where $H$ is a Hilbert space. We do not assume that the domain of the operator $A(t)$ is constant in time $t$, but that $A(t)$ is associated with a sesquilinear form $a(t)$. Under sufficient time regularity of the forms $a(t)$ we prove well-posedness with maximal regularity in $L^2([0,T];H)$. Our regularity assumption is significantly weaker than those from previous results inasmuch as we only require a fractional Sobolev regularity with arbitrary small Sobolev index.
- Publication:
-
arXiv e-prints
- Pub Date:
- January 2016
- DOI:
- 10.48550/arXiv.1601.05213
- arXiv:
- arXiv:1601.05213
- Bibcode:
- 2016arXiv160105213D
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35K90;
- 35K50;
- 35K45;
- 47D06
- E-Print:
- 24 pages