Previous |  Up |  Next


Full entry | Fulltext not available (moving wall 24 months)      Feedback
sesquilinear form; non-autonomous evolution equation; maximal regularity; convex set
We prove $L^2$-maximal regularity of the linear non-autonomous evolutionary Cauchy \rlap {problem} $$ \dot {u} (t)+A(t)u(t)=f(t) \quad \text {for a.e.\ } t\in [0,T],\quad u(0)=u_0, $$ where the operator $A(t)$ arises from a time depending sesquilinear form $\mathfrak {a}(t,\cdot ,\cdot )$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of $H.$
[1] Arendt, W., Batty, C. J. K., Hieber, M., Neubrander, F.: Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics 96 Birkhäuser, Basel (2011). MR 2798103 | Zbl 1226.34002
[2] Arendt, W., Chill, R.: Global existence for quasilinear diffusion equations in isotropic nondivergence form. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 9 (2010), 523-539. MR 2722654 | Zbl 1223.35202
[3] Arendt, W., Dier, D., Laasri, H., Ouhabaz, E. M.: Maximal regularity for evolution equations governed by non-autonomous forms. Adv. Differ. Equ. 19 (2014), 1043-1066. MR 3250762
[4] Arendt, W., Dier, D., Ouhabaz, E. M.: Invariance of convex sets for non-autonomous evolution equations governed by forms. J. Lond. Math. Soc., II. Ser. 89 (2014), 903-916. DOI 10.1112/jlms/jdt082 | MR 3217655
[5] Bardos, C.: A regularity theorem for parabolic equations. J. Funct. Anal. 7 (1971), 311-322. DOI 10.1016/0022-1236(71)90038-3 | MR 0283433 | Zbl 0214.12302
[6] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Universitext. Springer New York (2011). MR 2759829 | Zbl 1220.46002
[7] Dautray, R., Lions, J.-L.: Analyse mathématique et calcul numérique pour les sciences et les techniques. Volume 8: Évolution: semi-groupe, variationnel. Collection Enseignement Masson, Paris French (1988). MR 1016604 | Zbl 0749.35004
[8] El-Mennaoui, O., Keyantuo, V., Laasri, H.: Infinitesimal product of semigroups. Ulmer Seminare. Heft 16 (2011), 219-230.
[9] Haak, B., Maati, O. El: Maximal regularity for non-autonomous evolution equations. Version available at:
[10] Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics Springer, Berlin (1992). MR 1335452
[11] Laasri, H.: Problèmes d'évolution et intégrales produits dans les espaces de Banach. Thèse de Doctorat Faculté des science, Agadir (2012), French DOI 10.1007/s00208-015-1199-7. DOI 10.1007/s00208-015-1199-7
[12] Laasri, H., El-Mennaoui, O.: Stability for non-autonomous linear evolution equations with $L^p$-maximal regularity. Czech. Math. J. 63 887-908 (2013). DOI 10.1007/s10587-013-0060-y | MR 3165503
[13] Lions, J. L.: Équations différentielles opérationnelles et problèmes aux limites. Die Grundlehren der mathematischen Wissenschaften 111 Springer, Berlin French (1961). MR 0153974 | Zbl 0098.31101
[14] Ouhabaz, E. M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series 31 Princeton University Press, Princeton (2005). MR 2124040 | Zbl 1082.35003
[15] Ouhabaz, E. M.: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5 (1996), 611-625. DOI 10.1007/BF00275797 | MR 1437587 | Zbl 0868.47029
[16] Ouhabaz, E. M., Spina, C.: Maximal regularity for non-autonomous Schrödinger type equations. J. Differ. Equations 248 (2010), 1668-1683. DOI 10.1016/j.jde.2009.10.004 | MR 2593603 | Zbl 1190.35132
[17] Showalter, R. E.: Monotone Operators in Banach Space and Nonlinear Partial Differential Equations. Mathematical Surveys and Monographs 49, AMS Providence (1997). MR 1422252 | Zbl 0870.35004
[18] Slavík, A.: Product Integration, Its History and Applications. Dějiny Matematiky/History of Mathematics 29; Nečas Center for Mathematical Modeling Lecture Notes 1 Matfyzpress, Praha (2007). MR 2917851 | Zbl 1216.28001
[19] Tanabe, H.: Equations of Evolution. Monographs and Studies in Mathematics 6 Pitman, London (1979). MR 0533824 | Zbl 0417.35003
[20] Thomaschewski, S.: Form Methods for Autonomous and Non-Autonomous Cauchy Problems. PhD Thesis, Universität Ulm (2003).
Partner of
EuDML logo