Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations
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- by Matthias Hieber and Sylvie Monniaux PDF
- Proc. Amer. Math. Soc. 128 (2000), 1047-1053 Request permission
Abstract:
In this paper, we show that a pseudo-differential operator associated to a symbol $a\in L^{\infty }(\mathbb {R}\times \mathbb {R},\mathcal {L}(H))$ ($H$ being a Hilbert space) which admits a holomorphic extension to a suitable sector of $\mathbb {C}$ acts as a bounded operator on $L^{2}(\mathbb {R},H)$. By showing that maximal $L^{p}$-regularity for the non-autonomous parabolic equation $u’(t) + A(t)u(t) = f(t), u(0)=0$ is independent of $p\in (1,\infty )$, we obtain as a consequence a maximal $L^{p}([0,T],H)$-regularity result for solutions of the above equation.References
- Paolo Acquistapace and Brunello Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova 78 (1987), 47–107. MR 934508
- Herbert Amann, Linear and quasilinear parabolic problems. Vol. I, Monographs in Mathematics, vol. 89, Birkhäuser Boston, Inc., Boston, MA, 1995. Abstract linear theory. MR 1345385, DOI 10.1007/978-3-0348-9221-6
- Herbert Amann, Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56. MR 1461211, DOI 10.1002/mana.3211860102
- Thierry Coulhon and Damien Lamberton, Régularité $L^p$ pour les équations d’évolution, Séminaire d’Analyse Fonctionelle 1984/1985, Publ. Math. Univ. Paris VII, vol. 26, Univ. Paris VII, Paris, 1986, pp. 155–165 (French). MR 941819
- de Simon, L.: Un applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratta del primo ordine. Rend. Sem. Mat. Univ. Padova, 34 (1964), 547-558.
- Hieber, M., Monniaux, S.: Heat-Kernels and Maximal $L^{p}-L^{q}$- Estimates: The Non-Autonomous Case. Preprint, 1998.
- Matthias Hieber and Jan Prüss, Heat kernels and maximal $L^p$-$L^q$ estimates for parabolic evolution equations, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1647–1669. MR 1469585, DOI 10.1080/03605309708821314
- Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Progress in Nonlinear Differential Equations and their Applications, vol. 16, Birkhäuser Verlag, Basel, 1995. MR 1329547, DOI 10.1007/978-3-0348-9234-6
- Monniaux, S., Rhandi, A.: Semigroup methodes to solve non-autonomous evolution equations. Semigroup Forum, to appear.
- Jan Prüss, Evolutionary integral equations and applications, Monographs in Mathematics, vol. 87, Birkhäuser Verlag, Basel, 1993. MR 1238939, DOI 10.1007/978-3-0348-8570-6
- José L. Rubio de Francia, Francisco J. Ruiz, and José L. Torrea, Calderón-Zygmund theory for operator-valued kernels, Adv. in Math. 62 (1986), no. 1, 7–48. MR 859252, DOI 10.1016/0001-8708(86)90086-1
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Atsushi Yagi, Parabolic evolution equations in which the coefficients are the generators of infinitely differentiable semigroups. II, Funkcial. Ekvac. 33 (1990), no. 1, 139–150. MR 1065472
Additional Information
- Matthias Hieber
- Affiliation: Mathematisches Institut I, Englerstr. 2, Universität Karlsruhe, D-76128 Karlsruhe, Germany
- MR Author ID: 270487
- Email: matthias.hieber@math.uni-karlsruhe.de
- Sylvie Monniaux
- Affiliation: Abteilung Mathematik V, Universität Ulm, D-89069 Ulm, Germany
- Address at time of publication: Laboratoire de Mathématiques Fondamentales et Appliquées, Centre de Saint-Jérôme, Case Cour A, Avenue Escadrille Normandie-Niemen, 13397 Marseille Cédex 20, France
- Email: monniaux@mathematik.uni-ulm.de, sylvie.monniaux@math.u-3mrs.fr
- Received by editor(s): May 18, 1998
- Published electronically: July 28, 1999
- Communicated by: Christopher D. Sogge
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1047-1053
- MSC (1991): Primary 35K22, 35S05, 47D06
- DOI: https://doi.org/10.1090/S0002-9939-99-05145-X
- MathSciNet review: 1641630