Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Pseudo-differential operators
and maximal regularity results
for non-autonomous parabolic equations


Authors: Matthias Hieber and Sylvie Monniaux
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
Published electronically: July 28, 1999
MathSciNet review: 1641630
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. ACQUISTAPACE, P., TERRENI, B.: A unified approach to abstract linear nonautonomous parabolic equations. Rend. Sem. Mat. Univ. Padova 78, (1987), 47-107. MR 89e:34099
  • 2. AMANN, H.: Linear and Quasilinear Parabolic Problems. Birkhäuser, Basel, (1995). MR 96g:34088
  • 3. AMANN, H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186 (1997), 5-56. MR 98h:46033
  • 4. COULHON,T., LAMBERTON, D.: Régularité $L^{p}$ pour les équations d'évolution. In: B. Beauzaumy, B. Maurey, G. Pisier (eds.): Seminaire d'Analyse Fonctionelle 1984/85; Publ. Math. Univ. Paris VII , 26,(1986), 155-165. MR 89e:34100
  • 5. 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.
  • 6. HIEBER, M., MONNIAUX, S.: Heat-Kernels and Maximal $L^{p}-L^{q}$- Estimates: The Non-Autonomous Case. Preprint, 1998.
  • 7. HIEBER, M., PRÜSS, J.: Heat kernels and maximal $L^{p}-L^{q}$ estimates for parabolic evolution equations.Commun. in Partial Differential Equations 22, (1997), 1647-1669. MR 98k:34096
  • 8. LUNARDI, A.: Analytic Semigroups and Optimal Regularity in Parabolic Equations. Birkhäuser, Basel, (1995). MR 96e:47039
  • 9. MONNIAUX, S., RHANDI, A.: Semigroup methodes to solve non-autonomous evolution equations. Semigroup Forum, to appear.
  • 10. PRÜSS, J.: Evolutionary Integral Equations and Applications. Birkhäuser, Basel, (1993). MR 94h:45010
  • 11. RUBIO DE FRANCIA, J.L., RUIZ, F.J., TORREA, J.L.: Calderón-Zygmund theory for operator-valued kernels. Adv. Math. 62, (1986), 7-48. MR 88f:42035
  • 12. STEIN, E.M.: Harmonic Analysis: Real-Variables Methods, Orthogonality and Oscillartory Integrals. Princeton University Press, Princeton, (1993). MR 95c:42002
  • 13. YAGI, A.: Parabolic evolution equations in which coefficients are the generators of infinitely differentiable semigroups, II. Funk. Ekvacioj 33,(1990), 139-150. MR 91h:47039

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K22, 35S05, 47D06

Retrieve articles in all journals with MSC (1991): 35K22, 35S05, 47D06


Additional Information

Matthias Hieber
Affiliation: Mathematisches Institut I, Englerstr. 2, Universität Karlsruhe, D-76128 Karlsruhe, Germany
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

DOI: https://doi.org/10.1090/S0002-9939-99-05145-X
Received by editor(s): May 18, 1998
Published electronically: July 28, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society