MAXIMAL $L^1$-REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES

doi:10.1007/s10473-022-0401-8

Abstract

Abstract: In this paper, we prove that the generator of any bounded analytic semigroup in $(\theta,1)$ -type real interpolation of its domain and underlying Banach space has maximal $L^1$ -regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$ -regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$ -type Besov spaces on domains of $\mathbb R^n$ , $n\geq 2$ .

Key words: Maximal $L^1$ -regularity, sectorial operator, Stokes operator

CLC Number:

35K90

Myong-Hwan RI, Reinhard FARWIG. MAXIMAL $L^1$ -REGULARITY OF GENERATORS FOR BOUNDED ANALYTIC SEMIGROUPS IN BANACH SPACES[J].Acta mathematica scientia,Series B, 2022, 42(4): 1261-1272.

Trendmd

References

[1] Amann H. Parabolic evolution equations in interpolation and extrapolation spaces. J Funct Anal, 1988, 78:233-270
[2] Amann H. Linear and Quasilinear Parabolic Problems. I. Abstract Linear Theory. Birkhäuser, 1995
[3] Amann H. On the strong solvability of the Navier-Stokes equations. J Math Fluid Mech, 2000, 2:16-98
[4] Amann H. Navier-Stokes equations with nonhomogeneous Dirichlet data. J Nonlinear Math Phys, 2003, 10(Suppl 1):1-11
[5] Amann H. Vector-valued distributions and Fourier multipliers. www.math.uzh.ch/amann/books, 2003
[6] Angenent S. Nonlinear analytic semiflows. Proc Roy Soc Edinburgh Sect A, 1990, 115:91-107
[7] Bergh J, Löfström J. Interpolation Spaces. Berlin Heidelberg New York:Springer-Verlag, 1976
[8] Cannarsa P, Vespri V. On maximal

$L_p$ regularity for the abstract Cauchy problem. Boll Unione Mat Ital B, 1986, 5:165-175
[9] Clément P, Simonett G. Maximal regularity in continuous interpolation spaces and quasilinear parabolic equations. J Evol Equ, 2001, 1:39-67
[10] Cowling M, Doust I, McIntosh A, Yagi A. Banach space operators with a bounded

$H^\infty$ functional calculus. J Aust Math Soc, 1996, 60A:51-89
[11] Da Prato, Grisvard P. Equations d'évolution abstraites non linéaires de type parabolique. Ann Mat Pura Appl, 1979, 120:329-396
[12] Danchin R, Mucha P B. A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space. J Funct Anal, 2009, 256:881-927
[13] Danchin R, Mucha P B. Critical functional framework and maximal regularity in action on systems of incompressible flows. Mém Soc Math Fr (NS), 2015, 143
[14] Denk R, Hieber M, Prüss J.

$R$ -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem Amer Math Soc, 2003, 788
[15] Dore G.

$L^p$ Regularity for abstract differential equations//Komatsu H. Functional analysis and related topics. Kyoto, 1991:Lecture Notes in Math, 1540. Berlin:Springer, 1993:25-38
[16] Dore G. Maximal regularity in

$L_p$ spaces for an abstract Cauchy problem. Adv Differential Equations, 2000, 5:293-322
[17] Dore G, Venni A. On the closedness of the sum of two closed operators. Math Z, 1987, 196:189-201
[18] Farwig R, Kozono H, Sohr H. An

$L^q$ -approach to Stokes and Navier-Stokes equations in general domains. Acta Math, 2005, 195:21-53
[19] Farwig R, Ri M -H. The resolvent problem and

$H^\infty$ -calculus of the Stokes operator in unbounded cylinders with several exits to infinity. J Evol Equ, 2007, 7:497-528
[20] Giga Y, Saal J.

$L_1$ maximal regularity for the Laplacian and applications, Dynamical systems, differential equations and applications. 8th AIMS Conference. Discrete Contin Dyn Syst, 2011, I(Suppl):495-504
[21] Guerre-Delabrière S.

$L_p$ -regularity of the Cauchy problem and the geometry of Banach spaces. Illinois J Math, 1995, 39(4):556-566
[22] Kalton N J, Portal P. Remarks on

$l^1$ - and

$l^\infty$ -maximal regularity for powerbounded operators. J Aust Math Soc, 2008, 84:345-365
[23] LeCrone J, Simonett G. Continuous maximal reguarity and anaytic semigroups. Dynamical systems, differential equations and applications. 8th AIMS Conference. Discrete Contin Dyn Syst, 2011, II(Suppl):963-970
[24] Lindenstrauss J, Tzafriri L. Classical Banach spaces I. Berlin-Heidelberg-New York:Springer-Verlag, 1977
[25] Prüss J. Maximal regularity for evolution equations in

$L_p$ -spaces. Conf Semin Mat Univ Bari, 2002, 285:1-39
[26] Prüss J, Simonett G. Maximal regularity for evolution equations in weighted

$L_p$ -spaces. Arch Math, 2004, 82:415-431
[27] Ri M -H, Zhang P, Zhang Z. Global well-posedness for Navier-Stokes equations with small initial value in

$B^0_{n,\infty}(Ω)$ . J Math Fluid Mech, 2016, 18:103-131
[28] de Simon L. Un applicazione della teoria degli integrali singolari allo studio delle equazioni lineari astratte del primo ordine. Rend Sem Mat Univ Padova, 1964, 34:547-558
[29] Sobolevskii P E. Coerciveness inequalities for abstract parabolic equations. Soviet Math Dokl, 1964, 5:894-897
[30] Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam:North Holland, 1983