Acta Mathematica

Analyticity of the Stokes semigroup in spaces of bounded functions

Ken Abe and Yoshikazu Giga

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Abstract

The analyticity of the Stokes semigroup with the Dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the Helmholtz decomposition. The proof is based on a blow-up argument. This is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years.

Article information

Source
Acta Math., Volume 211, Number 1 (2013), 1-46.

Dates
Received: 6 July 2011
Revised: 24 October 2012
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.acta/1485892722

Digital Object Identifier
doi:10.1007/s11511-013-0098-6

Mathematical Reviews number (MathSciNet)
MR3118304

Zentralblatt MATH identifier
1288.35383

Rights
2013 © Institut Mittag-Leffler

Citation

Abe, Ken; Giga, Yoshikazu. Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211 (2013), no. 1, 1--46. doi:10.1007/s11511-013-0098-6. https://projecteuclid.org/euclid.acta/1485892722


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References

  • Abe, K. & Giga, Y., The L-Stokes semigroup in exterior domains. Hokkaido University Preprint Series in Mathematics, 1011. Sapporo, 2012. To appear in J. Evol. Equ.
  • Abe, K., Giga, Y. & Hieber, M., Stokes resolvent estimates in spaces of bounded functions. Hokkaido University Preprint Series in Mathematics, 1022. Sapporo, 2012.
  • Abe T., Shibata Y.: On a resolvent estimate of the Stokes equation on an infinite layer. J. Math. Soc. Japan, 55, 469–497 (2003)
  • Abels H.: Nonstationary Stokes system with variable viscosity in bounded and unbounded domains. Discrete Contin. Dyn. Syst. Ser. S, 3, 141–157 (2010)
  • Abels H., Terasawa Y.: On Stokes operators with variable viscosity in bounded and unbounded domains. Math. Ann., 344, 381–429 (2009)
  • Acquistapace P., Terreni B.: Hölder classes with boundary conditions as interpolation spaces. Math. Z., 195, 451–471 (1987)
  • Adams, R. A. & Fournier, J. J. F., Sobolev Spaces. Pure and Applied Mathematics (Amsterdam), 140. Elsevier, Amsterdam, 2003.
  • Arendt, W. & Schätzle, R., Semigroups generated by elliptic operators in non-divergence on C0(Ω). Preprint, 2010.
  • Bae H.-O., Jin B. J.: Existence of strong mild solution of the Navier–Stokes equations in the half space with nondecaying initial data. J. Korean Math. Soc., 49, 113–138 (2012)
  • Bogovskiĭ, M. E., Solution of the first boundary value problem for an equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR, 248 (1979), 1037–1040 (Russian); English translation in Soviet Math. Dokl., 20 (1979), 1094–1098.
  • Bogovskiĭ, M. E., Decomposition of Lp(Ω; Rn) into a direct sum of subspaces of solenoidal and potential vector fields. Dokl. Akad. Nauk SSSR, 286 (1986), 781–786 (Russian); English translation in Soviet Math. Dokl., 33 (1986), 161–165.
  • Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in Lq-spaces. Math. Z., 196, 415–425 (1987)
  • De Giorgi, E., Frontiere Orientate di Misura Minima. Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960–61. Editrice Tecnico Scientifica, Pisa, 1961.
  • Desch W., Hieber M., Prüss J.: Lp-theory of the Stokes equation in a half space. J. Evol. Equ., 1, 115–142 (2001)
  • Evans, L. C., Partial Differential Equations. Graduate Studies in Mathematics, 19. Amer. Math. Soc., Providence, RI, 2010.
  • Farwig R., Kozono H., Sohr H.: An Lq-approach to Stokes and Navier–Stokes equations in general domains. Acta Math., 195, 21–53 (2005)
  • Farwig R., Kozono H., Sohr H.: On the Helmholtz decomposition in general unbounded domains. Arch. Math. (Basel), 88, 239–248 (2007)
  • Farwig R., Kozono H., Sohr H.: On the Stokes operator in general unbounded domains. Hokkaido Math. J., 38, 111–136 (2009)
  • Farwig R., Sohr H.: resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan, 46, 607–643 (1994)
  • Farwig R., Sohr H.: Helmholtz decomposition and Stokes resolvent system for aperture domains in Lq-spaces. Analysis (Munich), 16, 1–26 (1996)
  • Farwig R., Taniuchi Y.: On the energy equality of Navier–Stokes equations in general unbounded domains. Arch. Math. (Basel), 95, 447–456 (2010)
  • Galdi, G. P., An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Vol. I. Springer Tracts in Natural Philosophy, 38. Springer, New York, 1994
  • Geißert, M., Heck, H. & Hieber, M., On the equation div u = g and Bogovskiĭ’s operator in Sobolev spaces of negative order, in Partial Differential Equations and Functional Analysis, Operator Theory: Advances and Applications, 168, pp. 113–121. Birkhäuser, Basel, 2006.
  • Geissert M., Heck H., Hieber M., Sawada O.: Weak Neumann implies Stokes. J. Reine Angew. Math., 669, 75–100 (2012)
  • Gidas B., Spruck J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations, 6, 883–901 (1981)
  • Giga, M.-H., Giga, Y. & Saal, J., Nonlinear Partial Differential Equations. Progress in Nonlinear Differential Equations and their Applications, 79. Birkhäuser, Boston, MA, 2010.
  • Giga Y.: Analyticity of the semigroup generated by the Stokes operator in Lr spaces. Math. Z., 178, 297–329 (1981)
  • Giga Y.: A bound for global solutions of semilinear heat equations. Comm. Math. Phys., 103, 415–421 (1986)
  • Giga, Y., Surface Evolution Equations. Monographs in Mathematics, 99. Birkhäuser, Basel, 2006.
  • Giga, Y., Inui, K. & Matsui, S., On the Cauchy problem for the Navier–Stokes equations with nondecaying initial data, in Advances in Fluid Dynamics, Quaderni di Matematica, 4, pp. 27–68. Dept. Math., Seconda Univ. Napoli, Caserta, 1999.
  • Giga Y., Kohn R.V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J., 36, 1–40 (1987)
  • Giga Y., Matsui S., Sawada O.: Global existence of two-dimensional Navier–Stokes flow with nondecaying initial velocity. J. Math. Fluid Mech., 3, 302–315 (2001)
  • Giga Y., Matsui S., Shimizu Y.: estimates in Hardy spaces for the Stokes flow in a half space. Math. Z., 231, 383–396 (1999)
  • Giga Y., Miura H.: On vorticity directions near singularities for the Navier–Stokes flows with infinite energy. Comm. Math. Phys., 303, 289–300 (2011)
  • Giga Y., Sohr H.: On the Stokes operator in exterior domains. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36, 103–130 (1989)
  • Gilbarg, D. & Trudinger, N. S., Elliptic Partial Differential Equations of Second Order. Grundlehren der Mathematischen Wissenschaften, 224. Springer, Berlin–Heidelberg, 1983.
  • Heck H., Hieber M., Stavrakidis K.: L-estimates for parabolic systems with VMO-coefficients. Discrete Contin. Dyn. Syst. Ser. S, 3, 299–309 (2010)
  • Koch G., Nadirashvili N., Seregin G.A., Šverák V.: theorems for the Navier–Stokes equations and applications. Acta Math., 203, 83–105 (2009)
  • Krantz, S. G. & Parks, H. R., The Implicit Function Theorem: History, Theory, and Applications. Birkhäuser, Boston, MA, 2002.
  • Krylov, N. V., Lectures on Elliptic and Parabolic Equations in Hölder Spaces. Graduate Studies in Mathematics, 12. Amer. Math. Soc., Providence, RI, 1996.
  • Ladyzhenskaya, O. A., Solonnikov, V. A. & Uraltseva, N. N., Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs, 23. Amer. Math. Soc., Providence, RI, 1968.
  • Lunardi, A., Analytic Semigroups and Optimal Regularity in Parabolic Problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser, Basel, 1995.
  • Maremonti P.: Pointwise asymptotic stability of steady fluid motions. J. Math. Fluid Mech., 11, 348–382 (2009)
  • Maremonti, P., On the Stokes problem: the maximum modulus theorem. To appear in Discrete Contin. Dyn. Syst.
  • Maremonti, P. & Starita, G., On the nonstationary Stokes equations in half-space with continuous initial data. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 295 (2003) 118–167, 246 (Russian); English translation in J. Math. Sci. (N.Y.), 127 (2005), 1886–1914.
  • Maslennikova, V. N. & Bogovskiĭ, M. E., Elliptic boundary value problems in unbounded domains with noncompact and nonsmooth boundaries. Rend. Sem. Mat. Fis. Milano, 56 (1986), 125–138.
  • Masuda, K., On please, update if possible the generation of analytic semigroups by elliptic differential operators with unbounded coefficients. Unpublished note, 1972.
  • Masuda, K., On the generation of analytic semigroups of higher-order elliptic operators in spaces of continuous functions, in Proc. Katata Symposium on Partial Differential Equations (Katata, 1972), pp. 144–149. Sugaku Shinkokai, Tokyo, 1973 (Japanese).
  • Masuda, K., Evolution Equations. Kinokuniya Shoten, Tokyo, 1975 (Japanese).
  • Poláčik P., Quittner P., Souplet P.: and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations. Indiana Univ. Math. J., 56, 879–908 (2007)
  • Quittner, P. & Souplet, P., Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel, 2007.
  • Seregin G., Šverák V.: On type I singularities of the local axi-symmetric solutions of the Navier–Stokes equations. Comm. Partial Differential Equations, 34, 171–201 (2009)
  • Simon, L., Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Australian National University, 3. Australian National University Centre for Mathematical Analysis, Canberra, 1983.
  • Sohr, H., The Navier–Stokes Equations: An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Basel, 2001.
  • Solonnikov, V. A., Estimates of the solution of a certain initial-boundary value problem for a linear nonstationary system of Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel Mat. Inst. Steklov. (LOMI), 59 (1976), 178–254,257 (Russian); English translation in J. Soviet. Math., 8 (1977), 467–529.
  • Solonnikov, V. A., On the theory of nonstationary hydrodynamic potentials, in The Navier–Stokes Equations: Theory and Numerical Methods (Varenna, 2000), Lecture Notes in Pure and Applied Mathematics, 223, pp. 113–129. Marcel Dekker, New York, 2002.
  • Solonnikov, V. A., Potential theory for the nonstationary Stokes problem in nonconvex domains, in Nonlinear Problems in Mathematical Physics and Related Topics, I, International Mathematical Series (New York), 1, pp. 349–372. Kluwer/Plenum, New York, 2002.
  • Solonnikov V.A.: On nonstationary Stokes problem and Navier–Stokes problem in a half-space with initial data nondecreasing at infinity. J. Math. Sci. (N.Y.), 114, 1726–1740 (2003)
  • Solonnikov V.A.: Weighted Schauder estimates for evolution Stokes problem. Ann. Univ. Ferrara Sez. VII Sci. Mat., 52, 137–172 (2006)
  • Solonnikov, V. A., Schauder estimates for the evolutionary generalized Stokes problem, in Nonlinear Equations and Spectral Theory, American Mathematical Society Translations, Series 2, 220, pp. 165–200. Amer. Math. Soc., Providence, RI, 2007.
  • Stewart H.B.: Generation of analytic semigroups by strongly elliptic operators. Trans. Amer. Math. Soc., 199, 141–162 (1974)
  • Stewart H. B.: eneration of analytic semigroups by strongly elliptic operators under general boundary conditions. Trans. Amer. Math. Soc., 259, 299–310 (1980)
  • Taira, K., Semigroups, Boundary Value Problems and Markov Processes. Springer Monographs in Mathematics. Springer, Berlin–Heidelberg, 2004.
  • Tanabe, H., Functional Analytic Methods for Partial Differential Equations. Monographs and Textbooks in Pure and Applied Mathematics, 204. Marcel Dekker, New York, 1997.
  • Vasil′ev, V. N. & Solonnikov, V. A., Bounds for the maximum modulus of the solution of a linear nonstationary system of Navier–Stokes equations. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 69 (1977), 34–44 (Russian); English translation in J. Soviet Math., 10 (1978), 22–29.
  • Yosida K.: On holomorphic Markov processes. Proc. Japan Acad., 42, 313–317 (1966)