The L ∞-Stokes semigroup in exterior domains
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K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions, Acta Math., to appear
K. Abe, Y. Giga, M. Hieber, Stokes resolvent estimates in spaces of bounded functions, Hokkaido University Preprint Series in Mathematics, no.1022 (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)
R. A. Adams, J. J. F. Fournier, Sobolev spaces. Second edition, Elsevier, Amsterdam 2003.
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)
M. E. Bogovskiǐ, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl, Akad. Nauk. SSSR 248 (1979), 1037–1040; translation: Soviet Math. Dokl. 20 (1979), 1094–1098.
M. E. Bogovskiǐ, Decomposition of L p (Ω, R n ) into the direct sum of subspaces of solenoidal and potential vector fields, Dokl. Akad. Nauk. SSSR 286 (1986), 781–786; translation: Soviet Math. Dokl. 33 (1986), 161–165.
Borchers W., Sohr H.: On the semigroup of the Stokes operator for exterior domains in L q -spaces. Math. Z. 196, 415–425 (1987)
W. Borchers, W. Varnhorn, On the boundedness of the Stokes semigroup in two-dimensional exterior domains, Math. Z. 213 (1993), 275–299.
Cattabriga L.: Su un problema al contorno relativo al sistema di equazioni di Stokes (Italian). Rend. Sem. Mat. Univ. Padova 31, 308–340 (1961)
W. Desch, M. Hieber, J. Prüss, L p -theory of the Stokes equation in a half space, J. Evol. Equ. 1 (2001), 115–142.
R. Farwig, H. Kozono, H. Sohr, An L q -approach to Stokes and Navier-Stokes equations in general domains, Acta Math. 195 (2005), 21–53.
Farwig R., Kozono H., Sohr H.: On the Helmholtz decomposition in general unbounded domains. Arch Math. 88, 239–248 (2007)
R. Farwig, H. Kozono, H. Sohr, On the Stokes operator in general unbounded domains, Hokkaido Math. J. 38 (2009), 111–136.
Farwig R., Sohr H.: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46, 607–643 (1994)
R. Farwig, H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in L q spaces, Analysis 16 (1996), 1–26.
G. P. Galdi, An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I. Linearized steady problems, Springer Tracts in Natural Philosophy. 38., Springer, New York 1994.
G. P. Galdi, P. Maremonti, Y. Zhou, On the Navier-Stokes Problem in Exterior Domains with Non Decaying Initial Data, J. Math. Fluid Mech. 14 (2012), 633–652.
Geissert M., Heck H., Hieber M., Sawada O.: Weak Neumann implies Stokes. J. Reine. angew. Math. 669, 75–100 (2012)
M.-H. Giga, Y. Giga, J. Saal, Nonlinear partial differential equations: Asymptotic behavior of solutions and self-similar solutions, Birkhäuser, Boston-Basel-Berlin 2010.
Y. Giga, Analyticity of the semigroup generated by the Stokes operator in L r spaces, Math. Z. 178 (1981), 297–329.
Giga Y.: A bound for global solutions of semilinear heat equations. Comm. Math. Phys. 103, 415–421 (1986)
Y. Giga, Surface evolution equitions: a level set approach, Birkhäuser, Basel-Boston-Berlin 2006.
Giga Y., Inui K., Matsui S.: On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data. Quad. Mat. 4, 27–68 (1999)
Giga Y., Kohn R. V.: Characterizing blowup using similarity variables. Indiana Univ. Math. J. 36, 1–40 (1987)
Y. Giga, S. Matsui, O. Sawada, Global existence of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech 3 (2001), 302–315.
Giga Y., Miura H.: On vorticity directions near singularities for the Navier-Stokes flow with infinite energy. Comm. Math. Phys. 303, 289–300 (2011)
Y. Giga, H. Sohr, On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 36 (1989), 103–130.
C. E. Kenig, F. Lin, Z. Shen, Homogenization of elliptic systems with Neumann boundary conditions, preprint
G. Koch, N. Nadirashvili, G. A. Seregin, V. Šverák, Liouville theorems for the Navier-Stokes equations and applications, Acta Math. 203 (2009), 83–105.
O. A. Ladyženskaja, V. A. Solonnikov, N. N. Ural’ceva, Linear and quasilinear equations of parabolic Type, Transl. Math. Monogr. vol. 23., American Mathematical Society, Providence. R. I. 1968.
A. Lunardi, Analytic semigroup and optimal regularity in parabolic problems, Birkhäuser, Basel 1995.
Maremonti P.: Pointwise asymptotic stability of steady fluid motions. J. Math. Fluid Mech. 11, 348–382 (2009)
P. Maremonti, A remark on the Stokes problem with initial data in L 1, J. Math. Fluid Mech. 13 (2011), 469–480.
P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorem, preprint
P. Maremonti, G. Starita, Nonstationary Stokes equations in a half-space with continuous initial data, Zapiski Nauchnykh Seminarov POMI, 295 (2003) 118–167; translation: J. Math. Sci. (N.Y.) 127 (2005), 1886–1914.
V. N. Maslennikova, M. E. Bogovskiǐ, Elliptic boundary value problems in unbounded domains with noncompact and non smooth boundaries, Rend. Sem. Mat. Fis. Milano, 56 (1986), 125–138.
P. Polácik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems. Part II. Parabolic equations., Indiana Univ. Math. J. 56 (2007), 879–908.
P. Quittner, P. Souplet, Superlinear parabolic problems: Blow-up, global existence and steady states, Birkhäuser, Basel-Boston-Berlin 2007.
J. Saal, The Stokes operator with Robin boundary conditions in solenoidal subspaces of $${L^1(\mathbb{R}^n_+)}$$ L 1 ( R + n ) and $${L^\infty(\mathbb{R}^n_+)}$$ L ∞ ( R + n ) , Commun. Partial Differ. Equations 32 (2007), 343–373.
Sawada O., Taniuchi Y.: A remark on L ∞ solutions to the 2-D Navier-Stokes equations. J. Math. Fluid Mech. 9, 533–542 (2007)
Shibata Y., Shimada R.: On a generalized resolvent for the Stokes system with Robin boundary condition. J. Math. Soc. Japan, 59, 469–519 (2007)
C. G. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in L q -spaces for bounded and exterior domains, Mathematical Problems Relating to the Navier-Stokes Equation, Ser. Adv. Math. Appl. Sci. 11 (1992), 1–35.
C. G. Simader, H. Sohr, The Dirichlet problem for the Laplacian in bounded and unbounded domains. A new approach to weak, strong and (2 + k)-solutions in Sobolev-type spaces, Pitman Research Notes in Mathematics Series 360, Longman, Harlow 1996.
L. Simon, Lectures on geometric measure theory, Proc. Centre Math. Anal., Australian National University. 3., Australian National University, Canberra 1983.
Sinestrari E.: On the abstract Cauchy problem of parabolic type in spaces of continuous functions. J. Math. Anal. Appl. 107, 16–66 (1985)
Solonnikov V. A.: Estimates for solutions of nonstationary Navier-Stokes equations. J. Soviet Math. 8, 467–529 (1977)
V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, J. Math. Sci. (N. Y.) 114 (2003), 1726–1740.
Solonnikov V. A.: Weighted Schauder estimates for evolution Stokes problem. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, 137–172 (2006)
V. A. Solonnikov, Schauder estimates for the evolutionary generalized Stokes problem, Amer. Math. Soc. Transl. Ser. 2. 220 (2007), 165–199.