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Discrete and Continuous Dynamical Systems - Series A (DCDS-A)
 

On the Stokes problem in exterior domains: The maximum modulus theorem
Pages: 2135 - 2171, Issue 5, May 2014

doi:10.3934/dcds.2014.34.2135      Abstract        References        Full text (665.8K)           Related Articles

Paolo Maremonti - Dipartimento di Matematica, Università degli Studi di Napoli, via Vivaldi, 43, I-81100 Caserta, Italy (email)

1 K. Abe and Y. Giga, Analyticity of the stokes semigroup in spaces of bounded functions, Acta Math., to appear.
2 K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains, submitted for the publication.
3 F. Crispo and P. Maremonti, An interpolation inequality in exterior domains, Rend. Semin. Mat. Univ. Padova, 112 (2004), 11-39.       
4 W. Desch, M. Hieber and J. Prüss, $L^p$-Theory of the Stokes equation in a half space, J. Evol. Equation, 1 (2001), 115-142.       
5 G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 2011.       
6 G. P. Galdi, P. Maremonti and Y. Zhou, On the Navier-Stokes problem in exterior domains with non decaying initial data, J. Math. Fluid Mech., 14 (2012), 633-–652.       
7 G. P. Galdi and S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity, Lecture Notes in Math., 1134, Springer-Verlag, Berlin, 1985.       
8 Y. Giga, K. Inui and Sh. Matsui, On the Cauchy problem for the Navier-Stokes equations with nondecaying initial data, Quad. Mat., 4 (1999), 27-68.       
9 Y. Giga, Sh. Matsui and O. Sawada, Global existenceof of two-dimensional Navier-Stokes flow with nondecaying initial velocity, J. Math. Fluid Mech., 3 (2001), 302-315.       
10 Y. Giga and H. Sohr, On the Stokes operator in exterior domain, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103-130.       
11 Y. Giga and H. Sohr, Abstract $L^p$ estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.       
12 Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system, Func. Analysis and Related Topics, 102 (1991), 55-67.
13 H. Iwashita, $L^q-L^r$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problem in $L^q$ spaces, Math. Ann., 285 (1989), 265-288.       
14 O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2 Gordon and Breach, Science Publishers, New York-London-Paris 1969 xviii+224 pp.       
15 P. Maremonti, On the Stokes equations: The maximum modulus theorem, Math. Models Methods Appl. Sci., 10 (2000), 1047-1072.       
16 P. Maremonti, Stokes and Navier-Stokes problem in the half-space: Existence and uniqueness of solutions a priori non convergent to a limit at infinity, Zapiski Nauch. Sem. POMI, 362 (2008), 176-240; (trasl.) J. of Math., Science, 159 (2009), 486-523.       
17 P. Maremonti, A remark on the Stokes problem with initial data in $L^1$, J. of Math. Fluid Mech., 13 (2011), 469-480.       
18 P. Maremonti, Pointwise asymptotic stability of steady fluid maotion, J. Math. Fluid Mech., 11 (2009), 348-382, Addendum to: Pointwise Asymptotic Stability of Steady Fluid Motions, J. Math. Fluid Mech., 14 (2012) 197-200.
19 P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 24 (1997), 395-449.       
20 P. Maremonti and V. A. Solonnikov, Estimates for solutions of the nonstationary Stokes problem in anisotropic Sobolev spaces with mixed norm, Zap. Nauchn. Semin. POMI, 222 (1995), 124-150; (trasl.) J. Math. Sci., New York, 87 (1997), 3859-3877.       
21 T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain, Hiroshima Math. J., 12 (1982), 115-140.       
22 O. Sawada and Y. Taniuki, A remark on $L^\infty$ solutions to the 2-D Navier-Stokes equations, J. Math. Fluid Mech., 9 (2007), 533-542.       
23 V. A. Solonnikov, Estimates for the solutions of a nonstationary linearized system of Navier-Stokes equations, Trudy Mat. Inst. Steklov, 70 (1964), 213-317.       
24 V. A. Solonnikov, On the differential properties of the solutions of the first boundary-value problem for a nonstationary system of Navier-Stokes equations, Trudy Mat. Inst. Steklov, 73 (1964), 221-291.       
25 V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations, J. Soviet Math., 8 (1977), 467-529.
26 V. A. Solonnikov, On nonstationary Stokes problem and Navier-Stokes problem in a half-space with initial data nondecreasing at infinity, Function theory and applications. J. Math. Sci., 114 (2003), 1726-1740.       
27 V. A. Solonnikov, On the estimates of the solution of the evolution Stokes problem in weighted Hölder norms, Annali dell'Univ. di Ferrara, (Sez. VII, Sci. Mat.), 52 (2006), 137-172.       
28 R. Temam, Navier-Stokes Equations, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, 2. North-Holland Pub. Co., Amsterdam-New York-Tokyo, 1984.       

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