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May  2014, 34(5): 2135-2171. doi: 10.3934/dcds.2014.34.2135

## On the Stokes problem in exterior domains: The maximum modulus theorem

 1 Dipartimento di Matematica, Università degli Studi di Napoli, via Vivaldi, 43, I-81100 Caserta

Received  March 2012 Revised  October 2012 Published  October 2013

We study the Stokes initial boundary value problem, in $(0,T) \times Ω$, where $Ω \subseteq \mathbb{R}^n$, $n\geq3$, is an exterior domain, assuming that the initial data belongs to $L^\infty(Ω)$ and has null divergence in weak sense. We prove the maximum modulus theorem for the corresponding solutions. Crucial for the proof of this result is the analogous one proved by Abe-Giga for bounded domains. Our proof is developed by duality arguments and employing the semigroup properties of the resolving operator defined on $L^1(Ω)$. Our results are similar to the ones proved by Solonnikov by means of the potential theory.
Citation: Paolo Maremonti. On the Stokes problem in exterior domains: The maximum modulus theorem. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2135-2171. doi: 10.3934/dcds.2014.34.2135
##### References:
 [1] K. Abe and Y. Giga, Analyticity of the stokes semigroup in spaces of bounded functions,, Acta Math., (). Google Scholar [2] K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains,, submitted for the publication., (). Google Scholar [3] F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, Rend. Semin. Mat. Univ. Padova, 112 (2004), 11. Google Scholar [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. doi: 10.1007/PL00001362. Google Scholar [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, (2011). Google Scholar [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). doi: 10.1007/s00021-011-0083-9. Google Scholar [7] G. P. Galdi and S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity,, Lecture Notes in Math., (1134). Google Scholar [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. Google Scholar [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. doi: 10.1007/PL00000973. Google Scholar [10] Y. Giga and H. Sohr, On the Stokes operator in exterior domain,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103. Google Scholar [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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar [12] Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, Func. Analysis and Related Topics, 102 (1991), 55. Google Scholar [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. doi: 10.1007/BF01443518. Google Scholar [14] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969). Google Scholar [15] P. Maremonti, On the Stokes equations: The maximum modulus theorem,, Math. Models Methods Appl. Sci., 10 (2000), 1047. doi: 10.1142/S0218202500000537. Google Scholar [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. doi: 10.1007/s10958-009-9458-3. Google Scholar [17] P. Maremonti, A remark on the Stokes problem with initial data in $L^1$,, J. of Math. Fluid Mech., 13 (2011), 469. doi: 10.1007/s00021-010-0036-8. Google Scholar [18] P. Maremonti, Pointwise asymptotic stability of steady fluid maotion,, J. Math. Fluid Mech., 11 (2009), 348. Google Scholar [19] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 395. Google Scholar [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. doi: 10.1007/BF02355828. Google Scholar [21] T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115. Google Scholar [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. doi: 10.1007/s00021-005-0212-4. Google Scholar [23] V. A. Solonnikov, Estimates for the solutions of a nonstationary linearized system of Navier-Stokes equations,, Trudy Mat. Inst. Steklov, 70 (1964), 213. Google Scholar [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. Google Scholar [25] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Soviet Math., 8 (1977), 467. doi: 10.1007/BF01084616. Google Scholar [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. doi: 10.1023/A:1022317029111. Google Scholar [27] V. A. Solonnikov, On the estimates of the solution of the evolution Stokes problem in weighted Hölder norms,, Annali dell'Univ. di Ferrara, 52 (2006), 137. doi: 10.1007/s11565-006-0012-7. Google Scholar [28] R. Temam, Navier-Stokes Equations,, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, (1984). Google Scholar

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##### References:
 [1] K. Abe and Y. Giga, Analyticity of the stokes semigroup in spaces of bounded functions,, Acta Math., (). Google Scholar [2] K. Abe and Y. Giga, The $L^{\infty}$-Stokes semigroup in exterior domains,, submitted for the publication., (). Google Scholar [3] F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, Rend. Semin. Mat. Univ. Padova, 112 (2004), 11. Google Scholar [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. doi: 10.1007/PL00001362. Google Scholar [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, (2011). Google Scholar [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). doi: 10.1007/s00021-011-0083-9. Google Scholar [7] G. P. Galdi and S. Rionero, Weighted Energy Methods in Fluid Dynamics and Elasticity,, Lecture Notes in Math., (1134). Google Scholar [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. Google Scholar [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. doi: 10.1007/PL00000973. Google Scholar [10] Y. Giga and H. Sohr, On the Stokes operator in exterior domain,, J. Fac. Sci. Univ. Tokyo, 36 (1989), 103. Google Scholar [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. doi: 10.1016/0022-1236(91)90136-S. Google Scholar [12] Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, Func. Analysis and Related Topics, 102 (1991), 55. Google Scholar [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. doi: 10.1007/BF01443518. Google Scholar [14] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969). Google Scholar [15] P. Maremonti, On the Stokes equations: The maximum modulus theorem,, Math. Models Methods Appl. Sci., 10 (2000), 1047. doi: 10.1142/S0218202500000537. Google Scholar [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. doi: 10.1007/s10958-009-9458-3. Google Scholar [17] P. Maremonti, A remark on the Stokes problem with initial data in $L^1$,, J. of Math. Fluid Mech., 13 (2011), 469. doi: 10.1007/s00021-010-0036-8. Google Scholar [18] P. Maremonti, Pointwise asymptotic stability of steady fluid maotion,, J. Math. Fluid Mech., 11 (2009), 348. Google Scholar [19] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Ann. Sc. Norm. Super. Pisa, 24 (1997), 395. Google Scholar [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. doi: 10.1007/BF02355828. Google Scholar [21] T. Miyakawa, On non-stationary solutions of the Navier-Stokes equations in an exterior domain,, Hiroshima Math. J., 12 (1982), 115. Google Scholar [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. doi: 10.1007/s00021-005-0212-4. Google Scholar [23] V. A. Solonnikov, Estimates for the solutions of a nonstationary linearized system of Navier-Stokes equations,, Trudy Mat. Inst. Steklov, 70 (1964), 213. Google Scholar [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. Google Scholar [25] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Soviet Math., 8 (1977), 467. doi: 10.1007/BF01084616. Google Scholar [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. doi: 10.1023/A:1022317029111. Google Scholar [27] V. A. Solonnikov, On the estimates of the solution of the evolution Stokes problem in weighted Hölder norms,, Annali dell'Univ. di Ferrara, 52 (2006), 137. doi: 10.1007/s11565-006-0012-7. Google Scholar [28] R. Temam, Navier-Stokes Equations,, Theory and numerical analysis. With an appendix by F. Thomasset. Third edition. Studies in Mathematics and its Applications, (1984). Google Scholar
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