# American Institute of Mathematical Sciences

February  2016, 9(1): 255-267. doi: 10.3934/dcdss.2016.9.255

## A note on the Navier-Stokes IBVP with small data in $L^n$

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

Received  September 2014 Revised  February 2015 Published  December 2015

We study existence and uniqueness of regular solutions to the Navier-Stokes initial boundary value problem in bounded or exterior domains $\Omega$ ($\partial\Omega$ sufficiently smooth) under the assumption $v_\circ$ in $L^n(\Omega)$, sufficiently small, and we prove global in time existence. The results are known in literature (see Remark 3), however the proof proposed here seems shorter, and we give a result concerning the behavior in time of the $L^q$-norm ($q\in[n,\infty]$) of the solutions and of the $L^n$-norm of the time derivative, with a sort of continuous dependence on the data, which, as far as we know, are new, and are close to the ones of the solution to the Stokes problem. Moreover, the constant for the $L^q$-estimate is independent of $q$.
Citation: Paolo Maremonti. A note on the Navier-Stokes IBVP with small data in $L^n$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 255-267. doi: 10.3934/dcdss.2016.9.255
##### References:
 [1] H. Beirăo da Veiga and P. Secchi, $L^p$-stability for the strong solutions of the Navier-Stokes equations in the whole space,, Arch. for Rational Mech. and Anal., 98 (1987), 65. doi: 10.1007/BF00279962. Google Scholar [2] F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11. Google Scholar [3] Y. Enomoto and Y. Shibata, On a stability theorem of the Navier-Stokes equation in an exterior domain,, in Hyperbolic Problems, (2006), 383. Google Scholar [4] R. Farwig, H. Kozono and H. Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general domains,, Acta Math., 195 (2005), 21. doi: 10.1007/BF02588049. Google Scholar [5] C. Foias, Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension $n$,, Bull. Soc. Math. France, 89 (1961), 1. Google Scholar [6] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, I,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar [7] G. P. Galdi, J. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest,, Arch. Rational Mech. Anal., 138 (1997), 307. doi: 10.1007/s002050050043. Google Scholar [8] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system,, J. Differential Equations, 62 (1986), 186. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [9] Y. Giga and T. Miyakawa, Solutions in $L^r$ of the Navier-Stokes initial value problem,, Arch. Rational Mech. Anal., 89 (1985), 267. Google Scholar [10] T. Kato, Strong $L^p$-solution of the Navier-Stokes equation in $\mathbbR^n$, with applications to weak solutions,, Math. Z., 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar [11] H. Kozono and T. Ogawa, Decay properties of strong solutions for the Navier-Stokes equations in two dimensional unbounded domains,, Arch. Rational Mech. Anal., 122 (1993), 1. doi: 10.1007/BF01816552. Google Scholar [12] H. Kozono and T. Ogawa, On stability of Navier-Stokes flows in exterior domains,, Arch. Rational Mech. Anal., 128 (1994), 1. doi: 10.1007/BF00380792. Google Scholar [13] P. Maremonti, Stabilità asintotica in media per moti fluidi viscosi in domini esterni,, Annali di Matematica Pura ed Applicata, 142 (1985), 57. Google Scholar [14] P. Maremonti, On the asymptotic behavior of the $L^2$-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains,, Comm. Math. Phys., 118 (1988), 385. doi: 10.1007/BF01466723. Google Scholar [15] P. Maremonti, Some results on the asymptotic behavior of Hopf weak solutions to the Navier-Stokes equations in unbounded domains,, Math. Z., 210 (1992), 1. doi: 10.1007/BF02571780. Google Scholar [16] P. Maremonti, A remark on the Stokes problem with initial data in $L^1$,, J. Math. Fluid Mech., 13 (2011), 469. doi: 10.1007/s00021-010-0036-8. Google Scholar [17] P. Maremonti, A remark on the Stokes problem in Lorentz spaces,, Discrete and Continuous Dynamical Systems, 6 (2013), 1323. doi: 10.3934/dcdss.2013.6.1323. Google Scholar [18] P. Maremonti, On the Stokes problem in exterior domains: The maximum modulus theorem,, Discrete and Continuous Dynamical Systems, 34 (2014), 2135. doi: 10.3934/dcds.2014.34.2135. Google Scholar [19] P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes system in exterior domains,, Zap. Nauch. Sem. LOMI, 180 (1990), 105. doi: 10.1007/BF01249337. Google Scholar [20] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 395. Google Scholar [21] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Soviet Math., 8 (1977), 467. doi: 10.1007/BF01084616. Google Scholar [22] M. Yamazaki, The Navier-Stokes equations in the weak-Ln external force,, Math. Ann., 317 (2000), 635. doi: 10.1007/PL00004418. Google Scholar

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##### References:
 [1] H. Beirăo da Veiga and P. Secchi, $L^p$-stability for the strong solutions of the Navier-Stokes equations in the whole space,, Arch. for Rational Mech. and Anal., 98 (1987), 65. doi: 10.1007/BF00279962. Google Scholar [2] F. Crispo and P. Maremonti, An interpolation inequality in exterior domains,, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11. Google Scholar [3] Y. Enomoto and Y. Shibata, On a stability theorem of the Navier-Stokes equation in an exterior domain,, in Hyperbolic Problems, (2006), 383. Google Scholar [4] R. Farwig, H. Kozono and H. Sohr, An Lq-approach to Stokes and Navier-Stokes equations in general domains,, Acta Math., 195 (2005), 21. doi: 10.1007/BF02588049. Google Scholar [5] C. Foias, Une remarque sur l'unicité des solutions des équations de Navier-Stokes en dimension $n$,, Bull. Soc. Math. France, 89 (1961), 1. Google Scholar [6] G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, I,, Springer-Verlag, (1994). doi: 10.1007/978-1-4612-5364-8. Google Scholar [7] G. P. Galdi, J. Heywood and Y. Shibata, On the global existence and convergence to steady state of Navier-Stokes flow past an obstacle that is started from rest,, Arch. Rational Mech. Anal., 138 (1997), 307. doi: 10.1007/s002050050043. Google Scholar [8] Y. Giga, Solutions for semilinear parabolic equations in $L^p$ and regularity of weak solutions of the Navier-Stokes system,, J. Differential Equations, 62 (1986), 186. doi: 10.1016/0022-0396(86)90096-3. Google Scholar [9] Y. Giga and T. Miyakawa, Solutions in $L^r$ of the Navier-Stokes initial value problem,, Arch. Rational Mech. Anal., 89 (1985), 267. Google Scholar [10] T. Kato, Strong $L^p$-solution of the Navier-Stokes equation in $\mathbbR^n$, with applications to weak solutions,, Math. Z., 187 (1984), 471. doi: 10.1007/BF01174182. Google Scholar [11] H. Kozono and T. Ogawa, Decay properties of strong solutions for the Navier-Stokes equations in two dimensional unbounded domains,, Arch. Rational Mech. Anal., 122 (1993), 1. doi: 10.1007/BF01816552. Google Scholar [12] H. Kozono and T. Ogawa, On stability of Navier-Stokes flows in exterior domains,, Arch. Rational Mech. Anal., 128 (1994), 1. doi: 10.1007/BF00380792. Google Scholar [13] P. Maremonti, Stabilità asintotica in media per moti fluidi viscosi in domini esterni,, Annali di Matematica Pura ed Applicata, 142 (1985), 57. Google Scholar [14] P. Maremonti, On the asymptotic behavior of the $L^2$-norm of suitable weak solutions to the Navier-Stokes equations in three-dimensional exterior domains,, Comm. Math. Phys., 118 (1988), 385. doi: 10.1007/BF01466723. Google Scholar [15] P. Maremonti, Some results on the asymptotic behavior of Hopf weak solutions to the Navier-Stokes equations in unbounded domains,, Math. Z., 210 (1992), 1. doi: 10.1007/BF02571780. Google Scholar [16] P. Maremonti, A remark on the Stokes problem with initial data in $L^1$,, J. Math. Fluid Mech., 13 (2011), 469. doi: 10.1007/s00021-010-0036-8. Google Scholar [17] P. Maremonti, A remark on the Stokes problem in Lorentz spaces,, Discrete and Continuous Dynamical Systems, 6 (2013), 1323. doi: 10.3934/dcdss.2013.6.1323. Google Scholar [18] P. Maremonti, On the Stokes problem in exterior domains: The maximum modulus theorem,, Discrete and Continuous Dynamical Systems, 34 (2014), 2135. doi: 10.3934/dcds.2014.34.2135. Google Scholar [19] P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes system in exterior domains,, Zap. Nauch. Sem. LOMI, 180 (1990), 105. doi: 10.1007/BF01249337. Google Scholar [20] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Ann. Sc. Norm. Sup. Pisa, 24 (1997), 395. Google Scholar [21] V. A. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equations,, J. Soviet Math., 8 (1977), 467. doi: 10.1007/BF01084616. Google Scholar [22] M. Yamazaki, The Navier-Stokes equations in the weak-Ln external force,, Math. Ann., 317 (2000), 635. doi: 10.1007/PL00004418. Google Scholar
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