# American Institute of Mathematical Sciences

October  2013, 6(5): 1323-1342. doi: 10.3934/dcdss.2013.6.1323

## A remark on the Stokes problem in Lorentz spaces

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

Received  December 2011 Revised  February 2012 Published  March 2013

We study the Stokes initial boundary value problem with an initial data in a Lorentz space. We develop a suitable technique able to solve the problem and to prove the semigroup properties of the resolving operator in the case of the ''limit exponents''. The results are a completion of the ones related to the usual $L^p$-theory, of the ones already known and they are also useful tool to study some questions related to the Navier-Stokes equations.
Citation: Paolo Maremonti. A remark on the Stokes problem in Lorentz spaces. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1323-1342. doi: 10.3934/dcdss.2013.6.1323
##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, (1976). [2] W. Borchers and T. Miyakawa, Algebraic $L^2$-decay for Navier-Stokes flows in exterior domains,, Acta Math., 165 (1990), 189. doi: 10.1007/BF02391905. [3] W. Dan and Y. Shibata, On the $L_q-L_r$ estimates of the Stokes semigroup in a two-dimensional exterior domain,, J. Math. Soc. Japan, 51 (1999), 181. doi: 10.2969/jmsj/05110181. [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. [5] R. Farwig, H. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains,, Acta Math., 195 (2005), 21. doi: 10.1007/BF02588049. [6] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,", Second edition, (2011). doi: 10.1007/978-0-387-09620-9. [7] 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. doi: 10.1007/s00021-011-0083-9. [8] Y. Giga and H. Sohr, On the Stokes operator in exterior domain,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103. [9] Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, Func. Analysis and Related Topics, 102 (1991), 55. [10] R. A. Hunt, On L(p,q) spaces,, Enseignement Mathématique (2), 12 (1966), 249. [11] 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. [12] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data,, Commun. in Partial Diff. Eq., 19 (1994), 959. doi: 10.1080/03605309408821042. [13] P. Maremonti, Some interpolation inequalities involving Stokes operator and first order derivatives,, Ann. Mat. Pura Appl. (4), 175 (1998), 59. doi: 10.1007/BF01783676. [14] P. Maremonti, Pointwise asymptotic stability of steady fluid motions,, J. Math. Fluid Mech., 11 (2009), 348. doi: 10.1007/s00021-007-0262-x. [15] 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. [16] P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains,, J. Math. Sci., 68 (1994), 229. doi: 10.1007/BF01249337. [17] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Annali Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 395. [18] E. T. Oklander, $L_{pq}$ interpolators and the theorem of Marcinkiewicz,, Bull. A. M. S., 72 (1966), 49. [19] C. Simader and E. Sohr, A new approach to the Helmholtz decomposition and the Neuomann problem in $L^q$-spaces for bounded and exterior domains,, in, 11 (1992). [20] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1990). [21] V. Šverák and T.-P. Tsai, On the spatial decay of 3-D steady-state Navier-Stokes flows,, Commun. Part. Diff. Eq., 25 (2000), 2107. doi: 10.1080/03605300008821579. [22] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978). [23] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force,, Math. Ann., 317 (2000), 635. doi: 10.1007/PL00004418.

show all references

##### References:
 [1] J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction,", Grundlehren der Mathematischen Wissenschaften, (1976). [2] W. Borchers and T. Miyakawa, Algebraic $L^2$-decay for Navier-Stokes flows in exterior domains,, Acta Math., 165 (1990), 189. doi: 10.1007/BF02391905. [3] W. Dan and Y. Shibata, On the $L_q-L_r$ estimates of the Stokes semigroup in a two-dimensional exterior domain,, J. Math. Soc. Japan, 51 (1999), 181. doi: 10.2969/jmsj/05110181. [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. [5] R. Farwig, H. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains,, Acta Math., 195 (2005), 21. doi: 10.1007/BF02588049. [6] G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,", Second edition, (2011). doi: 10.1007/978-0-387-09620-9. [7] 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. doi: 10.1007/s00021-011-0083-9. [8] Y. Giga and H. Sohr, On the Stokes operator in exterior domain,, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103. [9] Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system,, Func. Analysis and Related Topics, 102 (1991), 55. [10] R. A. Hunt, On L(p,q) spaces,, Enseignement Mathématique (2), 12 (1966), 249. [11] 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. [12] H. Kozono and M. Yamazaki, Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data,, Commun. in Partial Diff. Eq., 19 (1994), 959. doi: 10.1080/03605309408821042. [13] P. Maremonti, Some interpolation inequalities involving Stokes operator and first order derivatives,, Ann. Mat. Pura Appl. (4), 175 (1998), 59. doi: 10.1007/BF01783676. [14] P. Maremonti, Pointwise asymptotic stability of steady fluid motions,, J. Math. Fluid Mech., 11 (2009), 348. doi: 10.1007/s00021-007-0262-x. [15] 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. [16] P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains,, J. Math. Sci., 68 (1994), 229. doi: 10.1007/BF01249337. [17] P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains,, Annali Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 395. [18] E. T. Oklander, $L_{pq}$ interpolators and the theorem of Marcinkiewicz,, Bull. A. M. S., 72 (1966), 49. [19] C. Simader and E. Sohr, A new approach to the Helmholtz decomposition and the Neuomann problem in $L^q$-spaces for bounded and exterior domains,, in, 11 (1992). [20] E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces,", Princeton Mathematical Series, (1990). [21] V. Šverák and T.-P. Tsai, On the spatial decay of 3-D steady-state Navier-Stokes flows,, Commun. Part. Diff. Eq., 25 (2000), 2107. doi: 10.1080/03605300008821579. [22] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,", North-Holland Mathematical Library, 18 (1978). [23] M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force,, Math. Ann., 317 (2000), 635. doi: 10.1007/PL00004418.
 [1] Yoshikazu Giga. A remark on a Liouville problem with boundary for the Stokes and the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1277-1289. doi: 10.3934/dcdss.2013.6.1277 [2] Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517 [3] Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195 [4] Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143 [5] Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602 [6] Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149 [7] Misha Perepelitsa. An ill-posed problem for the Navier-Stokes equations for compressible flows. Discrete & Continuous Dynamical Systems - A, 2010, 26 (2) : 609-623. doi: 10.3934/dcds.2010.26.609 [8] Hantaek Bae. Solvability of the free boundary value problem of the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 769-801. doi: 10.3934/dcds.2011.29.769 [9] Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 [10] Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 [11] Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 [12] Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845 [13] Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 [14] Hermenegildo Borges de Oliveira. Anisotropically diffused and damped Navier-Stokes equations. Conference Publications, 2015, 2015 (special) : 349-358. doi: 10.3934/proc.2015.0349 [15] Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433 [16] Vittorino Pata. On the regularity of solutions to the Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 747-761. doi: 10.3934/cpaa.2012.11.747 [17] C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403 [18] Igor Kukavica. On regularity for the Navier-Stokes equations in Morrey spaces. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1319-1328. doi: 10.3934/dcds.2010.26.1319 [19] Igor Kukavica. On partial regularity for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 717-728. doi: 10.3934/dcds.2008.21.717 [20] Susan Friedlander, Nataša Pavlović. Remarks concerning modified Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 269-288. doi: 10.3934/dcds.2004.10.269

2017 Impact Factor: 0.561

## Metrics

• PDF downloads (4)
• HTML views (0)
• Cited by (6)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]