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, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[2]

W. Borchers and T. Miyakawa, Algebraic $L^2$-decay for Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227. doi: 10.1007/BF02391905.  Google Scholar

[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), 181-207. doi: 10.2969/jmsj/05110181.  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), 115-142. doi: 10.1007/PL00001362.  Google Scholar

[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), 21-53. doi: 10.1007/BF02588049.  Google Scholar

[6]

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. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[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), 633-652. doi: 10.1007/s00021-011-0083-9.  Google Scholar

[8]

Y. Giga and H. Sohr, On the Stokes operator in exterior domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130.  Google Scholar

[9]

Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system, Func. Analysis and Related Topics, 102 (1991), 55-67. Google Scholar

[10]

R. A. Hunt, On L(p,q) spaces, Enseignement Mathématique (2), 12 (1966), 249-276.  Google Scholar

[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), 265-288. doi: 10.1007/BF01443518.  Google Scholar

[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), 959-1014. doi: 10.1080/03605309408821042.  Google Scholar

[13]

P. Maremonti, Some interpolation inequalities involving Stokes operator and first order derivatives, Ann. Mat. Pura Appl. (4), 175 (1998), 59-91. doi: 10.1007/BF01783676.  Google Scholar

[14]

P. Maremonti, Pointwise asymptotic stability of steady fluid motions, J. Math. Fluid Mech., 11 (2009), 348-382. doi: 10.1007/s00021-007-0262-x.  Google Scholar

[15]

P. Maremonti, A remark on the Stokes problem with initial data in $L^1$, J. Math. Fluid Mech., 13 (2011), 469-480. doi: 10.1007/s00021-010-0036-8.  Google Scholar

[16]

P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains, J. Math. Sci., 68 (1994), 229-239. doi: 10.1007/BF01249337.  Google Scholar

[17]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Annali Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 395-449.  Google Scholar

[18]

E. T. Oklander, $L_{pq}$ interpolators and the theorem of Marcinkiewicz, Bull. A. M. S., 72 (1966), 49-53.  Google Scholar

[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 "Mathematical Problems Relating to the Navier-Stokes Equation," Ser. Adv. Math. Appl. Sci., 11, World Scientific Publ., River Edge, NJ, 1992.  Google Scholar

[20]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1990.  Google Scholar

[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), 2107-2117. doi: 10.1080/03605300008821579.  Google Scholar

[22]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[23]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, "Interpolation Spaces. An Introduction," Grundlehren der Mathematischen Wissenschaften, No. 223, Springer-Verlag, Berlin-New York, 1976.  Google Scholar

[2]

W. Borchers and T. Miyakawa, Algebraic $L^2$-decay for Navier-Stokes flows in exterior domains, Acta Math., 165 (1990), 189-227. doi: 10.1007/BF02391905.  Google Scholar

[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), 181-207. doi: 10.2969/jmsj/05110181.  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), 115-142. doi: 10.1007/PL00001362.  Google Scholar

[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), 21-53. doi: 10.1007/BF02588049.  Google Scholar

[6]

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. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[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), 633-652. doi: 10.1007/s00021-011-0083-9.  Google Scholar

[8]

Y. Giga and H. Sohr, On the Stokes operator in exterior domain, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 36 (1989), 103-130.  Google Scholar

[9]

Y. Giga and H. Sohr, $L^p$ estimates for the Stokes system, Func. Analysis and Related Topics, 102 (1991), 55-67. Google Scholar

[10]

R. A. Hunt, On L(p,q) spaces, Enseignement Mathématique (2), 12 (1966), 249-276.  Google Scholar

[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), 265-288. doi: 10.1007/BF01443518.  Google Scholar

[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), 959-1014. doi: 10.1080/03605309408821042.  Google Scholar

[13]

P. Maremonti, Some interpolation inequalities involving Stokes operator and first order derivatives, Ann. Mat. Pura Appl. (4), 175 (1998), 59-91. doi: 10.1007/BF01783676.  Google Scholar

[14]

P. Maremonti, Pointwise asymptotic stability of steady fluid motions, J. Math. Fluid Mech., 11 (2009), 348-382. doi: 10.1007/s00021-007-0262-x.  Google Scholar

[15]

P. Maremonti, A remark on the Stokes problem with initial data in $L^1$, J. Math. Fluid Mech., 13 (2011), 469-480. doi: 10.1007/s00021-010-0036-8.  Google Scholar

[16]

P. Maremonti and V. A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains, J. Math. Sci., 68 (1994), 229-239. doi: 10.1007/BF01249337.  Google Scholar

[17]

P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Annali Scuola Normale Superiore Pisa Cl. Sci. (4), 24 (1997), 395-449.  Google Scholar

[18]

E. T. Oklander, $L_{pq}$ interpolators and the theorem of Marcinkiewicz, Bull. A. M. S., 72 (1966), 49-53.  Google Scholar

[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 "Mathematical Problems Relating to the Navier-Stokes Equation," Ser. Adv. Math. Appl. Sci., 11, World Scientific Publ., River Edge, NJ, 1992.  Google Scholar

[20]

E. M. Stein and G. Weiss, "Introduction to Fourier Analysis on Euclidean Spaces," Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ, 1990.  Google Scholar

[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), 2107-2117. doi: 10.1080/03605300008821579.  Google Scholar

[22]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators," North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[23]

M. Yamazaki, The Navier-Stokes equations in the weak-$L^n$ space with time-dependent external force, Math. Ann., 317 (2000), 635-675. doi: 10.1007/PL00004418.  Google Scholar

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