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2017 | OriginalPaper | Buchkapitel

Weak Solutions to the Navier–Stokes Equations with Data in \(\mathbb {L}(3,\infty )\)

verfasst von : P. Maremonti

Erschienen in: Mathematics for Nonlinear Phenomena — Analysis and Computation

Verlag: Springer International Publishing

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Abstract

The paper concerns the existence of weak solutions to the 3d-Navier–Stokes initial boundary value problem in exterior domains. The problem is considered with an initial data belonging to \(\mathbb L(3,\infty )\) which is a special subspace of the Lorentz’s space \(L(3,\infty )\). The nature of the domain and the initial data in \(L(3,\infty )\) make the result of existence not comparable with the usual Leray-Hopf theory of weak solutions. However, we are able to prove both that the weak solutions enjoy the partial regularity in the sense of Leray’s structure theorem and the asymptotic limit of \(|u(t)|_{3\infty }\).

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The results of this note were communicated in some meetings: http://www.math.sci.hokudai.ac.jp/sympo/sapporo/program150819_en.html, http://perso-math.univ-mlv.fr/users/danchin.raphael/Porq15/abstract15.pdf, https://www.maths.ox.ac.uk/events/past/2895.

K. Abe, Y. Giga, The \(L^\infty \)-Stokes semigroup in exterior domains. J. Evol. Eq. 14, 1–28 (2014)MathSciNetCrossRefMATH

K. Abe, Y. Giga, Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211, 1–46 (2013)MathSciNetCrossRefMATH

A. Alvino, Sulla diseguaglianza di Sobolev in spazi di Lorentz. Boll. Un. Mat. Ital. A 14, 148–156 (1977)MathSciNetMATH

M. Bolkart, M. Hieber, Pointwise upper bounds for the solution of the Stokes equation on \(L^\infty _\sigma (\Omega )\) and applications. J. Funct. Anal. 268, 1678–1710 (2015)MathSciNetCrossRefMATH

L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)MathSciNetCrossRefMATH

F. Crispo, P. Maremonti, An interpolation inequality in exterior domains. Rend. Sem. Mat. Univ. Padova 112, 11–39 (2004)MathSciNetMATH

F. Crispo, P. Maremonti, On the spatial asymptotic decay of a suitable weak solution to the Navier-Stokes Cauchy problem. Nonlinearity 29, 1355–1383 (2016). https://doi.org/10.1088/0951-7715/29/4/1355

R. Farwig, H. Kozono, H. Sohr, An \(L^q\)-approach to Stokes and Navier-Stokes equations in general domains. Acta Math. 195, 21–53 (2005)MathSciNetCrossRefMATH

G.P. Galdi, An introduction to the Mathematical theory of the Navier-Stokes Equations (Springer, Berlin, 1994)

10.

Y. Giga, T. Miyakawa, Solutions in \(L^r\) of the Navier-Stokes initial value problem. Arch. Ration. Mech. Anal. 89, 267–281 (1985)MathSciNetCrossRefMATH

11.

Y. Giga, H. Sohr, Abstract \(L^p\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal. 102, 72–94 (1991)MathSciNetCrossRefMATH

12.

J. Heywood, The Navier-Stokes equations: on the existence, regularity and decay of solutions. Indiana Univ. Math. J. 29, 639–681 (1980)MathSciNetCrossRefMATH

13.

R.A. Hunt, On \(L(p, q)\) spaces. L’enseignement Mathematique 12, 249–276 (1966)MATH

14.

T. Kato, Strong \(L^p\)-solutions of the Navier-Stokes equation in \(R^m\), with applications to weak solutions. Math. Z. 187, 471–480 (1984)MathSciNetCrossRefMATH

15.

H. Koba, On \(L^{3,\infty }\)-Stability of the Navier-Stokes System in Exterior Domains, arXiv:1504.0814301

16.

V. Kolyada, J. Soria, Hölder type inequalities in Lorentz spaces. Annali di Matematica 189, 523–538 (2010)MathSciNetCrossRefMATH

17.

H. Kozono, Global \(L^n\) -solution and its decay property for the Navier-Stokes equations in half-space \(R^n_+\). J. Diff. Eq. 79, 79–88 (1989)MathSciNetCrossRefMATH

18.

H. Kozono, M. Yamazaki, Local and global unique solvability of the Navier-Stokes exterior problem with Cauchy data in the space \(L^{n,\infty }\). Houston J. Math. 21, 755–799 (1995)MathSciNetMATH

19.

O.A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Viscous Fluid (NewYork, Gordono Breach, 1969)MATH

20.

J. Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63, 193–248 (1934)MathSciNetCrossRefMATH

21.

P. Maremonti, Partial regularity of a generalized solution to the Navier-Stokes equations in exterior domain. Commun. Math. Phys. 110, 75–87 (1987)MathSciNetCrossRefMATH

22.

P. Maremonti, A remark on the Stokes problem with initial data in \(L^1\). J. Math. Fluid Mech 13, 469–480 (2011)MathSciNetCrossRefMATH

23.

P. Maremonti, A remark on the Stokes problem in Lorentz spaces. Discret. Contin. Dyn. Syst. Ser. S 6, 1323–1342 (2013)MathSciNetCrossRefMATH

24.

P. Maremonti, On the Stokes problem in exterior domains: the maximum modulus theorem. Discret. Contin. Dyn. Syst. 34, 2135–2171 (2014)MathSciNetCrossRefMATH

25.

P. Maremonti, On weak D-solutions to the non-stationary Navier-Stokes equations in a three-dimensional exterior domain. Ann. Univ. Ferrara Sez. VII Sci. Mat. 60, 209–223 (2014)

26.

P. Maremonti, Regular solutions to the Navier-Stokes equation with an initial data in \(L(n,\infty )\), Ricerche di Matematica 66, 65–97 (2017). https://doi.org/10.1007/s11587-016-0287-7

27.

P. Maremonti, V.A. Solonnikov, An estimate for the solutions of Stokes equations in exterior domains, Zap. Nauch. Sem. LOMI, 180, 105–120, trasl. J. Math. Sci. 68(1994), 229–239 (1990)

28.

P. Maremonti, V.A. Solonnikov, On nonstationary Stokes problem in exterior domains. Ann. Sc. Norm. Sup. Pisa 24, 395–449 (1997)MathSciNetMATH

29.

R. O’Neil, Convolution operators and L(p, q) spaces. Duke Math. J. 30, 129–142 (1963)MathSciNetCrossRefMATH

30.

G. Seregin, Lecture Notes on Regularity Theory for the Navier-Stokes Equations (World Scientific Publishing Co Pte. Ltd., Hackensack, NJ, 2015)MATH

31.

G. Seregin, Selected Topics of Local Regularity Theory for Navier-Stokes Equations, Topics in Mathematical Fluid Mechanics, Lecture Notes in Math., 2073, (Springer, Heidelberg, 2013), pp. 239–313

32.

G. Seregin, V. Šverák, On global weak solutions to the Cauchy problem for the Navier-Stokes equations with large \(L_3\)-initial data, Nonlinear Analisys, online 2 March 2016

33.

J. Serrin, The Initial Value Problem for the Navier-Stokes Equations, ed. by R.E. Langer (University of Wisconsin press, 1963), pp. 69–98

34.

Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations. Quart. Appl. Math. 57, 117–155 (1999)MathSciNetCrossRefMATH

35.

E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press, Princeton, 1990)

36.

R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, 3rd rev. edn. (North-Holland, Amsterdam, 1984)

Titel: Weak Solutions to the Navier–Stokes Equations with Data in
verfasst von: P. Maremonti
Verlag: Springer International Publishing
Buch: Mathematics for Nonlinear Phenomena — Analysis and Computation
Print ISBN: 978-3-319-66762-1

Electronic ISBN: 978-3-319-66764-5

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-66764-5_8

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