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Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in ℝ n

Reinhard Farwig; Hermann Sohr

Czechoslovak Mathematical Journal (2009)

  • Volume: 59, Issue: 1, page 61-79
  • ISSN: 0011-4642

Abstract

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For a bounded domain Ω ⊂ ℝ n , n ≥ 3 , we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system - Δ u + u · ∇ u + ∇ p = f , div u = k , u | ∂ Ω = g with u ∈ L q , q ≥ n , and very general data classes for f , k , g such that u may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.

How to cite

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Farwig, Reinhard, and Sohr, Hermann. "Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb {R}^n$." Czechoslovak Mathematical Journal 59.1 (2009): 61-79. <http://eudml.org/doc/37908>.

@article{Farwig2009,
abstract = {For a bounded domain $\Omega \subset \mathbb \{R\} ^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\operatorname\{div\}u = k$, $u_\{|_\{\partial \Omega \}\}=g$ with $u \in L^q$, $q \ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.},
author = {Farwig, Reinhard, Sohr, Hermann},
journal = {Czechoslovak Mathematical Journal},
keywords = {stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions; stationary Stokes system; Navier-Stokes system, very weak solution; regularity class; higher dimension},
language = {eng},
number = {1},
pages = {61-79},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb \{R\}^n$},
url = {http://eudml.org/doc/37908},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Farwig, Reinhard
AU - Sohr, Hermann
TI - Existence, uniqueness and regularity of stationary solutions to inhomogeneous Navier-Stokes equations in $\mathbb {R}^n$
JO - Czechoslovak Mathematical Journal
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 59
IS - 1
SP - 61
EP - 79
AB - For a bounded domain $\Omega \subset \mathbb {R} ^n$, $n\ge 3,$ we use the notion of very weak solutions to obtain a new and large uniqueness class for solutions of the inhomogeneous Navier-Stokes system $-\Delta u + u \cdot \nabla u + \nabla p=f$, $\operatorname{div}u = k$, $u_{|_{\partial \Omega }}=g$ with $u \in L^q$, $q \ge n$, and very general data classes for $f$, $k$, $g$ such that $u$ may have no differentiability property. For smooth data we get a large class of unique and regular solutions extending well known classical solution classes, and generalizing regularity results. Moreover, our results are closely related to those of a series of papers by Frehse Růžička, see e.g. Existence of regular solutions to the stationary Navier-Stokes equations, Math. Ann. 302 (1995), 669–717, where the existence of a weak solution which is locally regular is proved.
LA - eng
KW - stationary Stokes and Navier-Stokes system; very weak solutions; existence and uniqueness in higher dimensions; regularity classes in higher dimensions; stationary Stokes system; Navier-Stokes system, very weak solution; regularity class; higher dimension
UR - http://eudml.org/doc/37908
ER -

References

top
  1. Adams, R. A., Sobolev Spaces, Academic Press, New York (1975). (1975) Zbl0314.46030MR0450957
  2. Amann, H., 10.1007/978-1-4615-0701-7_1, Int. Math. Ser., Kluwer Academic/Plenum Publishing, New York (2002), 1-28. (2002) MR1971987DOI10.1007/978-1-4615-0701-7_1
  3. Amann, H., 10.2991/jnmp.2003.10.s1.1, J. Nonlinear Math. Physics 10 (2003), 1-11. (2003) MR2063541DOI10.2991/jnmp.2003.10.s1.1
  4. Borchers, W., Miyakawa, T., 10.32917/hmj/1206128724, Hiroshima Math. J. 21 (1991), 621-640. (1991) MR1148998DOI10.32917/hmj/1206128724
  5. Bogovskij, M. E., Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Soviet Math. Dokl. 20 (1979), 1094-1098. (1979) Zbl0499.35022
  6. Cannone, M., Viscous flows in Besov spaces, Advances in Math. Fluid Mech., Springer, Berlin (2000), 1-34. (2000) Zbl0980.35125MR1863208
  7. Fabes, E. B., Jones, B. F., Rivière, N. M., 10.1007/BF00281533, Arch. Rational Mech. Anal. 45 (1972), 222-240. (1972) MR0316915DOI10.1007/BF00281533
  8. Farwig, R., Sohr, H., 10.2969/jmsj/04640607, J. Math. Soc. Japan 46 (1994), 607-643. (1994) Zbl0819.35109MR1291109DOI10.2969/jmsj/04640607
  9. Farwig, R., Galdi, G. P., Sohr, H., 10.1007/s00021-005-0182-6, J. Math. Fluid Mech. 8 (2006), 423-444. (2006) Zbl1104.35032MR2258419DOI10.1007/s00021-005-0182-6
  10. Frehse, J., Růžička, M., 10.1007/BF00995129, Acta Appl. Math. 37 53-66 (1994). (1994) MR1308745DOI10.1007/BF00995129
  11. Frehse, J., Růžička, M., 10.1007/BF00387714, Arch. Rational Mech. Anal. 128 361-380 (1994). (1994) MR1308859DOI10.1007/BF00387714
  12. Frehse, J., Růžička, M., On the regularity of the stationary Navier-Stokes equations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (IV) 21 63-95 (1994). (1994) MR1276763
  13. Frehse, J., Růžička, M., 10.1007/BF01444513, Math. Ann. 302 669-717 (1995). (1995) MR1343646DOI10.1007/BF01444513
  14. Frehse, J., Růžička, M., Existence of regular solutions to the steady Navier-Stokes equations in bounded six-dimensional domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. (IV) 23 701-719 (1996). (1996) MR1469571
  15. Frehse, J., Růžička, M., 10.1142/9789812816740_0013, Proc. 3rd Intern. Conf. Navier-Stokes Equations: theory and numerical methods. World Scientific Ser. Adv. Math. Appl. Sci., Singapore 47 159-178 (1998). (1998) MR1643033DOI10.1142/9789812816740_0013
  16. Frehse, J., Růžička, M., A new regularity criterion for steady Navier-Stokes equations, Differential Integral Equations 11 (1998), 361-368. (1998) MR1741851
  17. Fujiwara, D., Morimoto, H., An L r -theory of the Helmholtz decomposition of vector fields, J. Fac. Sci. Univ. Tokyo (1A) 24 (1977), 685-700. (1977) MR0492980
  18. Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Linearized Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York (1998). (1998) MR2808162
  19. Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations; Nonlinear Steady Problems, Springer Tracts in Natural Philosophy, Vol. 39, New York (1998). (1998) MR2808162
  20. Galdi, G. P., Simader, C. G., Sohr, H., 10.1007/s00208-004-0573-7, Math. Ann. 331 (2005), 41-74. (2005) Zbl1064.35133MR2107439DOI10.1007/s00208-004-0573-7
  21. Gerhardt, C., 10.1007/BF01182469, Math. Z. 165 (1979), 193-197. (1979) MR0520820DOI10.1007/BF01182469
  22. Giga, Y., 10.1007/BF01214869, Math. Z. 178 (1981), 287-329. (1981) Zbl0473.35064MR0635201DOI10.1007/BF01214869
  23. Giga, Y., 10.1007/BF00276874, Arch. Rational Mech. Anal. 89 (1985), 251-265. (1985) MR0786549DOI10.1007/BF00276874
  24. Giga, Y., Sohr, H., On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 103-130. (1989) MR0991022
  25. Giga, Y., Sohr, H., 10.1016/0022-1236(91)90136-S, J. Funct. Anal. 102 (1991), 72-94. (1991) Zbl0739.35067MR1138838DOI10.1016/0022-1236(91)90136-S
  26. Kato, T., 10.1007/BF01174182, Math. Z. 187 (1984), 471-480. (1984) MR0760047DOI10.1007/BF01174182
  27. Kozono, H., Yamazaki, M., Local and global solvability of the Navier-Stokes exterior problem with Cauchy data in the space L n , ∞ , Houston J. Math. 21 (1995), 755-799. (1995) MR1368344
  28. Nečas, J., Les Méthodes Directes en Théorie des Équations Elliptiques, Academia, Prague (1967). (1967) MR0227584
  29. Simader, C. G., Sohr, H., 10.1142/9789814503594_0001, Adv. Math. Appl. Sci., World Scientific 11 (1992), 1-35. (1992) MR1190728DOI10.1142/9789814503594_0001
  30. Solonnikov, V. A., 10.1007/BF01084616, J. Soviet Math. 8 (1977), 467-528. (1977) Zbl0404.35081DOI10.1007/BF01084616
  31. Sohr, H., The Navier-Stokes equations. An elementary functional analytic approach, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel (2001). (2001) Zbl0983.35004MR1928881
  32. Temam, R., Navier-Stokes Equations. Theory and numerical analysis, North-Holland, Amsterdam, New York, Tokyo (1984). (1984) Zbl0568.35002MR0769654
  33. Triebel, H., Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978). (1978) Zbl0387.46033MR0503903
  34. Wahl, W. von, Regularity of weak solutions of the Navier-Stokes equations, Proc. Symp. Pure Math. 45 (1986), 497-503. (1986) MR0843635

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