Necessary and sufficient conditions for local strong solvability of the Navier–Stokes equations in exterior domains

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• 作者：Christian Komo
• 关键词：Primary ; 76D05 ; Secondary ; 35D35 ; 35Q30
• 刊名：Journal of Evolution Equations
• 出版年：2014
• 出版时间：September 2014
• 年：2014
• 卷：14
• 期：3
• 页码：713-725
• 全文大小：234 KB
• 参考文献：1. Amann H.: On the strong solvability of the Navier–Stokes equations. J. Math. Fluid. Mech. 2, 16-8 (2000) CrossRef
  2. Borchers W., Miyakawa T.: Algebraic / L ²-decay for Navier–Stokes flow in exterior domains. Acta Math. 165, 189-27 (1990) CrossRef
  3. W. Borchers and H. Sohr, / On the semigroup of the Stokes operator for exterior domains, Math. Z. 196 (1987), 415-25
  4. R. Farwig and C. Komo, / Regularity of weak solutions to the Navier–Stokes equations in exterior domains, Nonlinear Differ. Equ. Appl. 17 (2010), 303-21
  5. R. Farwig and C. Komo, / Optimal initial value conditions for local strong solutions of the Navier–Stokes equations in exterior domains, Analysis (Munich) 33 (2013), 101-19
  6. R. Farwig, H. Kozono and H. Sohr, / Very weak solutions of the Navier–Stokes equations in exterior domains with nonhomogeneous data, J. Math. Soc. Japan 59 (2007), 127-50
  7. R. Farwig, H. Sohr and W. Varnhorn, / On optimal initial value conditions for local strong solutions of the Navier–Stokes equations, Ann. Univ. Ferrara 55 (2009), 89-10
  8. R. Farwig, H. Sohr and W. Varnhorn, / Necessary and sufficient conditions on local strong solvability of the Navier–Stokes system. Appl. Anal. 90 (2011), 47-8
  9. H. Fujita and T. Kato, / On the Navier–Stokes initial value problem, Arch. Rational Mech. Anal. 16 (1964), 269-15
  10. Y. Giga, / Solution for semilinear parabolic equations in / L ^{/ p} / and regularity of weak solutions for the Navier–Stokes system, J. Differential Equations 61 (1986), 186-12
  11. Y. Giga and H. Sohr, / On the Stokes operator in exterior domains, J. Fac. Sci. Univ. Tokyo, Sec. IA 36 (1989), 103-30
  12. Y. Giga and H. Sohr, / Abstract L ^{/ p} / estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains, J. Funct. Anal. 102 (1991), 72-4
  13. J. Heywood, / The Navier–Stokes equations: on the existence, regularity and decay of solutions, Indiana Univ. Math. J. 29 (1980), 639-81
  14. H. Iwashita, / L _{/ q} ? / L _{/ r} / estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier–Stokes initial value problems in / L _{/ q} spaces, Math. Ann. 285 (1989), 265-88
  15. T. Kato, Strong / L ^{/ p} / solutions of the Navier–Stokes equation in \({\mathbb{R} ^m}\) , / with applications to weak solutions, Math. Z. 187 (1984), 471-80
  16. A. Kiselev and O. Ladyzenskaya, / On the existence and uniqueness of solutions of the non-stationary problems for flows of non-compressible fluids, Amer. Math. Soc. Transl. Ser. 2, 24 (1963), 79-06
  17. T. Miyakawa, / On the initial value problem for the Navier–Stokes equations in / L ^{/ p} -spaces, Hiroshima Math. J. 11 (1981), 9-0
  18. H. Sohr, / The Navier–Stokes Equations: An elementary functional analytic approach, Birkh?user Verlag, Basel, 2001
  19. Solonnikov V.: Estimates for solutions of nonstationary Navier–Stokes equations. J. Soviet Math. 8, 467-29 (1977) CrossRef
  20. E. Stein, / Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, 1970
  
• 作者单位：Christian Komo (1)
  
  1. Department of Mathematics, Darmstadt University of Technology, 64289, Darmstadt, Germany
  
• ISSN：1424-3202

文摘

We consider the instationary Navier–Stokes equations in a smooth exterior domain \({\Omega \subseteq \mathbb{R}^3}\) with initial value u 0, external force f =?div? F and viscosity ν. It is an important question to characterize the class of initial values \({u_0\in L^2_{\sigma}(\Omega)}\) that allow a strong solution \({u \in L^s(0,T; L^q(\Omega))}\) in some interval \({[0,T[ \, , 0 where s, q with 3 q ?/em> and \({\frac{2}{s} + \frac{3}{q} =1}\) are so-called Serrin exponents. In Farwig and Komo (Analysis (Munich) 33:101-19, 2013) it is proved that \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t is necessary and sufficient for the existence of a strong solution \({u \in L^s(0,T ; L^q(\Omega)) \, , 0 , if additionally 3 q ?8; here, A denotes the Stokes operator. In this paper, we will show that this result remains true if q >?8, and consequently, \({\int_0^{\infty} \| e^{-\nu t A} u_0 \|_q^{s} \, {d}t is the optimal initial value condition to obtain such a strong solution for all possible Serrin exponents s, q.