A large class of solutions for the instationary Navier–Stokes system

1. R. A. Adams and J. J. Fournier, Sobolev Spaces, Elsevier, Oxford, 2nd ed., 2003.

2. H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I: Abstract Linear Theory, Monographs in Mathematics, 89, Birkhäuser, Basel, Boston, Berlin, 1995.

3. H. Amann, Nonhomogeneous Navier–Stokes equations with integrable low-regularity data, in Nonlinear Problems in Mathematical Physics and Related Problems II. In Honor of Professor O. A. Ladyzhenskaya, M. S. Birman, S. Hildebrandt, V. A. Solonnikov, and N. N. Uraltseva, eds., International Mathematical Series, 2:1–26, Kluwer Academic/Plenum Publishers, New York, 2002.

4. Amrouche C., Girault V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech. Math. J. 44, 109–140 (1994)

   MATH  MathSciNet  Google Scholar 

5. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Grundlehren der mathematischen Wissenschaften, 223, Springer, Berlin, Heidelberg, New York, 1976.

6. R. Denk, M. Hieber, and J. Prüss, \({\mathcal{R}}\) -Boundedness, Fourier Multipliers and Problems of Elliptic and Parabolic Type, Mem. Amer. Math. Soc., 166, American Mathematical Society, Providence, 2003.

7. R. Farwig, G. Galdi, and H. Sohr, Very weak solutions of stationary and instationary Navier–Stokes equations with nonhomogeneous data, in Nonlinear Elliptic and Parabolic Problems, M. Chipot and J. Escher, eds., Progress in Nonlinear Differential Equations and Their Applications, 64:113–136, Birkhäuser, Basel, Boston, Berlin, 2005.

8. R. Farwig, G. Galdi, and H. Sohr, Very weak solutions and large uniqueness classes of stationary Navier–Stokes equations in bounded domains of \({\mathbb{R}^2}\) ., J. Differential Equations, 227 (2006), pp. 564–580.

   Google Scholar 

9. Farwig R., Galdi G.P., Sohr H.: A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data. J. Math. Fluid Mech., 8, 423–444 (2006)

   Article  MATH  MathSciNet  Google Scholar 

10. R. Farwig, H. Kozono, and H. Sohr, Very weak, weak and strong solutions to the instationary Navier–Stokes system, in Topics on partial differential equations, Lecture Notes, 2:1–54, P. Kaplický and Š. Nečasová, eds., J. Nečas Center Math. Model., Prague, 2007.

11. G. P. Galdi, C. G. Simader, and H. Sohr, A class of solutions to stationary Stokes and Navier–Stokes equations with boundary data in \({W^{-\frac{1}{q},q}}\) , Math. Ann., 331 (2005), pp. 41–74.

12. Giga Y.: Analyticity of the semigroup generated by the Stokes operator in L _r spaces. Math. Z. 178, 297–329 (1981)

    Article  MATH  MathSciNet  Google Scholar 

13. Schumacher K.: Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces. Ann. Univ. Ferrara Sez. VII Sci. Math. 54, 123–144 (2008)

    Article  MathSciNet  Google Scholar 

14. Schumacher K.: The instationary Stokes equations in weighted Bessel-potential spaces. J. Evol. Equ. 9, 1–36 (2009)

    Article  MathSciNet  Google Scholar 

15. J. Serrin, The initial value problem for the Navier–Stokes equations, in Nonlinear Problems, 69–98, R. Langer, ed., Madison, 1963, University of Wisconsin.

16. Simader C.G., Sohr H.: A new approach to the Helmholtz decomposition and the Neumann problem in L ^q-spaces for bounded and exterior domains. Adv. Math. Appl. Sci. 11, 1–35 (1992)

    MathSciNet  Google Scholar 

Download references