Very weak solutions of the stationary Stokes equations in unbounded domains of half space type

Farwig, Reinhard ; Sauer, Jonas

Mathematica Bohemica, Tome 140 (2015) no. 1, pp. 81-109

Abstract (VO)

We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as $\mathbb R^n_+$, bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of $\mathbb R^n_+$, and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous Sobolev spaces. In addition to very weak solutions we also construct corresponding pressure functions in negative homogeneous Sobolev spaces.

Détail

DOI : 10.21136/MB.2015.144181

Classification : 35D30, 35J65, 35Q30, 35Q35, 76D05
Keywords: Stokes equation; very weak solution; strong solution; domain of half space type

Bibliographie

[1] Amann, H.: Nonhomogeneous Navier-Stokes equations with integrable low-regularity data. Nonlinear Problems in Mathematical Physics and Related Topics II. In Honour of Professor O. A. Ladyzhenskaya Int. Math. Ser. (N.Y.) 2 Kluwer Academic Publishers, New York (2002), 1-28 M. S. Birman et al. | MR | Zbl

[2] Amann, H.: On the strong solvability of the Navier-Stokes equations. J. Math. Fluid Mech. 2 (2000), 16-98. | DOI | MR | Zbl

[3] Nezza, E. Di, Palatucci, G., Valdinoci, E.: Hitchhiker's guide to the fractional Sobolev spaces. Bull. Sci. Math. 136 (2012), 521-573. | DOI | MR | Zbl

[4] Farwig, R.: Note on the flux condition and pressure drop in the resolvent problem of the Stokes system. Manuscr. Math. 89 (1996), 139-158. | DOI | MR | Zbl

[5] 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 (2006), 423-444. | DOI | MR | Zbl

[6] Farwig, R., Galdi, G. P., Sohr, H.: Very weak solutions and large uniqueness classes of stationary Navier-Stokes equations in bounded domains of {${\mathbb R}^2$}. J. Differ. Equations 227 (2006), 564-580. | DOI | MR

[7] Farwig, R., Galdi, G. P., Sohr, H.: Very weak solutions of stationary and instationary Navier-Stokes equations with nonhomogeneous data. Nonlinear Elliptic and Parabolic Problems. A Special Tribute to the Work of Herbert Amann, Zürich, Switzerland, 2004 Progr. Nonlinear Differential Equations Appl. 64 Birkhäuser, Basel (2005), 113-136 M. Chipot et al. | MR | Zbl

[8] Farwig, R., Kozono, H., Sohr, H.: Very weak solutions of the Navier-Stokes equations in exterior domains with nonhomogeneous data. J. Math. Soc. Japan 59 (2007), 127-150. | DOI | MR | Zbl

[9] Farwig, R., Kozono, H., Sohr, H.: Very weak, weak and strong solutions to the instationary Navier-Stokes system. Topics on Partial Differential Equations Jindřich Nečas Cent. Math. Model. Lect. Notes 2 Matfyzpress, Praha (2007), 1-54 P. Kaplický et al. | MR

[10] Farwig, R., Sohr, H.: Helmholtz decomposition and Stokes resolvent system for aperture domains in {$L^q$}-spaces. Analysis 16 (1996), 1-26. | DOI | MR

[11] Farwig, R., Sohr, H.: Generalized resolvent estimates for the Stokes system in bounded and unbounded domains. J. Math. Soc. Japan 46 (1994), 607-643. | DOI | MR | Zbl

[12] Fichera, G.: The trace operator. Sobolev and Ehrling lemmas. Linear Elliptic Differential Systems and Eigenvalue Problems Lecture Notes in Mathematics 8 Springer, Berlin (1965), 24-29.

[13] Focardi, M.: Aperiodic fractional obstacle problems. Adv. Math. 225 (2010), 3502-3544. | DOI | MR | Zbl

[14] Franzke, M.: Die Navier-Stokes-Gleichungen in "Offnungsgebieten. PhD thesis Shaker, Aachen German (2000).

[15] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38 Springer, New York (1994). | MR | Zbl

[16] Galdi, G. P., Simader, C. G., Sohr, H.: A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in {$W^{-1/q,q}$}. Math. Ann. 331 (2005), 41-74. | DOI | MR | Zbl

[17] Kudrjavcev, L. D.: An imbedding theorem for a class of functions defined in the whole space or in the half-space. I. Transl., Ser. 2, Am. Math. Soc. 74 (1968), 199-225 translation from Mat. Sb., N. Ser. 69 (1966), 616-639 Russian. | MR

[18] Kudrjavcev, L. D.: Imbedding theorems for classes of functions defined in the whole space or in the half-space. {II}. Transl., Ser. 2, Am. Math. Soc. 74 (1968), 227-260 translation from Mat. Sb., N. Ser. 70 3-35 (1966), Russian. | MR

[19] Riechwald, P. F.: Interpolation of sum and intersection spaces of {$L^q$}-type and applications to the Stokes problem in general unbounded domains. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 58 (2012), 167-181. | DOI | MR | Zbl

[20] Riechwald, P. F.: Very Weak Solutions to the Navier-Stokes Equations in General Unbounded Domains. PhD thesis TU Darmstadt, Darmstadt; Fachbereich Mathematik (Diss.), München (2011). | Zbl

[21] Schumacher, K.: Very weak solutions to the stationary Stokes and Stokes resolvent problem in weighted function spaces. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 54 (2008), 123-144. | DOI | MR | Zbl

[22] Schumacher, K.: The Navier-Stokes Equations with Low-Regularity Data in Weighted Function Spaces. PhD thesis TU Darmstadt, Fachbereich Mathematik (Diss.), Darmstadt (2007). | Zbl

[23] Temam, R.: Navier-Stokes Equations. Theory and Numerical Analysis. Studies in Mathematics and Its Applications 2 North-Holland Publishing, Amsterdam (1977). | MR | Zbl