Abstract
We consider the steady Stokes equations in bounded and exterior domains Ω of \({{\mathbb{R}^3}}\) with boundary data and forces in L 1. We prove existence and uniqueness of a weak solution with gradient in the Iwaniek–Sbordone grand Lebesgue space \({L^{\frac{3}{2})}}\)
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Brown R.M., Shen Z.: Estimates for the Stokes operator in Lipschitz domains. Indiana Univ. Math. J. 44, 1183–1206 (1995)
Fabes E.B., Jodeit M. Jr., Rivière N.M.: Potential techniques for boundary value problems on C 1 domains. Acta Math. 141, 165–186 (1978)
Fabes E.B., Kenig C.E., Verchota G.C.: Boundary value problems for the Stokes system on Lipschitz domains. Duke Math. J. 57, 769–793 (1988)
Fiorenza A., Sbordone C.: Existence and uniqueness results for solutions of nonlinear equations with right hand side in L 1. Studia Math. 127, 223–231 (1998)
Gagliardo E.: Caratterizzazione delle tracce sulla frontiera relative ad alcune classi di funzioni in n variabli. Rend. Sem. Mat. Padova 27, 284–305 (1957)
Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady State Problems, 2nd edn, Springer Monographs in Mathematics, Springer, Berlin (2011)
Galdi G.P., Simader C.G., Sohr H.: On the Stokes problem in Lipschitz domains. Ann. Mat. Pura Appl. 167, 147–163 (1994)
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, 41–74 (2005)
Greco L., Iwaniec T., Sbordone C.: Inverting the p-harmonic operator. Manuscripta Math. 92, 249–258 (1997)
Iwaniec T., Sbordone C.: On the integrability of the Jacobian under minimal hypotheses. Arch. Ration. Mech. Anal. 119, 129–143 (1992)
Ladyzhenskaia O.A.: The Mathematical Theory of Viscous Incompressible Fluid. Gordon and Breach, London (1969)
Miranda C.: Partial Differential Equations of Elliptic Type. Springer, Berlin (1970)
Mitrea M., Taylor M.: Navier–Stokes equations on Lipschitz domains in Riemannian manifolds. Math. Ann. 321, 955–987 (2001)
Nečas, J.: Les méthodes directes en théorie des équations élliptiques. Masson-Paris and Academie-Prague (1967)
Russo R.: On the existence of solutions to the stationary Navier–Stokes equations. Ricerche Mat. 52, 285–348 (2003)
Russo R., Tartaglione A.: On the Robin problem for Stokes and Navier–Stokes systems. Math. Models Methods Appl. Sci. 19, 701–716 (2006)
Russo A., Tartaglione A.: On the Oseen and Navier-Stokes systems with a slip boundary condition. Appl. Math. Lett. 22, 674–678 (2009)
Tartaglione A.: On the Stokes problem with slip boundary conditions. CAIM 1, 186–205 (2010)
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Russo, A., Tartaglione, A. On the Stokes problem with data in L 1 . Z. Angew. Math. Phys. 64, 1327–1336 (2013). https://doi.org/10.1007/s00033-012-0290-0
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DOI: https://doi.org/10.1007/s00033-012-0290-0