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Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p-Spaces

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In this article we consider the Stokes problem with Navier-type boundary conditions on a domain \({\Omega}\), not necessarily simply connected. Since, under these conditions, the Stokes problem has a non trivial kernel, we also study the solutions lying in the orthogonal of that kernel. We prove the analyticity of several semigroups generated by the Stokes operator considered in different functional spaces. We obtain strong, weak and very weak solutions for the time dependent Stokes problem with the Navier-type boundary condition under different hypotheses on the initial data u 0 and external force f. Then, we study the fractional and pure imaginary powers of several operators related with our Stokes operators. Using the fractional powers, we prove maximal regularity results for the homogeneous Stokes problem. On the other hand, using the boundedness of the pure imaginary powers, we deduce maximal \({L^{p}-L^{q}}\) regularity for the inhomogeneous Stokes problem.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et de leurs applications Pau, UMR, CNRS 5142, Batiment IPRA, Université de Pau et des pays de L’Adour, Avenue de L’université, Bureau 012, BP 1155, 64013, Pau cedex, France

    Hind Al Baba

  2. Laboratoire de Mathématiques et de leurs applications Pau, UMR, CNRS 5142, Batiment IPRA, Université de Pau et des pays de L’Adour, Avenue de L’université, Bureau 225, BP 1155, 64013, Pau cedex, France

    Chérif Amrouche

  3. Departamento de Matemáticas Facultad de Ciencias y Tecnología, Universidad del País Vasco, Barrio Sarriena s/n, 48940, Lejona, Vizcaya, Spain

    Miguel Escobedo

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  1. Hind Al Baba
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  2. Chérif Amrouche
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  3. Miguel Escobedo
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Al Baba, H., Amrouche, C. & Escobedo, M. Semi-Group Theory for the Stokes Operator with Navier-Type Boundary Conditions on L p-Spaces. Arch Rational Mech Anal 223, 881–940 (2017). https://doi.org/10.1007/s00205-016-1048-1

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