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$L^p$-theory for the Navier-Stokes equations with pressure boundary conditions

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  • We consider the Navier-Stokes equations with pressure boundary conditions in the case of a bounded open set, connected of class $\mathcal{C}^{\,1,1}$ of $\mathbb{R}^3$. We prove existence of solution by using a fixed point theorem over the type-Oseen problem. This result was studied in [5] in the Hilbertian case. In our study we give the $L^p$-theory for $1< p <\infty$.
    Mathematics Subject Classification: Primary: 35J25, 35J47, 35J50, 35J57; Secondary: 76D05, 76D07.

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  • [1]

    C. Amrouche and V. Girault, Decomposition of vector space and application to the Stokes problem in arbitrary dimension, Czechoslovak Math. J., 44 (1994), 109-140.

    [2]

    C. Amrouche and M. Ángeles Rodríguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data, Arch. Rational. Mech. Anal., 199 (2011), 597-651.doi: 10.1007/s00205-010-0340-8.

    [3]

    C. Amrouche and N. Seloula$L^p$-theory for vector potentials and Sobolev's inequalities for vector fields. Application to the Stokes equations with pressure boundary condition, to appear in M3AS.

    [4]

    C. Amrouche and N. Seloula, On the Stokes equations with the Navier-type boundary conditions, Differential Equations and Applications, 3 (2011), 581-607.doi: 10.7153/dea-03-36.

    [5]

    C. Conca, F. Murat and O. Pironneau, The Stokes and Navier-Stokes equations with boundary conditions involving the pressure, Japan. J. Math. (N.S.), 20 (1994), 263-318.

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