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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.


    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.


    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.


    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.


    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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