# American Institute of Mathematical Sciences

October  2013, 6(5): 1163-1172. doi: 10.3934/dcdss.2013.6.1163

## Divergence

 1 Université Paris-Est, LAMA, UMR 8050, 61 avenue du Général de Gaulle, 94010 Créteil Cedex, France 2 Warsaw University, Inst. of Applied Math. and Mech., ul. Banacha 2, 02-097 Warszawa, Poland

Received  December 2011 Revised  February 2012 Published  March 2013

This note is dedicated to a few questions related to the divergence equation which have been motivated by recent studies concerning the Neumann problem for the Laplace equation or the (evolutionary) Stokes system in domains of $\mathbb{R}^n.$ For simplicity, we focus on the classical Sobolev spaces framework in bounded domains, but our results have natural and simple extensions to the Besov spaces framework in more general domains.
Citation: Raphaël Danchin, Piotr B. Mucha. Divergence. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1163-1172. doi: 10.3934/dcdss.2013.6.1163
##### References:
 [1] M. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, (Russian) Dokl. Akad. Nauk SSSR, 248 (1979), 1037-1040.  Google Scholar [2] M. Costabel and A. McIntosch, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Zeitschrift, 265 (2010), 297-320. doi: 10.1007/s00209-009-0517-8.  Google Scholar [3] R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space, J. Funct. Anal., 256 (2009), 881-927. doi: 10.1016/j.jfa.2008.11.019.  Google Scholar [4] R. Danchin and P. B. Mucha, The divergence equation in rough spaces, Journal of Mathematical Analysis and Applications, 386 (2012), 9-31. doi: 10.1016/j.jmaa.2011.07.036.  Google Scholar [5] R. Danchin and P. B. Mucha, A Lagrangian approach for solving the incompressible Navier-Stokes equations with variable density, Communications on Pure and Applied Mathematics, 65 (2012), no. 10, 14581480. doi: 10.1002/cpa.21409.  Google Scholar [6] R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013) 9911023. doi: 10.1007/s00205-012-0586-4.  Google Scholar [7] R. Danchin and P. B. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows,, in progress., ().   Google Scholar [8] G. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar [9] M. Geissert, H. Heck and M. Hieber, On the equation div $u=g$ and Bogovskiĭ's operator in Sobolev spaces of negative order, in "Partial Differential Equations and Functional Analysis," Oper. Theory Adv. Appl., 168, Birkhäuser, Basel, (2006), 113-121. Google Scholar [10] D. Mitrea, M. Mitrea and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition on nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333. doi: 10.3934/cpaa.2008.7.1295.  Google Scholar [11] D. Mitrea, M. Mitrea and S. Monniaux, Weighted Sobolev space estimates for a class of singular integral operators, in "Around the research of Vladimir Maz'ya. III," Int. Math. Ser. (N. Y.), 13, Springer, New York, (2010), 179-200. doi: 10.1007/978-1-4419-1345-6_7.  Google Scholar [12] P. B. Mucha, W. M. Zajączkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Appl. Math. (Warsaw), 27 (2000), 319-333.  Google Scholar [13] P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework, Adv. Differential Equations, 10 (2005), 861-900.  Google Scholar [14] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford Univ. Press, Oxford, 2004.  Google Scholar [15] S. Sobolev, "Applications of Functional Analysis to Mathematical Physics," American Mathematical Society, Translation of Monographs, Vol. 7, 1964. Google Scholar [16] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [17] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar [18] W. M. Zajączkowski, Existence and regularity of solutions of some elliptic system in domains with edges, Dissertationes Math. (Rozprawy Mat.), 274 (1988), 95 pp.  Google Scholar

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##### References:
 [1] M. Bogovskiĭ, Solution of the first boundary value problem for an equation of continuity of an incompressible medium, (Russian) Dokl. Akad. Nauk SSSR, 248 (1979), 1037-1040.  Google Scholar [2] M. Costabel and A. McIntosch, On Bogovskiĭ and regularized Poincaré integral operators for de Rham complexes on Lipschitz domains, Math. Zeitschrift, 265 (2010), 297-320. doi: 10.1007/s00209-009-0517-8.  Google Scholar [3] R. Danchin and P. B. Mucha, A critical functional framework for the inhomogeneous Navier-Stokes equations in the half-space, J. Funct. Anal., 256 (2009), 881-927. doi: 10.1016/j.jfa.2008.11.019.  Google Scholar [4] R. Danchin and P. B. Mucha, The divergence equation in rough spaces, Journal of Mathematical Analysis and Applications, 386 (2012), 9-31. doi: 10.1016/j.jmaa.2011.07.036.  Google Scholar [5] R. Danchin and P. B. Mucha, A Lagrangian approach for solving the incompressible Navier-Stokes equations with variable density, Communications on Pure and Applied Mathematics, 65 (2012), no. 10, 14581480. doi: 10.1002/cpa.21409.  Google Scholar [6] R. Danchin and P. B. Mucha, Incompressible flows with piecewise constant density, Arch. Ration. Mech. Anal., 207 (2013) 9911023. doi: 10.1007/s00205-012-0586-4.  Google Scholar [7] R. Danchin and P. B. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows,, in progress., ().   Google Scholar [8] G. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems," Springer Tracts in Natural Philosophy, 38, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-5364-8.  Google Scholar [9] M. Geissert, H. Heck and M. Hieber, On the equation div $u=g$ and Bogovskiĭ's operator in Sobolev spaces of negative order, in "Partial Differential Equations and Functional Analysis," Oper. Theory Adv. Appl., 168, Birkhäuser, Basel, (2006), 113-121. Google Scholar [10] D. Mitrea, M. Mitrea and S. Monniaux, The Poisson problem for the exterior derivative operator with Dirichlet boundary condition on nonsmooth domains, Commun. Pure Appl. Anal., 7 (2008), 1295-1333. doi: 10.3934/cpaa.2008.7.1295.  Google Scholar [11] D. Mitrea, M. Mitrea and S. Monniaux, Weighted Sobolev space estimates for a class of singular integral operators, in "Around the research of Vladimir Maz'ya. III," Int. Math. Ser. (N. Y.), 13, Springer, New York, (2010), 179-200. doi: 10.1007/978-1-4419-1345-6_7.  Google Scholar [12] P. B. Mucha, W. M. Zajączkowski, On local existence of solutions of the free boundary problem for an incompressible viscous self-gravitating fluid motion, Appl. Math. (Warsaw), 27 (2000), 319-333.  Google Scholar [13] P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework, Adv. Differential Equations, 10 (2005), 861-900.  Google Scholar [14] A. Novotný and I. Straškraba, "Introduction to the Mathematical Theory of Compressible Flow," Oxford Lecture Series in Mathematics and its Applications, 27, Oxford Univ. Press, Oxford, 2004.  Google Scholar [15] S. Sobolev, "Applications of Functional Analysis to Mathematical Physics," American Mathematical Society, Translation of Monographs, Vol. 7, 1964. Google Scholar [16] R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis," Studies in Mathematics and its Applications, Vol. 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.  Google Scholar [17] H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar [18] W. M. Zajączkowski, Existence and regularity of solutions of some elliptic system in domains with edges, Dissertationes Math. (Rozprawy Mat.), 274 (1988), 95 pp.  Google Scholar
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