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 and 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.

[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.

[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.

[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.

[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.

[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.

[7]

R. Danchin and P. B. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows, in progress.

[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.

[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.

[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.

[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.

[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.

[13]

P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework, Adv. Differential Equations, 10 (2005), 861-900.

[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.

[15]

S. Sobolev, "Applications of Functional Analysis to Mathematical Physics," American Mathematical Society, Translation of Monographs, Vol. 7, 1964.

[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.

[17]

H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[7]

R. Danchin and P. B. Mucha, Critical functional framework and maximal regularity in action on systems of incompressible flows, in progress.

[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.

[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.

[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.

[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.

[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.

[13]

P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework, Adv. Differential Equations, 10 (2005), 861-900.

[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.

[15]

S. Sobolev, "Applications of Functional Analysis to Mathematical Physics," American Mathematical Society, Translation of Monographs, Vol. 7, 1964.

[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.

[17]

H. Triebel, "Theory of Function Spaces," Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983. doi: 10.1007/978-3-0346-0416-1.

[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.

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