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.   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.  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.  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.  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).  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).  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 (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, 168 (2006), 113.   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.  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, 13 (2010), 179.  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.   Google Scholar

[13]

P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework,, Adv. Differential Equations, 10 (2005), 861.   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 (2004).   Google Scholar

[15]

S. Sobolev, "Applications of Functional Analysis to Mathematical Physics,", American Mathematical Society, (1964).   Google Scholar

[16]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Studies in Mathematics and its Applications, (1977).   Google Scholar

[17]

H. Triebel, "Theory of Function Spaces,", Monographs in Mathematics, 78 (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).   Google Scholar

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.   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.  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.  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.  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).  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).  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 (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, 168 (2006), 113.   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.  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, 13 (2010), 179.  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.   Google Scholar

[13]

P. B. Mucha, On the Stefan problem with surface tension in the $L_p$ framework,, Adv. Differential Equations, 10 (2005), 861.   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 (2004).   Google Scholar

[15]

S. Sobolev, "Applications of Functional Analysis to Mathematical Physics,", American Mathematical Society, (1964).   Google Scholar

[16]

R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,", Studies in Mathematics and its Applications, (1977).   Google Scholar

[17]

H. Triebel, "Theory of Function Spaces,", Monographs in Mathematics, 78 (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).   Google Scholar

[1]

Angelo Favini, Rabah Labbas, Keddour Lemrabet, Stéphane Maingot, Hassan D. Sidibé. Resolution and optimal regularity for a biharmonic equation with impedance boundary conditions and some generalizations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4991-5014. doi: 10.3934/dcds.2013.33.4991

[2]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475

[3]

Rong Dong, Dongsheng Li, Lihe Wang. Regularity of elliptic systems in divergence form with directional homogenization. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 75-90. doi: 10.3934/dcds.2018004

[4]

Kenji Nakanishi, Hideo Takaoka, Yoshio Tsutsumi. Local well-posedness in low regularity of the MKDV equation with periodic boundary condition. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1635-1654. doi: 10.3934/dcds.2010.28.1635

[5]

Lianzhang Bao, Wenxian Shen. Logistic type attraction-repulsion chemotaxis systems with a free boundary or unbounded boundary. I. Asymptotic dynamics in fixed unbounded domain. Discrete & Continuous Dynamical Systems - A, 2020, 40 (2) : 1107-1130. doi: 10.3934/dcds.2020072

[6]

S. Raynor, G. Staffilani. Low regularity stability of solitons for the KDV equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 277-296. doi: 10.3934/cpaa.2003.2.277

[7]

Xavier Lamy, Petru Mironescu. Characterization of function spaces via low regularity mollifiers. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 6015-6030. doi: 10.3934/dcds.2015.35.6015

[8]

Takamori Kato. Global well-posedness for the Kawahara equation with low regularity. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1321-1339. doi: 10.3934/cpaa.2013.12.1321

[9]

Hyungjin Huh, Bora Moon. Low regularity well-posedness for Gross-Neveu equations. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1903-1913. doi: 10.3934/cpaa.2015.14.1903

[10]

Jianyu Chen, Hong-Kun Zhang. Statistical properties of one-dimensional expanding maps with singularities of low regularity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4955-4977. doi: 10.3934/dcds.2019203

[11]

Aram L. Karakhanyan. Lipschitz continuity of free boundary in the continuous casting problem with divergence form elliptic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 261-277. doi: 10.3934/dcds.2016.36.261

[12]

Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - A, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262

[13]

Jochen Merker. Generalizations of logarithmic Sobolev inequalities. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 329-338. doi: 10.3934/dcdss.2008.1.329

[14]

Magdalena Czubak, Nina Pikula. Low regularity well-posedness for the 2D Maxwell-Klein-Gordon equation in the Coulomb gauge. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1669-1683. doi: 10.3934/cpaa.2014.13.1669

[15]

Takafumi Akahori. Low regularity global well-posedness for the nonlinear Schrödinger equation on closed manifolds. Communications on Pure & Applied Analysis, 2010, 9 (2) : 261-280. doi: 10.3934/cpaa.2010.9.261

[16]

Olivier Glass, Franck Sueur. Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 2791-2808. doi: 10.3934/dcds.2013.33.2791

[17]

J. Colliander, A. D. Ionescu, C. E. Kenig, Gigliola Staffilani. Weighted low-regularity solutions of the KP-I initial-value problem. Discrete & Continuous Dynamical Systems - A, 2008, 20 (2) : 219-258. doi: 10.3934/dcds.2008.20.219

[18]

J. Colliander, Tristan Roy. Bootstrapped Morawetz estimates and resonant decomposition for low regularity global solutions of cubic NLS on $R^2$. Communications on Pure & Applied Analysis, 2011, 10 (2) : 397-414. doi: 10.3934/cpaa.2011.10.397

[19]

Hiroyuki Hirayama. Well-posedness and scattering for a system of quadratic derivative nonlinear Schrödinger equations with low regularity initial data. Communications on Pure & Applied Analysis, 2014, 13 (4) : 1563-1591. doi: 10.3934/cpaa.2014.13.1563

[20]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]