October  2014, 7(5): 901-916. doi: 10.3934/dcdss.2014.7.901

Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces

1. 

Laboratoire de Mathématiques et de leurs Applications, CNRS UMR 5142, Université de Pau et des Pays de l'Adour, 64013 Pau, France, France

2. 

Mathematical Institute, Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Praha 1

Received  April 2013 Published  May 2014

In this work, we study the linearized Navier-Stokes equations in $\mathbb{R}^3$, the Oseen equations. We are interested in the existence and the uniqueness of generalized and strong solutions in $L^p$-theory which makes analysis more difficult. Our approach rests on the use of weighted Sobolev spaces.
Citation: Chérif Amrouche, Mohamed Meslameni, Šárka Nečasová. Linearized Navier-Stokes equations in $\mathbb{R}^3$: An approach in weighted Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2014, 7 (5) : 901-916. doi: 10.3934/dcdss.2014.7.901
References:
[1]

F. Alliot and C. Amrouche, The Stokes problem in $\mathbbR^n$: An approach in weighted Sobolev spaces,, Math. Mod. Meth. Appl. Sci., 9 (1999), 723.  doi: 10.1142/S0218202599000361.  Google Scholar

[2]

C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$,, Communications in Mathematical Analysis, 10 (2011), 5.   Google Scholar

[3]

C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for the laplace equation in $\mathbbR^n$,, J. Math. Pures et Appl., 73 (1994), 579.   Google Scholar

[4]

C. Amrouche and M. A. Rodriguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data,, Archive for Rational Mechanics and Analysis, 199 (2011), 597.  doi: 10.1007/s00205-010-0340-8.  Google Scholar

[5]

M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$,, Indiana Univ. Math. J., 24 (1975), 897.   Google Scholar

[6]

R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z, 211 (1992), 409.  doi: 10.1007/BF02571437.  Google Scholar

[7]

R. Farwig, The stationary Navier-Stokes equations in a 3D-exterior domain,, in Recent Topics on Mathematical Theory of Viscous Incompressible Fluid (Tsukuba, (1996), 53.   Google Scholar

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems,, Springer Tracts in Natural Philosophy, (1994).   Google Scholar

[9]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems,, Springer Tracts in Natural Philosophy, (1994).   Google Scholar

[10]

B. Hanouzet, Espace de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace,, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227.   Google Scholar

show all references

References:
[1]

F. Alliot and C. Amrouche, The Stokes problem in $\mathbbR^n$: An approach in weighted Sobolev spaces,, Math. Mod. Meth. Appl. Sci., 9 (1999), 723.  doi: 10.1142/S0218202599000361.  Google Scholar

[2]

C. Amrouche and L. Consiglieri, On the stationary Oseen equations in $\mathbbR^{3}$,, Communications in Mathematical Analysis, 10 (2011), 5.   Google Scholar

[3]

C. Amrouche, V. Girault and J. Giroire, Weighted Sobolev spaces for the laplace equation in $\mathbbR^n$,, J. Math. Pures et Appl., 73 (1994), 579.   Google Scholar

[4]

C. Amrouche and M. A. Rodriguez-Bellido, Stationary Stokes, Oseen and Navier-Stokes equations with singular data,, Archive for Rational Mechanics and Analysis, 199 (2011), 597.  doi: 10.1007/s00205-010-0340-8.  Google Scholar

[5]

M. Cantor, Spaces of functions with asymptotic conditions on $\mathbbR^n$,, Indiana Univ. Math. J., 24 (1975), 897.   Google Scholar

[6]

R. Farwig, The stationary exterior 3D-problem of Oseen and Navier-Stokes equations in anisotropically weighted Sobolev spaces,, Math. Z, 211 (1992), 409.  doi: 10.1007/BF02571437.  Google Scholar

[7]

R. Farwig, The stationary Navier-Stokes equations in a 3D-exterior domain,, in Recent Topics on Mathematical Theory of Viscous Incompressible Fluid (Tsukuba, (1996), 53.   Google Scholar

[8]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I. Linearized Steady Problems,, Springer Tracts in Natural Philosophy, (1994).   Google Scholar

[9]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. II. Nonlinear Steady Problems,, Springer Tracts in Natural Philosophy, (1994).   Google Scholar

[10]

B. Hanouzet, Espace de Sobolev avec poids. Application au problème de Dirichlet dans un demi espace,, Rend. Sem. Mat. Univ. Padova, 46 (1971), 227.   Google Scholar

[1]

Elder J. Villamizar-Roa, Elva E. Ortega-Torres. On a generalized Boussinesq model around a rotating obstacle: Existence of strong solutions. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 825-847. doi: 10.3934/dcdsb.2011.15.825

[2]

T. Tachim Medjo. Existence and uniqueness of strong periodic solutions of the primitive equations of the ocean. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1491-1508. doi: 10.3934/dcds.2010.26.1491

[3]

Luiz Gustavo Farah. Local solutions in Sobolev spaces and unconditional well-posedness for the generalized Boussinesq equation. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1521-1539. doi: 10.3934/cpaa.2009.8.1521

[4]

Meili Li, Maoan Han, Chunhai Kou. The existence of positive periodic solutions of a generalized. Mathematical Biosciences & Engineering, 2008, 5 (4) : 803-812. doi: 10.3934/mbe.2008.5.803

[5]

Josep M. Olm, Xavier Ros-Oton. Existence of periodic solutions with nonconstant sign in a class of generalized Abel equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1603-1614. doi: 10.3934/dcds.2013.33.1603

[6]

Zijuan Wen, Meng Fan, Asim M. Asiri, Ebraheem O. Alzahrani, Mohamed M. El-Dessoky, Yang Kuang. Global existence and uniqueness of classical solutions for a generalized quasilinear parabolic equation with application to a glioblastoma growth model. Mathematical Biosciences & Engineering, 2017, 14 (2) : 407-420. doi: 10.3934/mbe.2017025

[7]

Tracy L. Stepien, Hal L. Smith. Existence and uniqueness of similarity solutions of a generalized heat equation arising in a model of cell migration. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3203-3216. doi: 10.3934/dcds.2015.35.3203

[8]

Tong Li, Hui Yin. Convergence rate to strong boundary layer solutions for generalized BBM-Burgers equations with non-convex flux. Communications on Pure & Applied Analysis, 2014, 13 (2) : 835-858. doi: 10.3934/cpaa.2014.13.835

[9]

Changchun Liu, Jingxue Yin, Juan Zhou. Existence of weak solutions for a generalized thin film equation. Communications on Pure & Applied Analysis, 2007, 6 (2) : 465-480. doi: 10.3934/cpaa.2007.6.465

[10]

Jiabao Su, Rushun Tian. Weighted Sobolev embeddings and radial solutions of inhomogeneous quasilinear elliptic equations. Communications on Pure & Applied Analysis, 2010, 9 (4) : 885-904. doi: 10.3934/cpaa.2010.9.885

[11]

Li Yang, Zeng Rong, Shouming Zhou, Chunlai Mu. Uniqueness of conservative solutions to the generalized Camassa-Holm equation via characteristics. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5205-5220. doi: 10.3934/dcds.2018230

[12]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809

[13]

Jun Yang, Yaotian Shen. Weighted Sobolev-Hardy spaces and sign-changing solutions of degenerate elliptic equation. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2565-2575. doi: 10.3934/cpaa.2013.12.2565

[14]

Shikuan Mao, Yongqin Liu. Decay of solutions to generalized plate type equations with memory. Kinetic & Related Models, 2014, 7 (1) : 121-131. doi: 10.3934/krm.2014.7.121

[15]

Yonggeun Cho, Tohru Ozawa. On small amplitude solutions to the generalized Boussinesq equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 691-711. doi: 10.3934/dcds.2007.17.691

[16]

J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467

[17]

Xiaoqian Liu, Yutian Lei. Existence of positive solutions for integral systems of the weighted Hardy-Littlewood-Sobolev type. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 467-489. doi: 10.3934/dcds.2020018

[18]

Shaoyong Lai, Qichang Xie, Yunxi Guo, YongHong Wu. The existence of weak solutions for a generalized Camassa-Holm equation. Communications on Pure & Applied Analysis, 2011, 10 (1) : 45-57. doi: 10.3934/cpaa.2011.10.45

[19]

Yixia Shi, Maoan Han. Existence of generalized homoclinic solutions for a modified Swift-Hohenberg equation. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020114

[20]

Ewa Schmeidel, Karol Gajda, Tomasz Gronek. On the existence of weighted asymptotically constant solutions of Volterra difference equations of nonconvolution type. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2681-2690. doi: 10.3934/dcdsb.2014.19.2681

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (25)
  • HTML views (0)
  • Cited by (3)

[Back to Top]