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An elementary approach to the 3D NavierStokes equations with Navier boundary conditions: Existence and uniqueness of various classes of solutions in the flat boundary case.
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On the very weak solution for the Oseen and NavierStokes equations
Loss of smoothness and energy conserving rough weak solutions for the $3d$ Euler equations
1.  Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, 175 Avenue du Chevaleret, Paris 75013 
2.  Department of Mathematics and Department of Mechanics and Aerospace Engineering, University of California, Irvine, CA 92697, United States 
[1] 
Igor Kukavica, Vlad C. Vicol. The domain of analyticity of solutions to the threedimensional Euler equations in a half space. Discrete & Continuous Dynamical Systems  A, 2011, 29 (1) : 285303. doi: 10.3934/dcds.2011.29.285 
[2] 
Christophe Cheverry, Mekki Houbad. A class of large amplitude oscillating solutions for three dimensional Euler equations. Communications on Pure & Applied Analysis, 2012, 11 (5) : 16611697. doi: 10.3934/cpaa.2012.11.1661 
[3] 
GuiQiang G. Chen, Hairong Yuan. Local uniqueness of steady spherical transonic shockfronts for the threedimensional full Euler equations. Communications on Pure & Applied Analysis, 2013, 12 (6) : 25152542. doi: 10.3934/cpaa.2013.12.2515 
[4] 
Xingwen Hao, Yachun Li, Zejun Wang. Nonrelativistic global limits to the three dimensional relativistic euler equations with spherical symmetry. Communications on Pure & Applied Analysis, 2010, 9 (2) : 365386. doi: 10.3934/cpaa.2010.9.365 
[5] 
Jian Su, Yinnian He. The almost unconditional convergence of the Euler implicit/explicit scheme for the three dimensional nonstationary NavierStokes equations. Discrete & Continuous Dynamical Systems  B, 2017, 22 (9) : 34213438. doi: 10.3934/dcdsb.2017173 
[6] 
Aibin Zang. Kato's type theorems for the convergence of EulerVoigt equations to Euler equations with Drichlet boundary conditions. Discrete & Continuous Dynamical Systems  A, 2019, 39 (9) : 49454953. doi: 10.3934/dcds.2019202 
[7] 
XueLi Song, YanRen Hou. Attractors for the threedimensional incompressible NavierStokes equations with damping. Discrete & Continuous Dynamical Systems  A, 2011, 31 (1) : 239252. doi: 10.3934/dcds.2011.31.239 
[8] 
Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the threedimensional regularized MHD equations. Discrete & Continuous Dynamical Systems  A, 2018, 38 (3) : 14611477. doi: 10.3934/dcds.2018060 
[9] 
Madalina Petcu, Roger Temam, Djoko Wirosoetisno. Averaging method applied to the threedimensional primitive equations. Discrete & Continuous Dynamical Systems  A, 2016, 36 (10) : 56815707. doi: 10.3934/dcds.2016049 
[10] 
Yeping Li, Jie Liao. Stability and $ L^{p}$ convergence rates of planar diffusion waves for threedimensional bipolar EulerPoisson systems. Communications on Pure & Applied Analysis, 2019, 18 (3) : 12811302. doi: 10.3934/cpaa.2019062 
[11] 
Tong Zhang, Yuxi Zheng. Exact spiral solutions of the twodimensional Euler equations. Discrete & Continuous Dynamical Systems  A, 1997, 3 (1) : 117133. doi: 10.3934/dcds.1997.3.117 
[12] 
Ju Ge, Wancheng Sheng. The two dimensional gas expansion problem of the Euler equations for the generalized Chaplygin gas. Communications on Pure & Applied Analysis, 2014, 13 (6) : 27332748. doi: 10.3934/cpaa.2014.13.2733 
[13] 
Yuxi Zheng. Absorption of characteristics by sonic curve of the twodimensional Euler equations. Discrete & Continuous Dynamical Systems  A, 2009, 23 (1&2) : 605616. doi: 10.3934/dcds.2009.23.605 
[14] 
Roman Shvydkoy. Lectures on the Onsager conjecture. Discrete & Continuous Dynamical Systems  S, 2010, 3 (3) : 473496. doi: 10.3934/dcdss.2010.3.473 
[15] 
Okihiro Sawada. Analytic rates of solutions to the Euler equations. Discrete & Continuous Dynamical Systems  S, 2013, 6 (5) : 14091415. doi: 10.3934/dcdss.2013.6.1409 
[16] 
Luigi Ambrosio. Variational models for incompressible Euler equations. Discrete & Continuous Dynamical Systems  B, 2009, 11 (1) : 110. doi: 10.3934/dcdsb.2009.11.1 
[17] 
Louis Tebou. Energy decay estimates for some weakly coupled EulerBernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 4560. doi: 10.3934/mcrf.2012.2.45 
[18] 
Jianwei Yang, Ruxu Lian, Shu Wang. Incompressible type euler as scaling limit of compressible EulerMaxwell equations. Communications on Pure & Applied Analysis, 2013, 12 (1) : 503518. doi: 10.3934/cpaa.2013.12.503 
[19] 
Yongcai Geng. Singularity formation for relativistic Euler and EulerPoisson equations with repulsive force. Communications on Pure & Applied Analysis, 2015, 14 (2) : 549564. doi: 10.3934/cpaa.2015.14.549 
[20] 
Franco Flandoli, Dejun Luo. EulerLagrangian approach to 3D stochastic Euler equations. Journal of Geometric Mechanics, 2019, 11 (2) : 153165. doi: 10.3934/jgm.2019008 
2018 Impact Factor: 0.545
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