American Institute of Mathematical Sciences

Advanced
2007, 2007(Special): 834-843. doi: 10.3934/proc.2007.2007.834

Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains

Lukáš Poul 1,

1. 

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

Received  September 2006 Revised  June 2007 Published  September 2007

We prove existence of a weak solution to the Navier-Stokes-Fourier system on a bounded Lipschitz domain in $\mathbb{R}^3$. The key tool is the existence theory for weak solutions developed by Feireisl for the case of bounded smooth domains. We prove our result by inserting an additional limit passage where smooth domains approximate the Lipschitz one. Results on sensitivity of solutions with respect to the convergence of spatial domains are shortly discussed at the end of the paper.
Keywords: Lipschitz domains., Navier-Stokes-Fourier system, existence, weak solutions.
Mathematics Subject Classification: 35Q30, 35A05, 76N1.
Citation: Lukáš Poul. Existence of weak solutions to the Navier-Stokes-Fourier system on Lipschitz domains. Conference Publications, 2007, 2007 (Special) : 834-843. doi: 10.3934/proc.2007.2007.834
[1]

Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic and Related Models, 2021, 14 (4) : 599-638. doi: 10.3934/krm.2021017
[2]

Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157
[3]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete and Continuous Dynamical Systems, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171
[4]

Sylvie Monniaux. Various boundary conditions for Navier-Stokes equations in bounded Lipschitz domains. Discrete and Continuous Dynamical Systems - S, 2013, 6 (5) : 1355-1369. doi: 10.3934/dcdss.2013.6.1355
[5]

Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[6]

Alessio Falocchi, Filippo Gazzola. Regularity for the 3D evolution Navier-Stokes equations under Navier boundary conditions in some Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1185-1200. doi: 10.3934/dcds.2021151
[7]

Michele Campiti, Giovanni P. Galdi, Matthias Hieber. Global existence of strong solutions for $2$-dimensional Navier-Stokes equations on exterior domains with growing data at infinity. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1613-1627. doi: 10.3934/cpaa.2014.13.1613
[8]

Jingrui Wang, Keyan Wang. Almost sure existence of global weak solutions to the 3D incompressible Navier-Stokes equation. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 5003-5019. doi: 10.3934/dcds.2017215
[9]

Jian-Guo Liu, Zhaoyun Zhang. Existence of global weak solutions of $ p $-Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 469-486. doi: 10.3934/dcdsb.2021051
[10]

Xiaoyu Chen, Jijie Zhao, Qian Zhang. Global existence of weak solutions for the 3D axisymmetric chemotaxis-Navier-Stokes equations with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2022  doi: 10.3934/dcds.2022062
[11]

Ling Liu, Jiashan Zheng, Gui Bao. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3437-3460. doi: 10.3934/dcdsb.2020068
[12]

Xin-Guang Yang, Rong-Nian Wang, Xingjie Yan, Alain Miranville. Dynamics of the 2D Navier-Stokes equations with sublinear operators in Lipschitz-like domains. Discrete and Continuous Dynamical Systems, 2021, 41 (7) : 3343-3366. doi: 10.3934/dcds.2020408
[13]

Xinguang Yang, Baowei Feng, Thales Maier de Souza, Taige Wang. Long-time dynamics for a non-autonomous Navier-Stokes-Voigt equation in Lipschitz domains. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 363-386. doi: 10.3934/dcdsb.2018084
[14]

John W. Barrett, Endre Süli. Existence of global weak solutions to Fokker-Planck and Navier-Stokes-Fokker-Planck equations in kinetic models of dilute polymers. Discrete and Continuous Dynamical Systems - S, 2010, 3 (3) : 371-408. doi: 10.3934/dcdss.2010.3.371
[15]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[16]

Zhendong Fang, Hao Wang. Convergence from two-species Vlasov-Poisson-Boltzmann system to two-fluid incompressible Navier-Stokes-Fourier-Poisson system. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021231
[17]

Peixin Zhang, Jianwen Zhang, Junning Zhao. On the global existence of classical solutions for compressible Navier-Stokes equations with vacuum. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1085-1103. doi: 10.3934/dcds.2016.36.1085
[18]

Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217
[19]

Zaihong Jiang, Li Li, Wenbo Lu. Existence of axisymmetric and homogeneous solutions of Navier-Stokes equations in cone regions. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4231-4258. doi: 10.3934/dcdss.2021126
[20]

Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75

 Impact Factor: 

Tools

Metrics

  • PDF downloads (52)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy