• Previous Article
    Asymptotic behavior of the best Sobolev trace constant in expanding and contracting domains
  • CPAA Home
  • This Issue
  • Next Article
    On existence and concentration behavior of ground state solutions for a class of problems with critical growth
September  2002, 1(3): 379-415. doi: 10.3934/cpaa.2002.1.379

In a horizontal layer with free upper surface

1. 

Department of Mathematics, University of Ferrara,44100 Ferrara, Italy, Italy

Received  August 2001 Revised  January 2002 Published  June 2002

We propose a new existence proof of global in time solutions of isothermal viscous gases in a layer bounded below by a horizontal plane, and above by a free upper surface, which are periodic in the two horizontal variables. Despite the importance of compressible fluids for physical applications, the problem of uniform in time estimates is scarcely explored. The rest state with a steady distribution of density in a rectangular domain is stable, without restrictions on initial data, in a "weak" norm provided the flows exist in a suitable regularity class. In this paper we show existence of regular global in time solutions, and the exponential decay of these solutions to the rest as time goes to $\infty$, when the initial data are small perturbation of the basic flow. The analysis presented here is based on estimates in Hilbert spaces.
Citation: Bum Ja Jin, Mariarosaria Padula. In a horizontal layer with free upper surface. Communications on Pure & Applied Analysis, 2002, 1 (3) : 379-415. doi: 10.3934/cpaa.2002.1.379
[1]

Yuan Gao, Hangjie Ji, Jian-Guo Liu, Thomas P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4433-4453. doi: 10.3934/dcdsb.2018170

[2]

Yingshan Chen, Shijin Ding, Wenjun Wang. Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5287-5307. doi: 10.3934/dcds.2016032

[3]

Marcelo M. Disconzi, Igor Kukavica. A priori estimates for the 3D compressible free-boundary Euler equations with surface tension in the case of a liquid. Evolution Equations & Control Theory, 2019, 8 (3) : 503-542. doi: 10.3934/eect.2019025

[4]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763

[5]

Rafael Granero-Belinchón, Martina Magliocca. Global existence and decay to equilibrium for some crystal surface models. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2101-2131. doi: 10.3934/dcds.2019088

[6]

Yoshihiro Shibata. Global well-posedness of unsteady motion of viscous incompressible capillary liquid bounded by a free surface. Evolution Equations & Control Theory, 2018, 7 (1) : 117-152. doi: 10.3934/eect.2018007

[7]

J. F. Toland. Non-existence of global energy minimisers in Stokes waves problems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (8) : 3211-3217. doi: 10.3934/dcds.2014.34.3211

[8]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361

[9]

Tae Gab Ha. Global existence and general decay estimates for the viscoelastic equation with acoustic boundary conditions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6899-6919. doi: 10.3934/dcds.2016100

[10]

Phoebus Rosakis. Continuum surface energy from a lattice model. Networks & Heterogeneous Media, 2014, 9 (3) : 453-476. doi: 10.3934/nhm.2014.9.453

[11]

Jihong Zhao, Qiao Liu, Shangbin Cui. Global existence and stability for a hydrodynamic system in the nematic liquid crystal flows. Communications on Pure & Applied Analysis, 2013, 12 (1) : 341-357. doi: 10.3934/cpaa.2013.12.341

[12]

Huanhuan Qiu, Shangjiang Guo. Global existence and stability in a two-species chemotaxis system. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1569-1587. doi: 10.3934/dcdsb.2018220

[13]

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

[14]

Dongfen Bian, Boling Guo. Global existence and large time behavior of solutions to the electric-magnetohydrodynamic equations. Kinetic & Related Models, 2013, 6 (3) : 481-503. doi: 10.3934/krm.2013.6.481

[15]

Risei Kano, Akio Ito. The existence of time global solutions for tumor invasion models with constraints. Conference Publications, 2011, 2011 (Special) : 774-783. doi: 10.3934/proc.2011.2011.774

[16]

Eduard Feireisl, Hana Petzeltová, Konstantina Trivisa. Multicomponent reactive flows: Global-in-time existence for large data. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1017-1047. doi: 10.3934/cpaa.2008.7.1017

[17]

Bilgesu A. Bilgin, Varga K. Kalantarov. Non-existence of global solutions to nonlinear wave equations with positive initial energy. Communications on Pure & Applied Analysis, 2018, 17 (3) : 987-999. doi: 10.3934/cpaa.2018048

[18]

Laiqing Meng, Jia Yuan, Xiaoxin Zheng. Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3413-3441. doi: 10.3934/dcds.2019141

[19]

Congming Peng, Dun Zhao. Global existence and blowup on the energy space for the inhomogeneous fractional nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3335-3356. doi: 10.3934/dcdsb.2018323

[20]

Hal L. Smith, Horst R. Thieme. Persistence and global stability for a class of discrete time structured population models. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4627-4646. doi: 10.3934/dcds.2013.33.4627

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]