American Institute of Mathematical Sciences

Advanced
September  2008, 22(3): 629-662. doi: 10.3934/dcds.2008.22.629

Stability of under-compressive waves with second and fourth order diffusions

Changbing Hu 1,

1. 

Department of Mathematics, University of Louisville, Louisville, KY 40292, United States

Received  September 2007 Revised  May 2008 Published  August 2008

This is a continuation of our work on the nonlinear stability of traveling shock fronts arising in multidimensional conservation laws with fourth order regularization only. Our motivating example is the thin film equation for which planar waves correspond with fluid coating a pre-wetted surface. Under only the fourth order regularization, we established the nonlinear stability of compressive waves for dimensions $d\geq 2$, and of under-compressive waves for dimensions $d\geq 3$ under general spectral conditions. The case of stability for under-compressive waves in the thin film equations for the critical dimensions $d=1,2$ remained open. In this paper we study the nonlinear stability of under-compressive waves by assuming both the second and fourth order regularization. We present a step toward the open problem by establishing the nonlinear stability of under-compressive waves in dimensions $d\geq 2$ under general spectral conditions. We emphasize the above mentioned stability question remains still open.
Keywords: under-compressive waves, Thin film equation, stability., conservation laws.
Mathematics Subject Classification: Primary: 35L67, 35B35; Secondary: 76L0.
Citation: Changbing Hu. Stability of under-compressive waves with second and fourth order diffusions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 629-662. doi: 10.3934/dcds.2008.22.629
[1]

Eric A. Carlen, Süleyman Ulusoy. Localization, smoothness, and convergence to equilibrium for a thin film equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4537-4553. doi: 10.3934/dcds.2014.34.4537
[2]

Marina Chugunova, Roman M. Taranets. New dissipated energy for the unstable thin film equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 613-624. doi: 10.3934/cpaa.2011.10.613
[3]

Richard S. Laugesen. New dissipated energies for the thin fluid film equation. Communications on Pure & Applied Analysis, 2005, 4 (3) : 613-634. doi: 10.3934/cpaa.2005.4.613
[4]

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
[5]

Jian-Guo Liu, Jinhuan Wang. Global existence for a thin film equation with subcritical mass. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1461-1492. doi: 10.3934/dcdsb.2017070
[6]

Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469
[7]

Daniel Ginsberg, Gideon Simpson. Analytical and numerical results on the positivity of steady state solutions of a thin film equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1305-1321. doi: 10.3934/dcdsb.2013.18.1305
[8]

Huiqiang Jiang. Energy minimizers of a thin film equation with born repulsion force. Communications on Pure & Applied Analysis, 2011, 10 (2) : 803-815. doi: 10.3934/cpaa.2011.10.803
[9]

Georges Bastin, B. Haut, Jean-Michel Coron, Brigitte d'Andréa-Novel. Lyapunov stability analysis of networks of scalar conservation laws. Networks & Heterogeneous Media, 2007, 2 (4) : 751-759. doi: 10.3934/nhm.2007.2.751
[10]

Alexander Bobylev, Mirela Vinerean, Åsa Windfäll. Discrete velocity models of the Boltzmann equation and conservation laws. Kinetic & Related Models, 2010, 3 (1) : 35-58. doi: 10.3934/krm.2010.3.35
[11]

María Santos Bruzón, Tamara María Garrido. Symmetries and conservation laws of a KdV6 equation. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 631-641. doi: 10.3934/dcdss.2018038
[12]

K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091
[13]

Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028
[14]

Minhajul, T. Raja Sekhar, G. P. Raja Sekhar. Stability of solutions to the Riemann problem for a thin film model of a perfectly soluble anti-surfactant solution. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3367-3386. doi: 10.3934/cpaa.2019152
[15]

Lihua Min, Xiaoping Yang. Finite speed of propagation and algebraic time decay of solutions to a generalized thin film equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 543-566. doi: 10.3934/cpaa.2014.13.543
[16]

Sergey Degtyarev. Classical solvability of the multidimensional free boundary problem for the thin film equation with quadratic mobility in the case of partial wetting. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3625-3699. doi: 10.3934/dcds.2017156
[17]

Maria Laura Delle Monache, Paola Goatin. Stability estimates for scalar conservation laws with moving flux constraints. Networks & Heterogeneous Media, 2017, 12 (2) : 245-258. doi: 10.3934/nhm.2017010
[18]

María Rosa, María de los Santos Bruzón, María de la Luz Gandarias. Lie symmetries and conservation laws of a Fisher equation with nonlinear convection term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1331-1339. doi: 10.3934/dcdss.2015.8.1331
[19]

Gianluca Crippa, Laura V. Spinolo. An overview on some results concerning the transport equation and its applications to conservation laws. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1283-1293. doi: 10.3934/cpaa.2010.9.1283
[20]

Cheng Wang, Xiaoming Wang, Steven M. Wise. Unconditionally stable schemes for equations of thin film epitaxy. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 405-423. doi: 10.3934/dcds.2010.28.405

2019 Impact Factor: 1.338

Tools

Metrics

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

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences