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

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

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.
Citation: Changbing Hu. Stability of under-compressive waves with second and fourth order diffusions. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 629-662. doi: 10.3934/dcds.2008.22.629
[1]

Andrew Comech, Elena Kopylova. Orbital stability and spectral properties of solitary waves of Klein–Gordon equation with concentrated nonlinearity. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021063

[2]

Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020

[3]

Pascal Noble, Sebastien Travadel. Non-persistence of roll-waves under viscous perturbations. Discrete & Continuous Dynamical Systems - B, 2001, 1 (1) : 61-70. doi: 10.3934/dcdsb.2001.1.61

[4]

Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021049

[5]

V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153

[6]

Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008

[7]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055

[8]

Simão Correia, Mário Figueira. A generalized complex Ginzburg-Landau equation: Global existence and stability results. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021056

[9]

Dmitry Treschev. Travelling waves in FPU lattices. Discrete & Continuous Dynamical Systems, 2004, 11 (4) : 867-880. doi: 10.3934/dcds.2004.11.867

[10]

Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035

[11]

Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, 2021, 15 (3) : 445-474. doi: 10.3934/ipi.2020075

[12]

Chiun-Chuan Chen, Hung-Yu Chien, Chih-Chiang Huang. A variational approach to three-phase traveling waves for a gradient system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021055

[13]

Skyler Simmons. Stability of Broucke's isosceles orbit. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3759-3779. doi: 10.3934/dcds.2021015

[14]

Yu-Hsien Liao. Solutions and characterizations under multicriteria management systems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021041

[15]

Miguel R. Nuñez-Chávez. Controllability under positive constraints for quasilinear parabolic PDEs. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021024

[16]

Rafael Luís, Sandra Mendonça. A note on global stability in the periodic logistic map. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4211-4220. doi: 10.3934/dcdsb.2020094

[17]

Lakmi Niwanthi Wadippuli, Ivan Gudoshnikov, Oleg Makarenkov. Global asymptotic stability of nonconvex sweeping processes. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1129-1139. doi: 10.3934/dcdsb.2019212

[18]

Michael Grinfeld, Amy Novick-Cohen. Some remarks on stability for a phase field model with memory. Discrete & Continuous Dynamical Systems, 2006, 15 (4) : 1089-1117. doi: 10.3934/dcds.2006.15.1089

[19]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005

[20]

Tomáš Roubíček. An energy-conserving time-discretisation scheme for poroelastic media with phase-field fracture emitting waves and heat. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 867-893. doi: 10.3934/dcdss.2017044

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (32)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]