American Institute of Mathematical Sciences

Advanced
May  2010, 27(2): 767-786. doi: 10.3934/dcds.2010.27.767

Degenerate diffusion with a drift potential: A viscosity solutions approach

Inwon C. Kim 1, and  Helen K. Lei 1,

1. 

UCLA Mathematics Department, Box 951555, Los Angeles, CA 90095-1555, United States, United States

Received  October 2009 Revised  February 2010 Published  February 2010

We introduce a notion of viscosity solutions for a nonlinear degenerate diffusion equation with a drift potential. We show that our notion of solutions coincide with the weak solutions defined via integration by parts. As an application of the viscosity solutions theory, we show that the free boundary uniformly converges to the equilibrium as $t$ grows. In the case of a convex potential, an exponential rate of free boundary convergence is obtained.
Keywords: well-posedness, porous medium equation, degenerate diffusion, viscosity solutions, convergence to equilibrium..
Mathematics Subject Classification: 35D40, 34K21, 35K6.
Citation: Inwon C. Kim, Helen K. Lei. Degenerate diffusion with a drift potential: A viscosity solutions approach. Discrete & Continuous Dynamical Systems, 2010, 27 (2) : 767-786. doi: 10.3934/dcds.2010.27.767
[1]

Boris P. Andreianov, Giuseppe Maria Coclite, Carlotta Donadello. Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network. Discrete & Continuous Dynamical Systems, 2017, 37 (11) : 5913-5942. doi: 10.3934/dcds.2017257
[2]

David M. Ambrose, Jerry L. Bona, David P. Nicholls. Well-posedness of a model for water waves with viscosity. Discrete & Continuous Dynamical Systems - B, 2012, 17 (4) : 1113-1137. doi: 10.3934/dcdsb.2012.17.1113
[3]

Luis Caffarelli, Juan-Luis Vázquez. Asymptotic behaviour of a porous medium equation with fractional diffusion. Discrete & Continuous Dynamical Systems, 2011, 29 (4) : 1393-1404. doi: 10.3934/dcds.2011.29.1393
[4]

Massimo Cicognani, Michael Reissig. Well-posedness for degenerate Schrödinger equations. Evolution Equations & Control Theory, 2014, 3 (1) : 15-33. doi: 10.3934/eect.2014.3.15
[5]

Mario Bukal. Well-posedness and convergence of a numerical scheme for the corrected Derrida-Lebowitz-Speer-Spohn equation using the Hellinger distance. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3389-3414. doi: 10.3934/dcds.2021001
[6]

Kenji Kimura, Yeong-Cheng Liou, Soon-Yi Wu, Jen-Chih Yao. Well-posedness for parametric vector equilibrium problems with applications. Journal of Industrial & Management Optimization, 2008, 4 (2) : 313-327. doi: 10.3934/jimo.2008.4.313
[7]

Nan-Jing Huang, Xian-Jun Long, Chang-Wen Zhao. Well-Posedness for vector quasi-equilibrium problems with applications. Journal of Industrial & Management Optimization, 2009, 5 (2) : 341-349. doi: 10.3934/jimo.2009.5.341
[8]

Lam Quoc Anh, Pham Thanh Duoc, Tran Quoc Duy. Existence and well-posedness for excess demand equilibrium problems. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021043
[9]

Giuseppe Floridia. Well-posedness for a class of nonlinear degenerate parabolic equations. Conference Publications, 2015, 2015 (special) : 455-463. doi: 10.3934/proc.2015.0455
[10]

Boris Kolev. Local well-posedness of the EPDiff equation: A survey. Journal of Geometric Mechanics, 2017, 9 (2) : 167-189. doi: 10.3934/jgm.2017007
[11]

Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
[12]

Jerry Bona, Nikolay Tzvetkov. Sharp well-posedness results for the BBM equation. Discrete & Continuous Dynamical Systems, 2009, 23 (4) : 1241-1252. doi: 10.3934/dcds.2009.23.1241
[13]

A. Alexandrou Himonas, Curtis Holliman. On well-posedness of the Degasperis-Procesi equation. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 469-488. doi: 10.3934/dcds.2011.31.469
[14]

Nils Strunk. Well-posedness for the supercritical gKdV equation. Communications on Pure & Applied Analysis, 2014, 13 (2) : 527-542. doi: 10.3934/cpaa.2014.13.527
[15]

Keyan Wang. Global well-posedness for a transport equation with non-local velocity and critical diffusion. Communications on Pure & Applied Analysis, 2008, 7 (5) : 1203-1210. doi: 10.3934/cpaa.2008.7.1203
[16]

Vo Van Au, Jagdev Singh, Anh Tuan Nguyen. Well-posedness results and blow-up for a semi-linear time fractional diffusion equation with variable coefficients. Electronic Research Archive, , () : -. doi: 10.3934/era.2021052
[17]

Kazuo Yamazaki, Xueying Wang. Global well-posedness and asymptotic behavior of solutions to a reaction-convection-diffusion cholera epidemic model. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1297-1316. doi: 10.3934/dcdsb.2016.21.1297
[18]

Peng Jiang. Global well-posedness and large time behavior of classical solutions to the diffusion approximation model in radiation hydrodynamics. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2045-2063. doi: 10.3934/dcds.2017087
[19]

Sofía Nieto, Guillermo Reyes. Asymptotic behavior of the solutions of the inhomogeneous Porous Medium Equation with critical vanishing density. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1123-1139. doi: 10.3934/cpaa.2013.12.1123
[20]

Gabriele Grillo, Matteo Muratori, Fabio Punzo. On the asymptotic behaviour of solutions to the fractional porous medium equation with variable density. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5927-5962. doi: 10.3934/dcds.2015.35.5927

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences