American Institute of Mathematical Sciences

Advanced
December  2005, 4(4): 735-742. doi: 10.3934/cpaa.2005.4.735

A variational principle for nonlinear transport equations

Nassif Ghoussoub 1,

1. 

Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2

Received  January 2005 Revised  June 2005 Published  September 2005

We verify -after appropriate modifications- an old conjecture of Brezis-Ekeland [4] concerning the feasibility of a global and variational approach to the problems of existence and uniqueness of solutions of non-linear transport equations, which do not normally fit in an Euler-Lagrange framework. Our method is based on a concept of "anti-self duality" that seems to be inherent in many problems, including gradient flows of convex energy functionals treated in [10] and other parabolic evolution equations ([7]).
Keywords: Transport equation, convex duality.
Mathematics Subject Classification: 35F30, 35A15.
Citation: Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure & Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735
[1]

Gang Li, Lipu Zhang, Zhe Liu. The stable duality of DC programs for composite convex functions. Journal of Industrial & Management Optimization, 2017, 13 (1) : 63-79. doi: 10.3934/jimo.2016004
[2]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[3]

Jianshe Yu, Honghua Bin, Zhiming Guo. Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 939-950. doi: 10.3934/dcds.2006.15.939
[4]

Yuying Zhou, Gang Li. The Toland-Fenchel-Lagrange duality of DC programs for composite convex functions. Numerical Algebra, Control & Optimization, 2014, 4 (1) : 9-23. doi: 10.3934/naco.2014.4.9
[5]

Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[6]

Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations & Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070
[7]

Elimhan N. Mahmudov. Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions. Evolution Equations & Control Theory, 2021, 10 (1) : 37-59. doi: 10.3934/eect.2020051
[8]

Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic & Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955
[9]

Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083
[10]

Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic & Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789
[11]

Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic & Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
[12]

Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114
[13]

Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
[14]

Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic & Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113
[15]

Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic & Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395
[16]

John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems & Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181
[17]

Lei Wu. Diffusive limit with geometric correction of unsteady neutron transport equation. Kinetic & Related Models, 2017, 10 (4) : 1163-1203. doi: 10.3934/krm.2017045
[18]

Xumin Gu. Global wellposedness for a transport equation with super-critial dissipation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 653-665. doi: 10.3934/cpaa.2011.10.653
[19]

Tomasz Komorowski. Long time asymptotics of a degenerate linear kinetic transport equation. Kinetic & Related Models, 2014, 7 (1) : 79-108. doi: 10.3934/krm.2014.7.79
[20]

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

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (77)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences