September  2008, 1(3): 405-426. doi: 10.3934/dcdss.2008.1.405

ODEs with Sobolev coefficients: The eulerian and the lagrangian approach

1. 

Universität Zürich, Institut für Mathematik, Winterthurerstrasse 190, CH–8057 Zürich, Switzerland

Received  November 2007 Revised  March 2008 Published  June 2008

In this paper we describe two approaches to the well-posedness of Lagrangian flows of Sobolev vector fields. One is the theory of renormalized solutions which was introduced by DiPerna and Lions in the eighties. In this framework the well-posedness of the flow is a corollary of an analogous result for the corresponding transport equation. The second approach has been recently introduced by Gianluca Crippa and the author and it is instead based on suitable estimates performed directly on the lagrangian formulation.
Citation: Camillo De Lellis. ODEs with Sobolev coefficients: The eulerian and the lagrangian approach. Discrete and Continuous Dynamical Systems - S, 2008, 1 (3) : 405-426. doi: 10.3934/dcdss.2008.1.405
[1]

Zhigang Wang, Lei Wang, Yachun Li. Renormalized entropy solutions for degenerate parabolic-hyperbolic equations with time-space dependent coefficients. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1163-1182. doi: 10.3934/cpaa.2013.12.1163

[2]

Hiroshi Watanabe. Existence and uniqueness of entropy solutions to strongly degenerate parabolic equations with discontinuous coefficients. Conference Publications, 2013, 2013 (special) : 781-790. doi: 10.3934/proc.2013.2013.781

[3]

Feng Zhou, Zhenqiu Zhang. Pointwise gradient estimates for subquadratic elliptic systems with discontinuous coefficients. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3137-3160. doi: 10.3934/cpaa.2019141

[4]

Pierpaolo Soravia. Uniqueness results for fully nonlinear degenerate elliptic equations with discontinuous coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 213-240. doi: 10.3934/cpaa.2006.5.213

[5]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177

[6]

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

[7]

Junjie Zhang, Shenzhou Zheng. Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients. Communications on Pure and Applied Analysis, 2017, 16 (3) : 899-914. doi: 10.3934/cpaa.2017043

[8]

Sergio Polidoro, Annalaura Rebucci, Bianca Stroffolini. Schauder type estimates for degenerate Kolmogorov equations with Dini continuous coefficients. Communications on Pure and Applied Analysis, 2022, 21 (4) : 1385-1416. doi: 10.3934/cpaa.2022023

[9]

Serena Dipierro, Aram Karakhanyan, Enrico Valdinoci. Classification of irregular free boundary points for non-divergence type equations with discontinuous coefficients. Discrete and Continuous Dynamical Systems, 2018, 38 (12) : 6073-6090. doi: 10.3934/dcds.2018262

[10]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure and Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155

[11]

Bingkang Huang, Lusheng Wang, Qinghua Xiao. Global nonlinear stability of rarefaction waves for compressible Navier-Stokes equations with temperature and density dependent transport coefficients. Kinetic and Related Models, 2016, 9 (3) : 469-514. doi: 10.3934/krm.2016004

[12]

Qiang Liu, Zhichang Guo, Chunpeng Wang. Renormalized solutions to a reaction-diffusion system applied to image denoising. Discrete and Continuous Dynamical Systems - B, 2016, 21 (6) : 1839-1858. doi: 10.3934/dcdsb.2016025

[13]

Virginie De Witte, Willy Govaerts. Numerical computation of normal form coefficients of bifurcations of odes in MATLAB. Conference Publications, 2011, 2011 (Special) : 362-372. doi: 10.3934/proc.2011.2011.362

[14]

Niklas Sapountzoglou, Aleksandra Zimmermann. Renormalized solutions for stochastic $ p $-Laplace equations with $ L^1 $-initial data: The case of multiplicative noise. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3979-4002. doi: 10.3934/dcds.2022041

[15]

Guillaume Bal, Alexandre Jollivet. Stability estimates in stationary inverse transport. Inverse Problems and Imaging, 2008, 2 (4) : 427-454. doi: 10.3934/ipi.2008.2.427

[16]

Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure and Applied Analysis, 2021, 20 (3) : 955-974. doi: 10.3934/cpaa.2021001

[17]

Gui-Qiang Chen, Bo Su. Discontinuous solutions for Hamilton-Jacobi equations: Uniqueness and regularity. Discrete and Continuous Dynamical Systems, 2003, 9 (1) : 167-192. doi: 10.3934/dcds.2003.9.167

[18]

Wuming Li, Xiaojun Liu, Quansen Jiu. The decay estimates of solutions for 1D compressible flows with density-dependent viscosity coefficients. Communications on Pure and Applied Analysis, 2013, 12 (2) : 647-661. doi: 10.3934/cpaa.2013.12.647

[19]

Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601

[20]

Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (84)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]