American Institute of Mathematical Sciences

Advanced
February  2009, 23(1&2): 49-64. doi: 10.3934/dcds.2009.23.49

Nonlocal heat flows preserving the L2 energy

Luis Caffarelli 1, and  Fanghua Lin 2,

1. 

Department of Mathematics, University of Texas at Austin, 1 University Station, C1200, Austin, TX 78712-1082
2. 

Department of Mathematics, New York University, 251 Mercer St. New York, NY 10012

Received  November 2007 Revised  April 2008 Published  September 2008

We shall study L2 energy conserved solutions to the heat equation.We shall first establish the global existence, uniqueness andregularity of solutions to such nonlocal heat flows. We then extend themethod to a family of singularly perturbed systems of nonlocal parabolicequations. The main goal is to show that solutions to these perturbedsystems converges strongly to some suitable weak-solutionsof the limiting constrained nonlocal heat flows of maps into a singularspace. It is then possible to study further properties of such suitableweak solutions and the corresponding free boundary problem, which willbe discussed in a forthcoming article.
Keywords: nonlocal heat equation, singularly perturbed parabolic equations, suitable weak solutions, global existence.
Mathematics Subject Classification: Primary: 35K05, 35K55; Secondary: 49N60.
Citation: Luis Caffarelli, Fanghua Lin. Nonlocal heat flows preserving the L2 energy. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 49-64. doi: 10.3934/dcds.2009.23.49
[1]

Minbo Yang, Yanheng Ding. Existence of solutions for singularly perturbed Schrödinger equations with nonlocal part. Communications on Pure and Applied Analysis, 2013, 12 (2) : 771-783. doi: 10.3934/cpaa.2013.12.771
[2]

Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209
[3]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Singularly perturbed degenerated parabolic equations and application to seabed morphodynamics in tided environment. Discrete and Continuous Dynamical Systems, 2011, 29 (3) : 1001-1030. doi: 10.3934/dcds.2011.29.1001
[4]

C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure and Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663
[5]

Xie Li, Zhaoyin Xiang. Existence and nonexistence of local/global solutions for a nonhomogeneous heat equation. Communications on Pure and Applied Analysis, 2014, 13 (4) : 1465-1480. doi: 10.3934/cpaa.2014.13.1465
[6]

Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005
[7]

Sergey Zelik. Asymptotic regularity of solutions of singularly perturbed damped wave equations with supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2004, 11 (2&3) : 351-392. doi: 10.3934/dcds.2004.11.351
[8]

Shihui Zhu. Existence and uniqueness of global weak solutions of the Camassa-Holm equation with a forcing. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 5201-5221. doi: 10.3934/dcds.2016026
[9]

Shijin Ding, Boling Guo, Junyu Lin, Ming Zeng. Global existence of weak solutions for Landau-Lifshitz-Maxwell equations. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 867-890. doi: 10.3934/dcds.2007.17.867
[10]

Wenji Chen, Jianfeng Zhou. Global existence of weak solutions to inhomogeneous Doi-Onsager equations. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021257
[11]

Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019
[12]

Michele Coti Zelati. Global and exponential attractors for the singularly perturbed extensible beam. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 1041-1060. doi: 10.3934/dcds.2009.25.1041
[13]

Daomin Cao, Norman E. Dancer, Ezzat S. Noussair, Shunsen Yan. On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 221-236. doi: 10.3934/dcds.1996.2.221
[14]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[15]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[16]

Valentin Butuzov, Nikolay Nefedov, Oleh Omel'chenko, Lutz Recke. Boundary layer solutions to singularly perturbed quasilinear systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021226
[17]

Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109
[18]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134
[19]

Bernhard Ruf, P. N. Srikanth. Hopf fibration and singularly perturbed elliptic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 823-838. doi: 10.3934/dcdss.2014.7.823
[20]

Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (135)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy