### Abstract

We shall study L^{2} energy conserved solutions to the heat equa- tion. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a fam- ily of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.

Original language | English (US) |
---|---|

Pages (from-to) | 49-64 |

Number of pages | 16 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 37 |

Issue number | 8 |

State | Published - Aug 1 2017 |

### Fingerprint

### Keywords

- Global existence
- Nonlocal heat equation
- Singularly perturbed parabolic equations
- Suitable weak solutions

### ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics

### Cite this

*Discrete and Continuous Dynamical Systems- Series A*,

*37*(8), 49-64.

**Nonlocal heat flows preserving the L2 energy.** / Caffarelli, Luis; Lin, Fang-Hua.

Research output: Contribution to journal › Article

*Discrete and Continuous Dynamical Systems- Series A*, vol. 37, no. 8, pp. 49-64.

}

TY - JOUR

T1 - Nonlocal heat flows preserving the L2 energy

AU - Caffarelli, Luis

AU - Lin, Fang-Hua

PY - 2017/8/1

Y1 - 2017/8/1

N2 - We shall study L2 energy conserved solutions to the heat equa- tion. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a fam- ily of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.

AB - We shall study L2 energy conserved solutions to the heat equa- tion. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a fam- ily of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.

KW - Global existence

KW - Nonlocal heat equation

KW - Singularly perturbed parabolic equations

KW - Suitable weak solutions

UR - http://www.scopus.com/inward/record.url?scp=85019421385&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85019421385&partnerID=8YFLogxK

M3 - Article

VL - 37

SP - 49

EP - 64

JO - Discrete and Continuous Dynamical Systems

JF - Discrete and Continuous Dynamical Systems

SN - 1078-0947

IS - 8

ER -