Paper · Mar 28, 2018

New differential Harnack inequalities for nonlinear heat equations

Jia-Yong Wu

math.DGarXiv:1803.10622
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Abstract

We prove constrained trace, matrix and constrained matrix Harnack inequalities for the nonlinear heat equation on closed manifolds. We also derive a new interpolated Harnack inequality for the equation on closed surfaces under the -Ricci flow. Finally we prove a new differential Harnack inequality for the equation under the Ricci flow without any curvature condition. Among these Harnack inequalities, the correction terms are all time-exponential functions, which are superior to time-polynomial functions.

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Submitted
Mar 28, 2018
Subjects
math.DG, math.AP

Comments: 19 pages, to appear in Chin. Ann. Math. Ser. B

Ricci flowHarnack inequalitiesSurfaces

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Elliptic gradient estimates for a nonlinear heat equation and applications (Jia-Yong Wu, 2016)8Wu 2016 Rate of curvature decay for the contracting cusp Ricci flow (Peter M. Topping and Hao Yin, 2016)4Topping 2016 Some new gradient estimates for two nonlinear parabolic equations under Ricci flow (Wen Wang and Hui Zhou, 2017)5Wang 2017 Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds (Songzi Li and Xiang-Dong Li, 2017)3Li 2017 Parabolic frequency monotonicity on compact manifolds (Xiaolong Li and Kui Wang, 2018)6Li 2018 Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds (Cyrille Combete, Serge Degla and Leonard Todjihounde, 2018)2Combete 2018 Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations (Xin-An Ren, 2019)1Ren 2019 Differential Harnack inequalities via Concavity of the arrival time (Theodora Bourni and Mat Langford, 2019)7Bourni 2019
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations

    Xin-An Ren

    Topics Harnack inequalities · Shared terms “nonlinear heat”, “heat equation”, “constrained”

  2. Harnack inequalities for positive solutions of the heat equation on closed Finsler manifolds

    Cyrille Combete, Serge Degla and Leonard Todjihounde

    Topics Harnack inequalities · Shared terms “heat equation”

  3. Hamilton differential Harnack inequality and -entropy for Witten Laplacian on Riemannian manifolds

    Songzi Li and Xiang-Dong Li

    Topics Harnack inequalities · Shared terms “differential Harnack”, “heat equation”

  4. Rate of curvature decay for the contracting cusp Ricci flow

    Peter M. Topping and Hao Yin

    Topics Harnack inequalities, Surfaces · Shared terms “differential Harnack”

  5. Some new gradient estimates for two nonlinear parabolic equations under Ricci flow

    Wen Wang and Hui Zhou

    Topics Harnack inequalities · Shared terms “heat equation”, “closed manifold”

  6. Parabolic frequency monotonicity on compact manifolds

    Xiaolong Li and Kui Wang

    Topics Harnack inequalities, Surfaces · Shared terms “matrix Harnack”, “heat equation”

  7. Differential Harnack inequalities via Concavity of the arrival time

    Theodora Bourni and Mat LangfordWider flows

    Topics Harnack inequalities · Shared terms “differential Harnack”

  8. Elliptic gradient estimates for a nonlinear heat equation and applications

    Jia-Yong Wu

    Same author Jia-Yong Wu · Shared terms “nonlinear heat”, “heat equation”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 6D.5
    Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation

    Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

  3. 11B.2
    Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities

    Comparing curvature across space and time.