# American Institute of Mathematical Sciences

• Previous Article
Lifespan of classical discontinuous solutions to the generalized nonlinear initial-boundary Riemann problem for hyperbolic conservation laws with small BV data: shocks and contact discontinuities
• CPAA Home
• This Issue
• Next Article
Uniform stability of the Boltzmann equation with an external force near vacuum
May  2015, 14(3): 793-809. doi: 10.3934/cpaa.2015.14.793

## Differential Harnack estimates for backward heat equations with potentials under geometric flows

 1 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China 2 School of Mathematics and physics, Jiangsu University of Technology, Changzhou, Jiangsu 213001, China

Received  January 2014 Revised  December 2014 Published  March 2015

In the paper we consider a closed Riemannian manifold $M$ with a time-dependent Riemannian metric $g_{i j}(t)$ evolving by $\partial_{t}g_{i j}=-2S_{i j}$ where $S_{i j}$ is a symmetric two-tensor on $(M,g(t))$. We prove some differential Harnack inequalities for positive solutions of backward heat equations with potentials when the metric satisfies the geometric flow. Some applications of these inequalities will be obtained. In particular, we show that the shrinking breathers of the Lorentzian mean curvature flow are the shrinking gradient solitons when the ambient Lorentzian manifold has nonnegative sectional curvature.
Citation: Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793
##### References:
 [1] B. Andrews, Harnack inequalities for evolving hypersurfaces,, \emph{Math. Z.}, 217 (1994), 179. doi: 10.1007/BF02571941. Google Scholar [2] S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow,, \emph{J. Diff. Geom.}, 82 (2009), 207. Google Scholar [3] H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow,, \emph{Invent. Math.}, 109 (1992), 247. doi: 10.1007/BF01232027. Google Scholar [4] H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds,, \emph{Math. Ann.}, 331 (2005), 795. doi: 10.1007/s00208-004-0605-3. Google Scholar [5] X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1024. doi: 10.1016/j.jfa.2008.05.009. Google Scholar [6] X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials,, \emph{Geom. Funct. Anal.}, 19 (2009), 989. doi: 10.1007/s00039-009-0024-4. Google Scholar [7] B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 469. doi: 10.1002/cpa.3160440405. Google Scholar [8] B. Chow, The Ricci flow on the 2-sphere,, \emph{J. Diff. Geom.}, 33 (1991), 325. Google Scholar [9] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 1003. doi: 10.1002/cpa.3160450805. Google Scholar [10] B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow,, \emph{Math. Res. Lett.}, 2 (1995), 701. doi: 10.4310/MRL.1995.v2.n6.a4. Google Scholar [11] B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces,, \emph{J. Geom. Anal.}, 11 (2001), 219. doi: 10.1007/BF02921963. Google Scholar [12] B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations,, \emph{Invent. Math.}, 129 (1997), 213. doi: 10.1007/s002220050162. Google Scholar [13] K. Ecker, A formula relating entropy monotonicity to Harnack inequalities,, \emph{Comm. Anal. Geom.}, 15 (2007), 1025. doi: 10.4310/CAG.2007.v15.n5.a5. Google Scholar [14] S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds,, \emph{Asian J. Math.}, 12 (2008), 545. doi: 10.4310/AJM.2008.v12.n4.a8. Google Scholar [15] S. W. Fang, Harnack estimates for curvature flows depending on mean curvature,, \emph{Appl. Math. J. Chinese Univ. B}, 24 (2009), 361. doi: 10.1007/s11766-009-2019-1. Google Scholar [16] S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow,, \emph{Acta Math. Sincia (Chinese Series)}, 53 (2010), 597. Google Scholar [17] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow,, \emph{Geom. Dedicata}, 161 (2012), 11. doi: 10.1007/s10711-011-9690-0. Google Scholar [18] S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow,, \emph{Adv. Geom.}, 13 (2013), 741. doi: 10.1515/advgeom-2013-0020. Google Scholar [19] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows,, \emph{Arch. Math.}, 100 (2013), 179. doi: 10.1007/s00013-013-0482-7. Google Scholar [20] C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics,, \emph{J. Geom. Anal.}, 12 (2002), 425. doi: 10.1007/BF02922048. Google Scholar [21] R. Hamilton, The Ricci flow on surfaces,, in \emph{Mathematics and General Relativity} (ed. James A. Isenberg), (1988), 237. doi: 10.1090/conm/071/954419. Google Scholar [22] R. Hamilton, A matrix Harnack estimate for the heat equation,, \emph{Comm. Anal. Geom.}, 1 (1993), 113. Google Scholar [23] R. Hamilton, The Harnack estimate for the Ricci flow,, \emph{J. Diff. Geom.}, 37 (1993), 225. Google Scholar [24] R. Hamilton, The Harnack estimate for the mean curvature flow,, \emph{J. Diff. Geom.}, 41 (1995), 215. Google Scholar [25] S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1008. doi: 10.1016/j.jfa.2008.05.014. Google Scholar [26] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, \emph{Acta. Math.}, 156 (1986), 153. doi: 10.1007/BF02399203. Google Scholar [27] R. Müller, Monotone volume formulas for geometric flows,, \emph{J. Reine Angew. Math.}, 643 (2010), 39. doi: 10.1515/CRELLE.2010.044. Google Scholar [28] L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow,, \emph{J. Diff. Geom.}, 75 (2007), 303. Google Scholar [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, (). Google Scholar [30] O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators,, \emph{J. Funct. Anal.}, 42 (1981), 110. doi: 10.1016/0022-1236(81)90050-1. Google Scholar [31] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature,, \emph{New York J. Math}, 3 (1997), 103. Google Scholar [32] K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow,, \emph{Calc. Var. Partial Differ. Equ.}, 8 (1999), 247. doi: 10.1007/s005260050125. Google Scholar [33] J. Wang, Harnack estimate for $H^k$-flow,, \emph{Science in China Series A: Mathematics}, 50 (2007), 1642. doi: 10.1007/s11425-007-0095-3. Google Scholar [34] J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space,, \emph{Asian J. Math.}, 13 (2009), 89. doi: 10.4310/AJM.2009.v13.n1.a5. Google Scholar [35] A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow,, \emph{J. Math. Anal. Appl.}, 406 (2013), 502. doi: 10.1016/j.jmaa.2013.05.004. Google Scholar

show all references

##### References:
 [1] B. Andrews, Harnack inequalities for evolving hypersurfaces,, \emph{Math. Z.}, 217 (1994), 179. doi: 10.1007/BF02571941. Google Scholar [2] S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow,, \emph{J. Diff. Geom.}, 82 (2009), 207. Google Scholar [3] H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow,, \emph{Invent. Math.}, 109 (1992), 247. doi: 10.1007/BF01232027. Google Scholar [4] H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds,, \emph{Math. Ann.}, 331 (2005), 795. doi: 10.1007/s00208-004-0605-3. Google Scholar [5] X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1024. doi: 10.1016/j.jfa.2008.05.009. Google Scholar [6] X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials,, \emph{Geom. Funct. Anal.}, 19 (2009), 989. doi: 10.1007/s00039-009-0024-4. Google Scholar [7] B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow,, \emph{Comm. Pure Appl. Math.}, 44 (1991), 469. doi: 10.1002/cpa.3160440405. Google Scholar [8] B. Chow, The Ricci flow on the 2-sphere,, \emph{J. Diff. Geom.}, 33 (1991), 325. Google Scholar [9] B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature,, \emph{Comm. Pure Appl. Math.}, 45 (1992), 1003. doi: 10.1002/cpa.3160450805. Google Scholar [10] B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow,, \emph{Math. Res. Lett.}, 2 (1995), 701. doi: 10.4310/MRL.1995.v2.n6.a4. Google Scholar [11] B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces,, \emph{J. Geom. Anal.}, 11 (2001), 219. doi: 10.1007/BF02921963. Google Scholar [12] B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations,, \emph{Invent. Math.}, 129 (1997), 213. doi: 10.1007/s002220050162. Google Scholar [13] K. Ecker, A formula relating entropy monotonicity to Harnack inequalities,, \emph{Comm. Anal. Geom.}, 15 (2007), 1025. doi: 10.4310/CAG.2007.v15.n5.a5. Google Scholar [14] S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds,, \emph{Asian J. Math.}, 12 (2008), 545. doi: 10.4310/AJM.2008.v12.n4.a8. Google Scholar [15] S. W. Fang, Harnack estimates for curvature flows depending on mean curvature,, \emph{Appl. Math. J. Chinese Univ. B}, 24 (2009), 361. doi: 10.1007/s11766-009-2019-1. Google Scholar [16] S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow,, \emph{Acta Math. Sincia (Chinese Series)}, 53 (2010), 597. Google Scholar [17] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow,, \emph{Geom. Dedicata}, 161 (2012), 11. doi: 10.1007/s10711-011-9690-0. Google Scholar [18] S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow,, \emph{Adv. Geom.}, 13 (2013), 741. doi: 10.1515/advgeom-2013-0020. Google Scholar [19] S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows,, \emph{Arch. Math.}, 100 (2013), 179. doi: 10.1007/s00013-013-0482-7. Google Scholar [20] C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics,, \emph{J. Geom. Anal.}, 12 (2002), 425. doi: 10.1007/BF02922048. Google Scholar [21] R. Hamilton, The Ricci flow on surfaces,, in \emph{Mathematics and General Relativity} (ed. James A. Isenberg), (1988), 237. doi: 10.1090/conm/071/954419. Google Scholar [22] R. Hamilton, A matrix Harnack estimate for the heat equation,, \emph{Comm. Anal. Geom.}, 1 (1993), 113. Google Scholar [23] R. Hamilton, The Harnack estimate for the Ricci flow,, \emph{J. Diff. Geom.}, 37 (1993), 225. Google Scholar [24] R. Hamilton, The Harnack estimate for the mean curvature flow,, \emph{J. Diff. Geom.}, 41 (1995), 215. Google Scholar [25] S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow,, \emph{J. Funct. Anal.}, 255 (2008), 1008. doi: 10.1016/j.jfa.2008.05.014. Google Scholar [26] P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator,, \emph{Acta. Math.}, 156 (1986), 153. doi: 10.1007/BF02399203. Google Scholar [27] R. Müller, Monotone volume formulas for geometric flows,, \emph{J. Reine Angew. Math.}, 643 (2010), 39. doi: 10.1515/CRELLE.2010.044. Google Scholar [28] L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow,, \emph{J. Diff. Geom.}, 75 (2007), 303. Google Scholar [29] G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, (). Google Scholar [30] O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators,, \emph{J. Funct. Anal.}, 42 (1981), 110. doi: 10.1016/0022-1236(81)90050-1. Google Scholar [31] K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature,, \emph{New York J. Math}, 3 (1997), 103. Google Scholar [32] K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow,, \emph{Calc. Var. Partial Differ. Equ.}, 8 (1999), 247. doi: 10.1007/s005260050125. Google Scholar [33] J. Wang, Harnack estimate for $H^k$-flow,, \emph{Science in China Series A: Mathematics}, 50 (2007), 1642. doi: 10.1007/s11425-007-0095-3. Google Scholar [34] J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space,, \emph{Asian J. Math.}, 13 (2009), 89. doi: 10.4310/AJM.2009.v13.n1.a5. Google Scholar [35] A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow,, \emph{J. Math. Anal. Appl.}, 406 (2013), 502. doi: 10.1016/j.jmaa.2013.05.004. Google Scholar
 [1] Wen Wang, Dapeng Xie, Hui Zhou. Local Aronson-Bénilan gradient estimates and Harnack inequality for the porous medium equation along Ricci flow. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1957-1974. doi: 10.3934/cpaa.2018093 [2] Aymen Jbalia. On a logarithmic stability estimate for an inverse heat conduction problem. Mathematical Control & Related Fields, 2019, 9 (2) : 277-287. doi: 10.3934/mcrf.2019014 [3] Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 [4] Michael Renardy. Backward uniqueness for linearized compressible flow. Evolution Equations & Control Theory, 2015, 4 (1) : 107-113. doi: 10.3934/eect.2015.4.107 [5] Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043 [6] Sylvain Ervedoza, Enrique Zuazua. Observability of heat processes by transmutation without geometric restrictions. Mathematical Control & Related Fields, 2011, 1 (2) : 177-187. doi: 10.3934/mcrf.2011.1.177 [7] John Sylvester. An estimate for the free Helmholtz equation that scales. Inverse Problems & Imaging, 2009, 3 (2) : 333-351. doi: 10.3934/ipi.2009.3.333 [8] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [9] Sebastián Ferrer, Martin Lara. Families of canonical transformations by Hamilton-Jacobi-Poincaré equation. Application to rotational and orbital motion. Journal of Geometric Mechanics, 2010, 2 (3) : 223-241. doi: 10.3934/jgm.2010.2.223 [10] Manuel de León, Juan Carlos Marrero, David Martín de Diego. Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics. Journal of Geometric Mechanics, 2010, 2 (2) : 159-198. doi: 10.3934/jgm.2010.2.159 [11] Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73 [12] Peng Gao. Global Carleman estimate for the Kawahara equation and its applications. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1853-1874. doi: 10.3934/cpaa.2018088 [13] Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46 [14] Y. Chen, S. Levine. The existence of the heat flow of H-systems. Discrete & Continuous Dynamical Systems - A, 2002, 8 (1) : 219-236. doi: 10.3934/dcds.2002.8.219 [15] Andrey Shishkov, Laurent Véron. Propagation of singularities of nonlinear heat flow in fissured media. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1769-1782. doi: 10.3934/cpaa.2013.12.1769 [16] Alberto Bressan, Khai T. Nguyen. Optima and equilibria for traffic flow on networks with backward propagating queues. Networks & Heterogeneous Media, 2015, 10 (4) : 717-748. doi: 10.3934/nhm.2015.10.717 [17] C. Brändle, E. Chasseigne, Raúl Ferreira. Unbounded solutions of the nonlocal heat equation. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1663-1686. doi: 10.3934/cpaa.2011.10.1663 [18] Arthur Ramiandrisoa. Nonlinear heat equation: the radial case. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 849-870. doi: 10.3934/dcds.1999.5.849 [19] Delio Mugnolo. Gaussian estimates for a heat equation on a network. Networks & Heterogeneous Media, 2007, 2 (1) : 55-79. doi: 10.3934/nhm.2007.2.55 [20] Sergei A. Avdonin, Sergei A. Ivanov, Jun-Min Wang. Inverse problems for the heat equation with memory. Inverse Problems & Imaging, 2019, 13 (1) : 31-38. doi: 10.3934/ipi.2019002

2018 Impact Factor: 0.925