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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
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 |
References:
[1] |
B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z., 217 (1994), 179-197.
doi: 10.1007/BF02571941. |
[2] |
S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow, J. Diff. Geom., 82 (2009), 207-227. |
[3] |
H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math., 109 (1992), 247-263.
doi: 10.1007/BF01232027. |
[4] |
H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann., 331 (2005), 795-807.
doi: 10.1007/s00208-004-0605-3. |
[5] |
X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal., 255 (2008), 1024-1038.
doi: 10.1016/j.jfa.2008.05.009. |
[6] |
X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal., 19 (2009), 989-1000.
doi: 10.1007/s00039-009-0024-4. |
[7] |
B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math., 44 (1991), 469-483.
doi: 10.1002/cpa.3160440405. |
[8] |
B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom., 33 (1991), 325-334. |
[9] |
B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math., 45 (1992), 1003-1014.
doi: 10.1002/cpa.3160450805. |
[10] |
B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Res. Lett., 2 (1995), 701-718.
doi: 10.4310/MRL.1995.v2.n6.a4. |
[11] |
B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. Anal., 11 (2001), 219-231.
doi: 10.1007/BF02921963. |
[12] |
B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math., 129 (1997), 213-238.
doi: 10.1007/s002220050162. |
[13] |
K. Ecker, A formula relating entropy monotonicity to Harnack inequalities, Comm. Anal. Geom., 15 (2007), 1025-1061.
doi: 10.4310/CAG.2007.v15.n5.a5. |
[14] |
S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds, Asian J. Math., 12 (2008), 545-552.
doi: 10.4310/AJM.2008.v12.n4.a8. |
[15] |
S. W. Fang, Harnack estimates for curvature flows depending on mean curvature, Appl. Math. J. Chinese Univ. B, 24 (2009), 361-369.
doi: 10.1007/s11766-009-2019-1. |
[16] |
S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow, Acta Math. Sincia (Chinese Series), 53 (2010), 597-606. |
[17] |
S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow, Geom. Dedicata, 161 (2012), 11-22.
doi: 10.1007/s10711-011-9690-0. |
[18] |
S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow, Adv. Geom., 13 (2013), 741-755.
doi: 10.1515/advgeom-2013-0020. |
[19] |
S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows, Arch. Math., 100 (2013), 179-189.
doi: 10.1007/s00013-013-0482-7. |
[20] |
C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal., 12 (2002), 425-436.
doi: 10.1007/BF02922048. |
[21] |
R. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity (ed. James A. Isenberg), Contemp. Math. 71, Amer. Math. Soc., Providence, RI, (1988), 237-262.
doi: 10.1090/conm/071/954419. |
[22] |
R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126. |
[23] |
R. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom., 37 (1993), 225-243. |
[24] |
R. Hamilton, The Harnack estimate for the mean curvature flow, J. Diff. Geom., 41 (1995), 215-226. |
[25] |
S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023.
doi: 10.1016/j.jfa.2008.05.014. |
[26] |
P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math., 156 (1986), 153-201.
doi: 10.1007/BF02399203. |
[27] |
R. Müller, Monotone volume formulas for geometric flows, J. Reine Angew. Math., 643 (2010), 39-57.
doi: 10.1515/CRELLE.2010.044. |
[28] |
L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow, J. Diff. Geom., 75 (2007), 303-358. |
[29] |
G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, ().
|
[30] |
O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal., 42 (1981), 110-120.
doi: 10.1016/0022-1236(81)90050-1. |
[31] |
K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math, 3 (1997), 103-118. |
[32] |
K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differ. Equ., 8 (1999), 247-258.
doi: 10.1007/s005260050125. |
[33] |
J. Wang, Harnack estimate for $H^k$-flow, Science in China Series A: Mathematics, 50 (2007), 1642-1650.
doi: 10.1007/s11425-007-0095-3. |
[34] |
J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space, Asian J. Math., 13 (2009), 89-100.
doi: 10.4310/AJM.2009.v13.n1.a5. |
[35] |
A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow, J. Math. Anal. Appl., 406 (2013), 502-510.
doi: 10.1016/j.jmaa.2013.05.004. |
show all references
References:
[1] |
B. Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z., 217 (1994), 179-197.
doi: 10.1007/BF02571941. |
[2] |
S. Brendle, A generalization of Hamilton's differential Harnack inequality for the Ricci flow, J. Diff. Geom., 82 (2009), 207-227. |
[3] |
H. D. Cao, On Harnack's inequalities for the Kähler-Ricci flow, Invent. Math., 109 (1992), 247-263.
doi: 10.1007/BF01232027. |
[4] |
H. D. Cao and L. Ni, Matrix Li-Yau-Hamilton estimates for the heat equation on Kähler manifolds, Math. Ann., 331 (2005), 795-807.
doi: 10.1007/s00208-004-0605-3. |
[5] |
X. D. Cao, Differential Harnack estimates for backward heat equations with potentials under the Ricci flow, J. Funct. Anal., 255 (2008), 1024-1038.
doi: 10.1016/j.jfa.2008.05.009. |
[6] |
X. D. Cao and R. Hamilton, Differential Harnack estimates for time-dependent heat equations with potentials, Geom. Funct. Anal., 19 (2009), 989-1000.
doi: 10.1007/s00039-009-0024-4. |
[7] |
B. Chow, On Harnack's inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math., 44 (1991), 469-483.
doi: 10.1002/cpa.3160440405. |
[8] |
B. Chow, The Ricci flow on the 2-sphere, J. Diff. Geom., 33 (1991), 325-334. |
[9] |
B. Chow, The Yamabe flow on locally conformally flat manifolds with positive Ricci curvature, Comm. Pure Appl. Math., 45 (1992), 1003-1014.
doi: 10.1002/cpa.3160450805. |
[10] |
B. Chow and S. C. Chu, A geometric interpretation of Hamilton's Harnack inequality for the Ricci flow, Math. Res. Lett., 2 (1995), 701-718.
doi: 10.4310/MRL.1995.v2.n6.a4. |
[11] |
B. Chow and S. C. Chu, Space-time formulation of Harnack inequalities for curvature flows of hypersurfaces, J. Geom. Anal., 11 (2001), 219-231.
doi: 10.1007/BF02921963. |
[12] |
B. Chow and R. Hamilton, Constrained and linear Harnack inequalities for parabolic equations, Invent. Math., 129 (1997), 213-238.
doi: 10.1007/s002220050162. |
[13] |
K. Ecker, A formula relating entropy monotonicity to Harnack inequalities, Comm. Anal. Geom., 15 (2007), 1025-1061.
doi: 10.4310/CAG.2007.v15.n5.a5. |
[14] |
S. W. Fang, Local Harnack estimate for Yamabe flow on locally conformally flat manifolds, Asian J. Math., 12 (2008), 545-552.
doi: 10.4310/AJM.2008.v12.n4.a8. |
[15] |
S. W. Fang, Harnack estimates for curvature flows depending on mean curvature, Appl. Math. J. Chinese Univ. B, 24 (2009), 361-369.
doi: 10.1007/s11766-009-2019-1. |
[16] |
S. W. Fang and F. Ye, Differential Harnack inequalities for heat equations with potentials under Kähler-Ricci flow, Acta Math. Sincia (Chinese Series), 53 (2010), 597-606. |
[17] |
S. W. Fang, Differential Harnack inequalities for heat equations with potentials under the Bernhard List's flow, Geom. Dedicata, 161 (2012), 11-22.
doi: 10.1007/s10711-011-9690-0. |
[18] |
S. W. Fang, Differential Harnack inequalities for backward heat equations with potentials under an extended Ricci flow, Adv. Geom., 13 (2013), 741-755.
doi: 10.1515/advgeom-2013-0020. |
[19] |
S. W. Fang, Differential Harnack inequalities for heat equations with potentials under geometric flows, Arch. Math., 100 (2013), 179-189.
doi: 10.1007/s00013-013-0482-7. |
[20] |
C. M. Guenther, The fundamental solution on manifolds with time-dependent metrics, J. Geom. Anal., 12 (2002), 425-436.
doi: 10.1007/BF02922048. |
[21] |
R. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity (ed. James A. Isenberg), Contemp. Math. 71, Amer. Math. Soc., Providence, RI, (1988), 237-262.
doi: 10.1090/conm/071/954419. |
[22] |
R. Hamilton, A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., 1 (1993), 113-126. |
[23] |
R. Hamilton, The Harnack estimate for the Ricci flow, J. Diff. Geom., 37 (1993), 225-243. |
[24] |
R. Hamilton, The Harnack estimate for the mean curvature flow, J. Diff. Geom., 41 (1995), 215-226. |
[25] |
S. L. Kuang and Q. S. Zhang, A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow, J. Funct. Anal., 255 (2008), 1008-1023.
doi: 10.1016/j.jfa.2008.05.014. |
[26] |
P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta. Math., 156 (1986), 153-201.
doi: 10.1007/BF02399203. |
[27] |
R. Müller, Monotone volume formulas for geometric flows, J. Reine Angew. Math., 643 (2010), 39-57.
doi: 10.1515/CRELLE.2010.044. |
[28] |
L. Ni, A matrix Li-Yau-Hamilton estimate for Kähler-Ricci flow, J. Diff. Geom., 75 (2007), 303-358. |
[29] |
G. Perelman, The entropy formula for the Ricci flow and its geometric applications,, preprint, ().
|
[30] |
O. Rothaus, Logarithmic Sobolev inequalities and the spectrum of Schrödinger operators, J. Funct. Anal., 42 (1981), 110-120.
doi: 10.1016/0022-1236(81)90050-1. |
[31] |
K. Smoczyk, Harnack inequalities for curvature flows depending on mean curvature, New York J. Math, 3 (1997), 103-118. |
[32] |
K. Smoczyk, Harnack inequality for the Lagrangian mean curvature flow, Calc. Var. Partial Differ. Equ., 8 (1999), 247-258.
doi: 10.1007/s005260050125. |
[33] |
J. Wang, Harnack estimate for $H^k$-flow, Science in China Series A: Mathematics, 50 (2007), 1642-1650.
doi: 10.1007/s11425-007-0095-3. |
[34] |
J. Wang, Local Harnack estimate for mean curvature flow in Euclidean space, Asian J. Math., 13 (2009), 89-100.
doi: 10.4310/AJM.2009.v13.n1.a5. |
[35] |
A. Q. Zhu, Differential Harnack inequalities for the backward heat equation with potential under the harmonic-Ricci flow, J. Math. Anal. Appl., 406 (2013), 502-510.
doi: 10.1016/j.jmaa.2013.05.004. |
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