1.Sobolev inequalities, heat kernels under Ricci flow and the Poincar\'e conjecture

Book published by Chapman Hall/CRC press (English) and Beijing University Press (Chinese).

Amazon Link We provide a treatment of Sobolev inequalities in various settings: the Euclidean case, the Riemannian case and especially the Ricci flow case. We also discuss several applications and ramifications, including heat kernel estimates, Perelman's $W$ entropies and Sobolev inequality with surgeries, and the proof of Hamilton's little loop conjecture with surgeries i.e. strong noncollapsing property of Ricci flow. Using these tools, we present a unified approach which clarifies and simplifies Perelman's proof of the Poincar\'e conjecture. The work is based on Perelman's papers and the works of Chen-Zhu, Kleiner-Lott, Morgan-Tian, and Tao and earlier results of Hamilton's.

2. Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries, arXiv:0712.1329v1 (Math DG), C.R.Academy Science, Paris, Volume 346, Issues 9-10, 2008, Pages 549-552.

3. (with Borislav Yordanov), Finite time blow up for critical wave equations in high dimensions, J. Functional Analysis, Feb. 2006

This paper solved the last piece of the Strauss conjecture on wave equations which has been open since late 70s. It is a problem that had been studied by several prominent mathematicians, Fritz John, e.g.

4. The boundary behavior of Dirichlet heat kernels,   J. Diff. Equations, Vol. 182, No2, pp416-430, 2002.

This paper establishes a lower bound for the Dirichelet heat kernel. The upper bound is well known since Davies' work in the 80's, however the existence of the lower bound had remained open in general. As an immediate consequence, a two sided bound for the parabolic Poisson kernel is obtained. We know that estimates of Poisson kernel has been a long running process starting with the work of S. Poisson himself over 200 years ago. Poisson kernel on the ball is a standard text book material. Two sided estimates on other smooth domains was only achieved in the 1980s by the work of Gruter-Widman, Zhao, Fabes-Cranston-Zhao. The parabolic case was settled by combining Davies's work and this paper. The results allows one to essentially characterize all positive solutions of the heat equation in smooth bounded domains. Applications to linear and nonlinear problems have emerged.

5. A strong regularity result for parabolic equations, Comm. Math. Phy., Vol. 244, p245-260, 2004

It is shown that divergence free condition on the drift term of a linear parabolic equation will result in a leap in allowable singularities while preserving local boundeness and mean value inequality of weak solutions.
 
 
PUBLICATIOINS IN REFEREED JOURNALS/Book


Years 2010-

65. (with Cao, Xiaodong), The conjugate heat equation and Ancient solutions of the Ricci flow, Advances in Mathematics 228 (2011) 2891-2919

64. (with Lei, Zhen), Structure of solutions of 3D Axi-symmetric Navier-Stokes Equations near Maximal Points, Pacific J. Math., to appear.

63. (with Lei, Zhen), A Liouville theorem for the axially-symmetric Navier-Stokes equations, Journal of Functional Analysis 261 (2011) 2323-2345.

62. Sobolev inequalities, heat kernels under Ricci flow, and the Poincaré conjecture. CRC Press, Boca Raton, FL, 2011. x+422 pp.

61. (with Burke Loftus, Jennifer), A priori bounds for the vorticity of axially symmetric solutions to the Navier-Stokes equations. Adv. Differential Equations 15 (2010), no. 5-6, 531-560.

60. Heat kernel bounds, ancient \kappa solutions and the Poincaré conjecture. J. Funct. Anal. 258 (2010), no. 4, 1225-1246.


Years 2005-2009

59. (with Wrkich, James ), Solvability conditions for some semi-linear parabolic equations. J. Differential Equations 247 (2009), no. 8, 2440-2454.

58. Strong noncollapsing and uniform Sobolev inequalities for Ricci flow with surgeries. Pacific J. Math. 239 (2009), no. 1, 179--200.

57.( with Kuang), Shilong; A gradient estimate for all positive solutions of the conjugate heat equation under Ricci flow. J. Funct. Anal. 255 (2008), no. 4, 1008--1023.

56. Strong non-collapsing and uniform Sobolev inequalities for Ricci flow with surgeries, arXiv:0712.1329v1 (Math DG), C.R.Academy Science Mathematics, Paris, Volume 346, Issues 9-10, 2008, Pages 549-552 .

55. A uniform Sobolev inequality under Ricci flow. Int. Math. Res. Not. IMRN 2007, no. 17, Art. ID rnm056, 17 pp, ibidi, Erratum, 2007, no. 19, Art. ID rnm096, 4 pp. ibidi, Addendum, rnm 138, pp1-12, 2007.

54. Some gradient estimates for the heat equation on domains and for an equation by Perelman, International Math. Research Notices, Vol. 2006, 39 pages.

53. (with Thierry Coulhon), Large time behavior of heat kernels on forms, J. Differential Geometry, vol 77, no3, p353-384, 2007.

52. (with Philippe Souplet) Global solutions of inhomogeneous Hamilton-Jacobi equations , J. d' Analyse Math. Vol. 99, p355-396, 2006.

51. (with Philippe Souplet) sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds, Bulletin LMS, vol. 38, no 6., p1045-1053, 2006.

50.(with Zoran Grujic) Space-time localization of a class of geometric criteria for preventing blow-up in the 3D NSE, Comm. Math. Phy. Vol. 262 (2006), no. 3, 555--564.

49. The ill-posedness of Navier-Stokes equation in connected sums of R^3, Complex Variables, Vol. 51, no. 8-11, p1059-1064; a special issue for Professor G. C. Wen's 75th birthday.

48. Local estimates on two linear parabolic equations with singular coefficients, Pacific J., Vol. 233, no2, p367-396, Feb. 2006

47. Stability result for the Cheng-Yau gradient estimate, Pacific Journal of Mathematics, June 2006. pdf file

46. (with Borislav Yordanov), Finite time blow up for critical wave equations in high dimensions, J. Functional Analysis, Vol. 231, no2, p361-374. 2006 pdf file

45. (with Borislav Yordanov), Finite time blow up for wave equations with a potential, SIAM Journal of Mathematical Analysis, Vol. 36 (2005), no. 5, 1426--1433. pdf file


Years 2000 - 2004

44. (with V. Liskevic), Extra regularity for parabolic equations with drift terms, Manuscrita Math. , Vol. 113, No 3, p191-209, 2004,

43. (with Bun Wong), Refined gradient bounds, Poisson equations and some applications to open Kahler manifolds, Asian J. Math., Vol. 7, No 3, p337-364, 2003, ps file

42. A strong regularity result for parabolic equations, Comm. Math. Phy., Vol. 244, p245-260, 2004 pdf file

41. Global solutions of Navier-Stokes equations with large $L^2$ norms in a new function space, Advances in Differential Equations., Vol. 9, No. 5-6, p587-624, 2004 pdf file

40. Positive solutions to DDu -V u + W u^p=0 and its parabolic counter part in noncompact manifolds, Pacific J. Math., Vol. 213, No 1, p163-200, 2004, dvi file

39. A Kazdan-Warner type condition and heat kernel estimate on noncompact manifolds, Indiana U. Math. J., Vol. 52, No4, p1075-p1111, 2003. dvi file

38. The global behavior of heat kernels on exterior domains, J. of Functional Analysis, 200 (2003), no. 1, 160--176.

37. Finite energy solutions to the Yamabe equation, Geom. Dedicata, Vol. 101, p153-165, 2003. dvi file

36.    A sharp comparison result concerning Schr\"odinger heat kernels, Bulletin London Math. Society, 35 (2003), no. 4, p461--472.

35. (with J. A. Goldstein),    Linear parabolic equations with strong singular potentials, Trans. AMS, Vol. 355, No.1, pp197-211, 2003.

34.    The boundary behavior of Dirichlet heat kernels,   J. Diff. Equations, Vol. 182, No2, pp416-430, 2002.

33. (with P. Souplet) ,    Stability for semilinear parabolic equations with decaying potential in R^n and dynamical approach to the existence of ground state,   Annales Inst. H. Poincar\'e, Nonlin\'eaire, Vol. 19, No. 5, pp683-703, 2002.

32.    A blow up result for a nonlinear wave equation with damping: the critical case ,   C. R. Acad. Sci. Paris, Vol. 333, No. 2, pp109-114, 2001 (this is a complete paper)

31. (with Ph. Souplet) ,   Existence of ground states for semilinear elliptic equations with decaying mass: a parabolic approach,   C. R. Acad. Sci. Paris, Vol. 332, pp515-520, 2001

30. (with J. A. Goldstein) ,   On a degenerate heat equation with a singular potential,      J. Functional Analysis, Vol. 186, No. 2, pp342-359, 2001

29.    A Liouville type theorem for some critical semilinear elliptic equations on noncompact manifolds,      Indiana U. Math. J., Vol. 50, No 4, Winter 2001, pp1915-1936

28.    Global bounds of Schr\"odinger Heat Kernels with negative potentials     J. Functional Analysis, vol. 182, no 2, pp344-370, 2001

27.    A critical behavior for some semilinear parabolic equations involving sign changing  solutions ,   Nonlinear Analysis, Vol. 50, pp967-980, 2002.

26.   Semilinear parabolic problems on manifolds and applications to
the  non-compact Yamabe problem  ,  Electron. J. Diff. Eqns, Vol 2000,no 46, pp1-30, 2000

25. A general blow up result on nonlinear boundary value problems on
exterior domains ,   Proceedings R. S. Edinburgh, Vol. 131, no. 2, pp251-275, 2001.

24 (with C. Bandle and H. A. Levine) , Critical exponents of Fujita type for in homogeneous  parabolic equations and systems, J. Math. Analysis and Appl., Vol. 251, No2, pp624-648, 2000

23.    The quantizing effect of potentials on the critical number of reaction diffusion equations,   J. Diff. Equations, Vol. 170, No.1, pp188-214, 2001.

22.   Global lower bound for the heat kernel of $-\Delta +\frac{c}{|x|^2}$, Proceedings AMS, Vol. 129, pp1105-1112, 2001

21. Large time behavior of Schr\"odinger heat kernels and application, Comm. Math. Physics,  Vol. 210, pp371-398, 2000

20. (with Z. Zhao) , Estimate of Global bounds  for some  Schr\"odinger heat kernels on manifolds, Illinois J. Math., Vol. 44, pp556-573, 2000

19.   An optimal  parabolic estimate and its applications in prescribing scalar curvature on open manifolds with Ricci >= 0, Math. Ann., Vol. 316, p703-731, 2000

18. (with H. A. Levine) , The critical Fujita number for a semilinear heat equation in exterior domains  with homogeneous Neumann boundary value, Proc. Royal Soc. Edin., Vol. 130 A, pp591-602, 2000

17. A new critical behavior for nonlinear wave equations, J. Computational Analysis and Applications, Vol.2 No 4, pp277-292, 2000

YEARS 1995-1999

16. A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem, J. Math. Analysis and Applications, Vol. 232 ,pp 413-427, 1999

15. (with Z. Zhao),  Existence results on prescribing zero scalar curvature,  Diff. and Int. Equations, Vol. 13, no. 4-6, pp779-790, 2000,

14.    Blow up results for nonlinear parabolic equations on manifolds, Duke Math. J., Vol. 97, pp515-540, 1999

13.   Blow up and global existence of solutions to an inhomogeneous  parabolic system, J. Diff. Equations, Vol. 147, pp155-183, July, 1998

12.   A new critical phenomenon for semilinear parabolic problems,
J. Math. Analysis and Applications, Vol. 129, 1998, pp125-139

11. Nonlinear parabolic problems on manifolds and non-existence result of the  non-compact Yamabe problem, Elect. Research Announcement AMS Vol 3. 1997, pp45-51.

10. (with Z. Zhao), Singular solutions of semilinear elliptic and parabolic equations , Math. Annalen,  Vol. 310, No. 4, pp 777-794, 1998

9. (with Z. Zhao), Global asymptotic behavior of solutions for a semilinear parabolic equation, Proceedings AMS, Vol. 126, pp 1491-1450, May 1998

8.   The critical exponent for a reaction diffusion equation on
some Lie groups,  Math. Zeit., Vol. 228, No 1, May 1998, pp 51-72

7.   Gaussian Bounds for the Fundamental Solutions of \nabla (A \nabla u)+ B \nabla u -u_{t}=0 Manuscripta Math., Vol. 93, 1997, pp381-391

6.   Global Existence and local continuity of solutions for  semilinear parabolic equations,  COMM. PDE, Vol. 22, 9 and 10, 1997, pp1529-1557

5. ,  On a parabolic equation with a singular lower order term, Part II,
Indiana University Math. Journal, Vol. 46, No 3, pp 989-1020, 1997.

4.   A Harnack inequality for the equation $\nabla( a \nabla u) + b \nabla u =0$ when $|b| \in K_{n+1}$,  Manuscripta Mathematica,  pp 61-78, 1996.

3    On a parabolic equation with a singular lower order term, Transactions AMS, Vol. 348, pp 2867-2899, 1996.

2      A Harnack inequality for Kolmogorov equations, J. Math. Analysis and Applications, vol. 190, pp 402-418, 1995

 
BEFORE 1994

1  Qi S. Zhang, On a complex equation arising in Hele-Shaw flow, Applied Mathematics Letters, vol.6, no. 5, pp 45-47, 1993.

 

Related publications and Conference Proceedings
 

3. WHEN DOES A SCHR\"ODINGER HEAT EQUATION PERMIT POSITIVE SOLUTIONS, Proceedings of International Conference on Boundary value problems, Beijing 2010, World Sci. Pub. Co. 2011. pdf file

2. An obstruction to prescribing positive scalar curvature on complete manifolds with Ricci >=0, p399-410 in Evolution Equations, Marcel Dekker 2003.

1. P. Souplet, A note on the paper by Qi S. Zhang: A priori estimates and the representation formula for all positive solutions to a semilinear parabolic problem, J. Math. Analysis and Applications, Vol. 243 ,pp 453-457, 2000