邓稼先创新研究中心

《现代调和分析及其应用》研究团队介绍

 

一、   主要研究方向(包括研究内容)

 

 (I). 现代调和分析及其在几何测度论、 数论中的应用;

 

调和分析领域的核心问题主要涉及:Kakeya猜想、Bochner-Riesz猜想、限制性猜想、局部光滑性猜想等。在研究上述四大猜想派生的各种方法已经成为解决调和分析、解析数论、偏微分方程、几何测度论、关联几何、数学物理等数学领域公开问题的重要工具。菲尔茨奖得主Bourgain等将许多看似不相关的研究领域如:调和分析、堆垒数论、关联几何学、几何测度论、拓扑学与代数几何、算术组合学等有机地联系起来。当然,研究与解决调和分析的四大猜想最引人入胜的是来自分析、数论和几何的多种技术间精细的相互作用。迄今为止所证明四大猜想的最好进展涉及到波包分解、尺度归纳、对垒组合乃至于高深的代数几何知识,这在客观上就需要更多年轻调和分析与PDE专家去探索诸如算术组合、堆垒数论、关联几何、几何测度论等研究领域。拟研究的问题涉及:

1.   非退化曲面(具非零Gauss曲率的超曲面)上的分离性定理与分离性猜想;

2.   离散性限制性估计与指数求和平均等价性;

3.   流形上的非线性色散方程;

4.   四大猜想的几何形式与组合数学版本的等价性;

5.   波包分解与正则性估计的几何实现;

6.   具非平凡位势的自伴算子对应的调和分析等.

 

(II). 流体动力学方程的数学理论

众所周知,流体动力学方程:如三维不可压缩Navier-Stokes方程光滑解的整体存在性是数学物理界最关注的公开问题, 至今尚未发现解决此该问题的有效方法, 著名数学家、Abel奖得主Nirenberg认为解决这些问题应该更多地使用调和分析方法。我们拟通过微局部分析、Littlewood-Paley理论等现代调和分析方法,研究如下问题:

1.   具有特殊结构的不可压Navier-Stokes方程与Euler方程光滑解的整体存在性(自相似结构、轴对称流体等)。

2.   可压缩Navier-Stokes方程高振荡函数的适定性与不适定性问题;

3.   Maxwell-Navier-Stokes 方程(流体在电磁场作用下的运动,耦合特点源于 NS方程中Lorentz力与Maxwell方程中的电流)及相关流体动力学方程的适定性问题。

4.   Navier-Stokes方程耦合上热流方程 (关于棒状液晶的指向方程)的适定性理论的研究。

 

(III). 非线性色散方程的散射理论.

众所周知,Morawetz估计在非线性色散()方程的散射理论中起着核心作用。 然而该估计无法处理临界问题,菲尔茨奖得主Bourgain开创了研究临界问题的方法(极小能量归纳)。Tao等发展了相互作用的 Morawetz 估计及其在频率空间的局部化技术, 彻底解决非聚焦临界临界Schrodinger方程的散射猜想。Kenig-MerleBourgain极小能量归纳的启示下, 发展了“广义变分方法”,(Profile分解与集中紧原理与刚性方法),解决了聚焦型非线性临界色散()方程的散射理论。然而,上述方法不适用于具有位势的非线性色散()方程。对于聚焦情形,当初始能量等于基态能量时,是否具有相应的刚性定理、孤立子猜想是否成立等问题也是公开的。

1.   解决了聚焦型非线性临界色散()方程的散射理论。

2.   具有位势项非线性色散方程的散射理论。

3.   非线性色散方程的“临界模猜想

4.   光滑紧流形上的非线性色散方程解的适定性与blow-up机制。

 

 

 

二、   团队主要成员

团队首席:苗长兴研究员

苗长兴研究员,曾荣获国家杰出青年基金、于敏数理科学奖、中国工程物理研究院杰出专家、中国工程物理研究院科技创新一等奖。近年来在国际一流的学术刊物(如:CPAMCMPARMAMZJFAJMPASIAMAIHPCPDEPLMS)上发表论文九十余篇, 主要贡献表现在调和分析、非线性色散方程的散射理论与流体动力学方程的数学理论等研究领域,解决了若干个具有国际影响的数学问题,得到了著名数学家KenigConstantin等国际同行的高度评价。先后出版了《调和分析及其在偏微分方程中的应用》、《偏微分方程的调和分析方法》、《非线性波动方程的现代方法》、《Littlewood-Paley理论及其在流体动力学方程中的应用》等四部专著。 对国内这一核心数学领域的研究与发展起到了基础性的作用。所领导的科研团队被国际数学联盟前主席Kenig称为“国际偏微分方程研究领域最具活力与影响力的团队之一”。

团队主要成员

1. 陈琼蕾,研究员、博士生导师,中国工程物理研究院科技创新一等奖。主要从事现代调和分析与流体动力学方程的数学理论等研究。 

2. 徐桂香,副研究员、硕士生导师, 中国工程物理研究院科技创新一等奖。主要从事现代调和分析、非线性色散方程的散射理论等研究。

3.   郑继强,主要从事现代调和分析、非线性色散方程的散射理论等研究。

 

 

三、  代表性论著

(I)                 专著:

1. 苗长兴.《调和分析及其在偏微分方程中的应用》 (第二版), 现代基础数学丛书No.89,科学出版社.2004.3.

2. 苗长兴、张波著.《偏微分方程的调和分析方法》, 现代基础数学丛书No.117,科学出版社.2008.3

3. 苗长兴著.《非线性波动方程的现代方法》,(第二版), 现代基础数学丛书No.133科学出版社.2010.4.

4. 苗长兴、吴家宏、章志飞著.Littlewood-Paley理论及其在流体动力学方程中的应用》, 现代基础数学丛书No.142科学出版社.2012.4.

  5. 苗长兴编著.《现代调和分析及其应用讲义》,高等教育出版社,2018.

 

学术论文:

1.       Q. Chen, C. Miao and Z. Zhang, Global well-posedness for the compressible Navier-Stokes equations with the highly oscillating initial velocity, Comm. Pure Appl. Math., Vol. LXIII (2010), 11731224. 

2.       Q. Chen, C. Miao and Z. Zhang, A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation, Comm. Math. Phys, 271(2007), 821-838.

3.       Q. Chen, C. Miao and Z. Zhang, Well-posedness in critical spaces for the compressible Navier-Stokes equations with density dependent viscosities, Revista Matematica Iberoam., 26(2010), 915-946.

4.       C. Miao, G. Xu and L. Zhao, The dynamics of the 3D radial NLS with the combined terms, Comm. Math. Phys., 318:3(2013), 767-808.

5.       C. Miao and X. Zheng, On the global well-posedness for the Boussinesq system with horizontal dissipation, Commun. Math. Phys. 321, 3367 (2013).

6.       Q. Chen, C. Miao and Z. Zhang, On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodyn- amics equations, Comm. Math. Phys. 284(2008), 919-930,

7.       Q. Chen, C. Miao and Z. Zhang, The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations, Comm. Math. Phys, 275(2007), 861-872.

8.       H. Jia, B. Liu and G. Xu, Long time dynamics of defocusing energy critical 3+1 dimensional wave equation with potential in the radial case, Comm. Math. Phys., 339(2015), 353-384.

9.       Q. Chen, C. Miao and Z. Zhang, On the well-posedness of the ideal MHD equations in the Triebel- Lizorkin spaces, Arch. Rational Mech. Anal. 195(2010), 561-578.

10.   R. Killip, C. Miao, M. Visan, J. Zhang and J. Zheng, Sobolev space adapted to the Schrodinger operator with inverse-square potential , Math. Z. (2017), doi: 10.1007/s00209-017-1934-8.

11.   H. Jia, B. Liu, W. Schlag and G. Xu Generic and non-generic behavior of solutions to defocusing energy critical wave equation with potential in the radial case, IMRN, 2016, 1-59.

12.   C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree eq- uation for radial data, J. Funct. Anal., 253(2007), 605-627.

13.   C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl., 91(2009) 49-79.

14.   C. Miao, J. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions, J. Funct. Anal., 267(2014), 1662-1724.

15.   C. Miao and X. Zheng, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. 101 (2014) 842872.

16.   Q. Chen, C. Miao and Z. Zhang, On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces, Revista Matemtica Iberoamericana, 31(2015)1375-1402.

17.   C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing H1/2-subcritical Hartree eq- uation in Rd, Ann. Inst. Henri Poincare-Nonlinear Analysis, 26(2009)18311852.

18.   Q. Chen, C. Miao and Z. Zhang, On the uniqueness of weak solutions for the 3D Navier-Stokes equations, Ann. Inst. Henri Poincare-Nonlinear Analysis, 26(2009)21652180.

19.   C. Miao, C. Sogge, Y. Xi and J. Yang, Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces, J. Func. Anal, 271(2016), 2752-2775.

20.   B. Dodson, C. Miao, J. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions, Ann. Inst. Henri Poincare-Nonlinear Analysis, 34 (2017), 759787.

21.   C. Miao, J. Zhang and J. Zheng, The defocusing energy-critical wave equation with a cubic convolution, Indiana University Mathematics Journal, 63(2014), 993-1015.

22.   C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in R1+n, Comm. PDEs, 36(2011)729 - 776.

23.   M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. London Math. Society, 106 (2013), 650674.

24.   W. Chen, C. Miao and X. Yao, Dispersive estimates with geometry of finite type, Comm. PDEs, 37(2012), 729 - 776.

25.   Q. Chen, C. Miao and Z. Zhang, Well-posedness for viscous shallow water equations in critical spaces, SIAM. J. Math. Anal., 40(2008), 443 - 474.

26.   M. Cannone, Q. Chen and C. Miao, A losing estimate for the ideal MHD equations with application to Blow-up criterion, SIAM J. Math. Anal., 38 (2007), 1847-859.

27.   G. Karch, C. Miao and X. Xu, About convergence of solutions of fractal Burgers equation toward rarefaction waves. SIAM. J. Math. Anal., 39(2008), 1536 - 1549.

28.   Q. Jiu, C. Miao, J. Wu and Z. Zhang, The 2D incompressible Boussinesq equations with general critical dissipation, SIAM J.Math.Anal46(2014) 3426 3454.

29.   Y. Li, Y. Wu and G. Xu, Low regularity global solutions for the focusing, mass-critical NLS equation in R. SIAM J. Math. Anal., 43:1 (2011), 322-340.

30.   C. Miao, X. Tand and G. Xu, Stability of the traveling waves for the derivative Schrodinger equation in the energy space, Calculus of Variations and PDEs, 56(2)(2017),  Paper No. 45, 48pp.

31.   C. Miao, T. Zhao and J. Zheng, On the 4D Nonlinear Schr\"odinger equation with combined terms under the energy threshold, Calculus of Variations and PDEs (2017) 56:179.

32.   C. Miao, J. Zhang and J. Zheng, Strichartz estimates for wave equation with inverse square potential, Contemp. Math., Vol. 15, No. 6 (2013) 1350026 (29 pages).

33.   D. Li, C. Miao and L. Xue, On the well-posedness of a 2D nonlinear and nonlocal system arising from the dislocation, Contemp. Math., Vol. 16, No. 2 (2014) 1350021 (36 pages).

34.   C. Miao and J. Zheng, Scattering theory for NLS below energy space in  dimension two, Comm. Contemp. Math., Vol. 17, No. 6 (2015) 1450052 (37 pages).

35.   D. Li, C. Miao and X. Zhang, On the isentropic compressible Euler equations with adiabatic index $\gamma=1$, Pacific J. Math., 262:1(2013), 109-128.

36.   C. Miao, J. Zhang and J. Zheng, Maximal estimates for Schrodinger equation with inverse-square potential Pacific Journal of mathematics,273(2015)1-20.

37.   Q. Chen, C. Miao and Z. Zhang, Global well-posedness for the 3D rotating Navier-Stokes equations with highly oscillating initial data, Pacific J. Math., 262:2(2013), 263-283.

38.   C. Miao, Y. Wu and G. Xu, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Mathematicum. 27, No. 2(2015), 373447.

39.   C. Miao, J. Yang and J. Zheng, On Wolff’s L2-Kakeya maximal inequality in R3, Forum Math. 27, No5(2015), 3035-3047.

40.   C. Miao, J. Zhang and J. Zheng, Sacttering theory for the radial H1/2-critical wave equation with a cubic convolution, J. Diff. Equat. 259 (2015), 71997237.

41.   C. Miao and J. Zheng, Energy scattering for a Klein-Gordon Equation with a cubic convolution, J. Diff. Equat.,257(2014), 2178-2224.

42.   C. Miao and L. Xue, Global well-posedness for a modified critical dissipative quasi-geostrophic equationJ. Diff. Equat., 252 (2012), 792 - 818.

43.   Q. Chen and C. MiaoGlobal well-posedness for the micropolar fluid system in critical Besov spacesJ. Diff. Equat., 252(2012), 2698 - 2724.

44.   C. Miao, J. Zhang and J. Zheng, A note on the cone restriction conjectureProc. AMS, 140(2012), 2091 - 2102.

45.   C. Miao and J. Zhang, On global solution to the KleinGordonHartree equation below energy space, J. Diff. Equat., 250 (2011), 3418 - 3447. 

46.   C. Miao, Y. Wu and G. Xu, Global well-posedness for Schrodinger equation with derivative in H1/2 (R), J. Diff. Equat., 251 (2011), 2164 - 2195.

47.   C. Miao and L. Xue, On the regularity of a class of generalized quasi- geostrophic equations, J. Diff. Equat., 251(2011), 2789-2821.

48.   C. Miao, G. Xu and L. Zhao, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\geq 9$, J. Diff. Equat., 251 (2011), 3381-3402.

49.   D. Li, C. Miao and X. Zhang, The focusing energy-critical Hartree equation, J.  Diff. Equat., 246 (2009), 1139-1163.

50.   C. Miao, G. Xu and L. Zhao, Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case, J.  Diff. Equat., 246 (2009), 37153749.

51.   C. Miao and G. Wu, Global well-posedness of the critical Burgers equation in critical Besov spaces, J. Diff. Equat., 247(2009), 1673 --1693.

52.   Y. Li, Y. Wu and G. Xu, Global well-posedness for periodic mass-critical nonlinear Schrodinger equation, J. Diff. Equat., 250:6, 15(2011), 2715-2736.

53.   X. Tang and G. Xu, Stability of the sum of two solitary waves for (gDNLS) in the energy space, Accepted by J. Diff. Equat.

54.   W. Chen, J. Li, C. Miao and J. Wu, Low regularity solutions of two fifth-order KdV type equations, Journal D’Analyse Mathematique, 107 (2009), 221-238.

55.   W. Chen, C. Miao and J. Li, On the low regularity of the fifth order KadomtsevPetviashvili I equation, J. Diff. Equat., 245(2008), 3433 - 3469.

56.   Q. Chen and C. Miao, Existence theorem and blow-up criterion of smooth solutions to the two-fluid MHD equations in R3, J. Diff. Equat., 239 (2007), 251 - 271.

57.   C. Miao and G. Xu, Global solutions of the Klein-Gordon-Schrodinger system with rough data in R2+1, J. Diff. Equat., 227(2006), 365405.

58.   W. Chen, Y. Han and C. Miao, Factorization theorem for product Hardy spaces.  Studia Mathematica, 177(2006), 255-249.

59.   C. Miao and J. Zheng, Scattering theory for the defocusing fourth-order Schrodinger Equations, Nonlinearity 29 (2016), 692736.

60.   C. Miao, J. Yang and J. Zheng, An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators, Studia Mathematica, 230 (2015), 121-165.

61.   C. Miao, J. Yang and J. Zheng, On local smoothing problems and Stein's maximal spherical means, Proc. of AMS, 145 (2017), 42694282).

62.   X. Cheng, C. Miao and L. Zhao, Global well-posedness and scattering for nonlinear Schrodinger equations with combined nonlinearities, J. Diff. Equat. 261(2016), 2881-2934 .

63.   C. Miao and Y. Wang, Regularity conditions for the suitable weak solutions of the Navier Stokes system from its rotation form, Pacific J. Math. 288(2017), 189-216.

64.   Q. Chen, C. Miao and Z. Zhang, The two-dimensional Euler equations in Yudovich type space and bmo-type space, To appear in Revista Matemtica Iberoamericana, (2017).

65.   C. Miao, X. Tang and G. Xu, Solitary waves for nonlinear Schrodinger equation with Derivative, Comm. Contemp. Math., (2017), doi: 10.1142 /S0219199717500493, 27pp.

66.   J. Lu, C. Miao and J. Murphy, Scattering in $H^1$ for the intercritical NLS with an inverse-square potential, J. Diff. Equat., doi:10.1016/j.jde.2017.11.015(2018).

67.   D. Li, G. Xu and X. Zhang, On the dispersive estimate for the Dirichlet Schrodinger propagator and applications to energy critical NLS, Canad. J. Math., 66(2014), 1110-1142.

68.   Q.Chen,and Z.Zhang,Regularity criterion via two components of vorticity on weak solutions to the Navier–Stokes equations in R^3,  J.of Differ.Equat.216 (2005) 470-481.

69.   Q.Chen, Z.Zhang, Regularity Criterion via the Pressure on Weak Solutions to the 3D Navier-Stokes Equations, Proc. of  Amer. Math. Soc. , 135(2007):1829-1837.