苗长兴
博士 研究员 博士生导师
北京应用物理与计算数学研究所
调和分析(局部光滑性猜想、Boncher-Riesz猜想、限制性猜想、Kakeya猜想)、;Fourier 积分算子与Decoupling方法、;非线性色散方程的散射理论、;流体动力学方程的数学理论、;紧流形上的PDEs与数论方法;Fourier分析与几何测度论
个性化签名
- 姓名:苗长兴
- 目前身份:
- 担任导师情况:博士生导师
- 学位:博士
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学术头衔:
国家杰出青年科学基金获得者, 博士生导师
- 职称:高级-研究员
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学科领域:
数学
- 研究兴趣:调和分析(局部光滑性猜想、Boncher-Riesz猜想、限制性猜想、Kakeya猜想)、;Fourier 积分算子与Decoupling方法、;非线性色散方程的散射理论、;流体动力学方程的数学理论、;紧流形上的PDEs与数论方法;Fourier分析与几何测度论
苗长兴, 北京应用物理与计算数学研究所研究员. 曾荣获国家杰出青年基金、于敏数理科学奖与中国工程物理研究院杰出专家,是我国自己培养的在国际偏微分方程领域有影响的数学家。主要贡献集中表现在调和分析、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》等五部专著, 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.与此同时,在他培养的一批年轻有为的才俊中,已有多位学生脱颖而出,在调和分析的前沿领域里取得了出色的研究成果,引起国际同行的广泛关注和重视。
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4972
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成果阅读
1551
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成果数
31
【期刊论文】The energy-critical nonlinear wave equation with an inverse-square potential
Changxing Miao, Jason .Murph, Jiqiang Zheng
Ann. Inst. Henri Poincare-Nonlinear Analysis,2020,37(2):417-456
2020年03月01日
We study the energy-critical nonlinear wave equation in the presence of an inverse-square potential in dimensions three and four. In the defocussingcase, we prove that arbitrary initial data in the energy space lead to global solutions that scatter. In the focusing case, we prove scattering below the ground state threshold.
Nonlinear wave equation, Inverse-square potential, Energy-critical, Scattering, Ground state threshold
调和分析、偏微分方程、自伴算子的谱理论
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66浏览
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【期刊论文】Linear adjoint restriction estimates for paraboloid
Changxing Miao, Junyong Zhang, Jiqiang Zheng
Mathematische Zeitschrift ,2019,292(2):427-451
2019年02月10日
We prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices. This result generalizes the paraboloid restriction estimate in radial case from [Shao, Rev. Mat. Iberoam. 25(2009), 1127-1168], as well as the result from [Miao et al. Proc. AMS 140(2012), 2091-2102]. As an application, we show a local smoothing estimate for a solution of the linear Schr\"odinger equation under the assumption that the initial datum has additional angular regularity.
Linear adjoint restriction estimate,, local restriction estimate,, , Bessel function,, spherical harmonics,, local smoothing.,
调和分析、自伴算子的谱理论、函数空间理论
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94浏览
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【期刊论文】The two-dimensional Euler equation in Yudovich and bmo-type spaces
Qionglei Chen, Changxing Miao, Xiaoxin Zheng
Rev. Mat. Iberoam. ,2019,35(1):195–240
2019年01月01日
We construct global-in-time, unique solutions of the two-dimensional Euler equations in a Yudovich type space and a $\rm bmo$-type space. First, we show the regularity of solutions for the two-dimensional Euler equations in the Spanne space involving an unbounded and non-decaying vorticity. Next, we establish an estimate with a logarithmic loss of regularity for the transport equation in a bmo-type space by developing classical analysis tool such as the John-Nirenberg inequality. We also optimize estimates of solutions to the vorticity-stream formulation of the two-dimensional Euler equations with a bi-Lipschitz vector field in bmo-type space by combining an observation introduced in \cite{Y1} by Yodovich with the so-called ``quasi-conformal property" of the incompressible flow.
Two-dimensional incompressible Euler equations,, Yudovich type data, John– Nirenberg inequality,, global existence and uniqueness of solutions.,
偏微分方程、调和分析、流体力学
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56浏览
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【期刊论文】On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion
Changxing Miao, Liutang Xue
SIAM J. Math. Anal.,2019,51(4): 2927-2970
2019年02月15日
In this paper we address the regularity issues of drift-diffusion equation with nonlocal diffusion, where the diffusion operator is in the realm of stable-type L\'evy operator and the velocity field is defined from the considered quantity by a zero-order pseudo-differential operator. Through using the method of nonlocal maximum principle in a unified way, we prove the eventual regularity result in the supercritical type cases and the global regularity at some logarithmically supercritical cases. The feature of these results is that the time after which the solution is smoothly regular in the supercritical type cases can be evaluated explicitly.
Drift-diffusion equation,, surface quasi-geostrophic equation,, nonlocal maximum principle, Levy operator,, regularity
偏微分方程、调和分析、流体力学
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76浏览
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【期刊论文】Forward self-similar solutions of the fractional Navier-Stokes equations
Baishun Lai, Changxing Miao, Xiaoxin Zheng
Advances in Mathematics,2019,352(2):981–1043
2019年06月28日
We study forward self-similar solutions to the 3-D Navier-Stokes equations with the fractional diffusion (−Δ)^α. First, we construct a global-time forward self-similar solutions to the fractional Navier-Stokes equations with 5/6 <α ≤1for arbitrarily large self-similar initial data by making use of the so called blow-up argument. Moreover, we prove that this so-lution is smooth in R^3×(0, +∞). In particular, when α =1, we prove that the solution constructed by Korobkov and Tsai (2016) [16]satisfies the decay estimate by establishing regu-larity of solution for the corresponding elliptic system, which implies this solution has the same properties as a solution which was constructed in Jia and Šverák (2014) [13].
Self-similar solution,, Nonlocal smoothing effect,, Blowup argument,, The weighted estimate
偏微分方程、调和分析、流体力学
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89浏览
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【期刊论文】Sobolev space adapted to the Schrodinger operator with inverse-square potential
苗长兴:R.Killip,C. Miao, M.Visan,J.Zhang.,J.Zheng
Mathematische Zeitschrift,2018,288(4):1273–1298
2018年01月30日
We study the $L^p$-theory for the Schr\"odinger operator with critical rough potential of $a|x|^{-2}$ type. The developed harmonic analysis tools, such as multiplier estimate, Littlewood-Paley theory and the equivalence of Sobolev norms, will be employed to study the scattering theory of energy-critical defocusing nonlinear Schr\"odinger equation with the inverse-square potential. These tools in $L^p$-theory are new and the range of $p$ depends on the parameter $a$, which is different from the classical theory. The main difficulty is raised from the failure of Gaussian boundedness of the heat kernel associated with the operator $P_a=-\Delta+a|x|^{-2}$ when $a$ is negative.
Riesz transforms · Inverse-square potential · Littlewood–Paley theory · Mikhlin multiplier theorem ·
调和分析、自伴算子的谱理论、函数空间理论
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70浏览
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【期刊论文】On the 4D nonlinear Schrödinger equation with combined terms under the energy threshold
Changxing Miao , Tengfei Zhao, Jiqiang Zheng
Calculus of Variations and PDEs,2017,56-179(1):1-39
2017年10月25日
In this paper, we consider the longtime dynamics of the solutions to focusing energy-critical Schrödinger equation with a defocusing energy-subcritical perturbation term under a ground state energy threshold in four spatial dimension. This extends the results in Miao et al. (Commun Math Phys 318(3):767–808, 2013, The dynamics of the NLS with the combined terms in five and higher dimensions. Some topics in harmonic analysis and applications, advanced lectures in mathematics, ALM34, Higher Education Press, Beijing, pp 265–298, 2015) to four dimension without radial assumption and the proof of scattering is based on the interactionMorawetz estimates developed in Dodson (Global well-posedness and scattering for the focusing, energy-critical nonlinear Schrödinger problem in dimension d = 4 for initial data below a ground state threshold, arXiv:1409.1950), the main ingredients of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions
Nonlinear Schrödinger equation · Longtime dynamics · Interaction Morawetz estimates · Scattering ·
偏微分方程、调和分析
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【期刊论文】Stability of the traveling waves for the derivative Schrödinger equation in the energy space
Changxing Miao, Xingdong Tang, Guixiang Xu
Calculus of Variations and PDEs,2017,56-54(1):1-48
2017年08月30日
In this paper, we continue the study of the dynamics of the traveling waves for nonlinear Schrödinger equation with derivative (DNLS) in the energy space. Under some technical assumptions on the speed of each traveling wave, the stability of the sum of two traveling waves for DNLS is obtained in the energy space by Martel–Merle–Tsai’s analytic approach in Martel et al. (Commun Math Phys 231(2):347–373, 2002, Duke Math J 133(3):405–466, 2006). As a by-product, we also give an alternative proof of the stability of the single travelingwave in the energy space in Colin andOhta (Ann InstHenri Poincaré Anal Non Linéaire 23(5):753–764, 2006), where Colin and Ohta made use of the concentration compactness argument.
traveling waves ,, Schrödinger equation with derivative,, concentration compactness argument.,
偏微分方程、调和分析
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33浏览
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【期刊论文】The defocusing quintic NLS in four space dimensions,
苗长兴: B.Dodson, C.Miao, J. Murphy, J.Zheng
Ann. Inst. Henri Poincare- Nonlinear Analysis,,2017,34(2):759–787
2017年03月15日
We consider the defocusing quintic nonlinear Schrödinger equation in four space dimensions. We prove that any solution that remains bounded in the critical Sobolev space must be global and scatter. We employ a space-localized interaction Morawetz inequality, the proof of which requires us to overcome the logarithmic failure in the double Duhamel argument in four dimensions.
Nonlinear Schrödinger equation, Concentration compactness, Scattering, Interaction Morawetz
调和分析、偏微分方程
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44浏览
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【期刊论文】Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces
C. Miao, C. Sogge, Y. Xi, J. Yang
Journal of Functional Analysis,2016,271(2):2752–2775
2016年03月15日
We obtain an improvement of the bilinear estimates of Burq, Gérard and Tzvetkov[6]in the spirit of the refined Kakeya–Nikodym estimates [2]of Blair and the second author. We do this by using microlocal techniques and a bilinear version of Hörmander’s oscillatory integral theorem in [7].
Eigenfunctions Bilinear estimates Kakeya–Nikodymaverages
调和分析、微分流形
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41浏览
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