Recruiting Fields

Fundamental Mathematics, Applied Mathematics

Research Directions

Modern Harmonic Analysis and Its Applications in Geometric Measure Theory and Number Theory; Scattering Theory of Nonlinear Dispersive Equations; Mathematical Theory of Fluid Dynamics Equations

Overview of Research Directions

1. Modern Harmonic Analysis and Its Applications in Geometric Measure Theory and Number Theory

2. Scattering Theory of Nonlinear Dispersive Equations

3. Mathematical Theory of Fluid Dynamics Equations

Monographs

[1] Changxing Miao. Harmonic Analysis and Its Applications in Partial Differential Equations (2nd Edition), Modern Basic Mathematics Series No. 89, Science Press, March 2004.

[2] Changxing Miao, Bo Zhang. Harmonic Analysis Methods for Partial Differential Equations, Modern Basic Mathematics Series No. 117, Science Press, March 2008.

[3] Changxing Miao. Modern Methods for Nonlinear Wave Equations (2nd Edition), Modern Basic Mathematics Series No. 133, Science Press, April 2010.

[4] Changxing Miao, Jiahong Wu, Zhifei Zhang. Littlewood-Paley Theory and Its Applications in Fluid Dynamics Equations, Modern Basic Mathematics Series No. 142, Science Press, April 2012.

[5] Changxing Miao. Lectures on Modern Harmonic Analysis and Its Applications, Higher Education Press, 2018.

Representative Publications

[1]  C.Miao, Forward self-similar solutions of the fractional Navier-Stokes Equations, Advances in Mathematics, 352(2019),981-1043.(with B.Lai and X.Zheng)

[2]  C. Miao, Sobolev space adapted to the Schrodinger operator with inverse-square potential , Math. Z. (2018) 288:1273-1298 (with R.Killip, M.Visan, J.Zhang and J.Zheng).

[3]  C.Miao, The energy-critical nonlinear wave equation with an inverse-square potential Ann. Inst. Henri Poincare-Nonlinear Analysis, 37 (2020) 417-456 (with J.Murpth, and J.Zheng)

[4]  C.Miao, The defocusing energy-supercritical NLS in four space dimensions, J.Functional Analysis, 267(2014), 1662-1724 (with J.Murphy and J.Zheng)

[5]  C. Miao, Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity, J. Math. Pures Appl. 101 (2014) 842–872(with X. Zheng)

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

[7]  C. Miao, Global well-posedness and scattering for the defocusing H1/2-subcritical Hartree eq- uation in Rd, Ann. Inst. Henri Poincare-Nonlinear Analysis, 26(2009)1831–1852.(with G. Xu and L. Zhao)

[8]  C. Miao, On the uniqueness of weak solutions for the 3D Navier-Stokes equations, Ann. Inst. Henri Poincare-Nonlinear Analysis, 26(2009)2165–2180. (with Q. Chen and Z. Zhang )

[9]  C.Miao, Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces, J. Functional. Analysis. 271(2016)2752-2775 (with C.Sogge, Y.Xi and J.Yang)

[10]  C. Miao, The defocusing quintic NLS in four space dimensions, Ann. Inst. Henri Poincare-Nonlinear Analysis, 34 (2017) 759–787 (B.Dodson, J.Murpth, and J.Zheng)

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

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

[13] C. Miao, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. London Math. Society, 106 (2013) 650–674, (with M. Cannone and L. Xue),

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

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

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

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

[18]  C.Miao, The 2D incompressible Boussinesq equations with general critical dissipation， SIAM J.Math.Anal，46(2014) 3426 –3454.(with Q.Jiu, J.Wu and Z.Zhang)

[19]  C.Miao, Stability of the traveling waves for the derivative Schrödinger equation in the energy space, Calculus of Variations and PDEs , 56:45(2017), Doi: 10.1007/s00526-017-1128-6 (with X.Tang and G.Xu)

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

[21] C. Miao, 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) (with D. Li and L. Xue)

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

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

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

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

[26]  C. Miao, Dynamics for the focusing, energy-critical nonlinear Hartree equation, Forum Mathematicum. 27 ,No. 2(2015), 373–447 (with Y.Wu and G.Xu)

[27]  C.Miao, On Wolff’s L^2-Kakeya maximal inequality in $R^3$, Forum Mathematicum. 27, No5(2015) 3035-3047 (with J.Yang and J.Zeng)

[28]  C.Miao, Sacttering theory for the radial H^{\frac12}-critical wave equation with a cubic convolution, J. of Diff. Equat. 259 (2015) 7199–7237.(with J.Zhang and J.Zheng)

[29]  C. Miao , Energy Scattering for a Klein-Gordon Equation with a Cubic Convolution, J. of Diff. Equat.,257(2014)2178-2224, (with J.Zheng)

[30]  C. Miao, Global well-posedness for a modified critical dissipative quasi-geostrophic equation，J. of Diff. Equat., 252 (2012), 792 - 818, (with L. Xue)

[31]  C. Miao，Global well-posedness for the micropolar fluid system in critical Besov spaces，J. of Diff. Equat., 252(2012), 2698 - 2724, (with Q. Chen).

[32]  C. Miao, A note on the cone restriction conjecture，Proc. AMS, 140(2012), 2091 - 2102,(with J. Zhang and J. Zheng)

[33]  C. Miao, On global solution to the Klein–Gordon–Hartree equation below energy space, J. of Diff. Equat., 250 (2011), 3418 - 3447, (with J. Zhang)

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

[35]  C. Miao, On the regularity of a class of generalized quasi- geostrophic equations, J. of Diff. Equat., 251(2011), 2789 – 2821, (with L. Xue)

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

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

[38]  C. Miao, Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case, J. of Diff. Equat., 246 (2009), 3715 – 3749, (with G. Xu and L. Zhao)

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

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

[41]  C. Miao, On the low regularity of the fifth order Kadomtsev–Petviashvili I equation, J. of Diff. Equat., 245(2008), 3433 - 3469 (with W. Chen and J. Li)

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

[43]  C. Miao, Global solutions of the Klein-Gordon-Schrodinger system with rough data in R2+1, J. of Diff. Equat., 227(2006), 365 – 405, (with G. Xu)

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

[45]  C.Miao, Scattering theory for the defocusing fourth-order Schrodinger Equations, Nonlinearity 29 (2016) 692–736(with J.Zheng),

[46]  C. Miao, An improved maximal inequality for 2D fractional order Schrödinger operators, Studia Mathematica, 230 (2015), 121-165. (with J.Yang and J.Zheng).

[47]  C.Miao, On local smoothing problems and Stein's maximal spherical means, Proc. of AMS, 145 (2017) 4269–4282 ( with J.Yang and J.Zheng).

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

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

[50]  Solitary waves for nonlinear Schrödinger equation with Derivative, Contemp. Math., Vol. 20, No. 4 (2018) 1750049. (X.Tang and G.Xu)

[51]  C.Miao, Scattering in H1 for the intercritical NLS with an inverse-square potential, JDE, 264(2018), 3174–3211. (with J.Lu and J.Murphy).

[52]  C. Miao, The two-dimensional Euler equations in Yudovich type space and bmo-type space, Revista Matemtica Iberoamericana, 195–240. (with Q.Chen and X.Zheng)

[53]  C. Miao, Linear adjoint restriction estimates for paraboloid, Math. Z. 292(2019) 427-451.( with J.Zhang and J.Zheng)

[54]  C.Miao, On the regularity issues of a class of drift-diffusion equations with nonlocal diffusion. SIAM. J.Math.Anal. Vol. 51, 4(2019) 2927—2970. (with L. Xue)

[55]  C. Miao, Square function inequality for a class of Fourier integral operators satisfying cinematic curvature conditions, Forum Math, 2020, (with C. Gao and J. Yang)

[56]  C. Miao, Global well-posedness for the defocusing Hartree equation with radial data in R4 Communications in Contemporary Mathematics, Vol. 22, No. 2 (2020) 1950004 (35 pages) (with G.Xu and J.Yang)

[57]  C. Miao,The global well-posedness and scattering for the 5-dimensional defocusing congormal invariant NLW with radial initial data in a critical Besov space, Pacific Journal of Mathematics, 305(1),2020, 251-290.( J. Yang and T.Zhao)