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研究兴趣：分析和偏微分方程，流体力学，色散方程，完全可积系统 （MSC2020：35Q30, 35Q35, 76D03, 35Q55, 37K10）

学术活动：

• 大连理工微分方程讨论班（2026年春季学期每周四上午9点，数学科学学院114报告厅，欢迎所有人！）

  DUT Differential Equations Seminar: https://dutdynamics.github.io/ddes/

• 大连偏微分方程分析研讨会（2026.05.25 - 2026.05.29）

  Workshop: Analysis of PDEs in Dalian:  https://jiashunliu.github.io/Analysis_in_Dalian/

教学活动：

• 2026 春季学期

  课程：复变函数与积分变换

• 2025 夏季学期

  学生讨论班：狄拉克算子(Dirac Operator)

• 2024 冬季学期

  课程：流体力学引论(Introduction to Fluid Mechanics)

  学生讨论班：自由界面问题(Free Interface Problem)

• 2024 夏季学期

  课程：计算方法(Numerical Methods)

  学生讨论班：水波问题(Water Waves Problem)

• 2023 冬季学期

  课程：高等数学3(Höhe Mathematik III )

• 2023 夏季学期

  课程：流体力学引论

  学生讨论班：逆散射变换(Inverse Scattering Transform)

• 2022 冬季学期

  课程：泛函分析(Functional Analysis, Notes)

  学生讨论班：数学分析(Proseminar Analysis)

• 2022 夏季学期

  课程：色散方程(Dispersive Equations)

• 2021 冬季学期

  课程：傅里叶分析及其在偏微分方程中的应用(Fourier Analysis and its Applications to PDEs)

• 2021 夏季学期

  课程：边值与特征值问题(Rand- und Eigenwertprobleme)

  学生讨论班：完全可积系统(Completely Integrable System I: Mathematical Methods of Classical Mechanics)

• 2020 冬季学期

  课程：偏微分方程的经典方法(Klassische Methoden für partielle Differentialgleichungen)

  学生讨论班：材料和流体力学中的微观结构(Microstructure in Materials and Fluid Dynamics)

• 2020 夏季学期

  学生讨论班：希尔伯特空间方法(Hilbert Space Methods)

博士生指导：

• Robert Wegner (2022.8-2026.2) On the NLS Hierarchy with Nonzero Boundary Condition

• Rebekka Zimmermann (2022.11-2025.10) Mathematical Analysis of Regularity Propagation in variable-viscosity Fluid Models 

• Zihui He (何子慧 2018.10-2022.4) Incompressible inhomogeneous viscous Fluid Flows: Existence, Uniqueness and Regularity   https://katalog.bibliothek.kit.edu/bib/1304878

论文发表：

Part I. Navier-Stokes equations related:

• X. Liao and R. Zimmermann,  Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient. arXiv:2409.06517

• X. Liao and S.M. Zodji, Global-in-time well-posedness of the compressible Navier–Stokes equations with striated density. Rev. Mat. Iberoam.: 2025, 41(6), 2167–2214.

• Z. He and X. Liao, Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient. Commun. Contemp. Math.: 2024,  26(7), 2350039.

• X. Liao and C. Zillinger, On variable viscosity and enhanced dissipation. Nonlinearity: 2023, 36(11),  6071–6103. 
  

• Z. He and X. Liao, On the two-dimensional Boussinesq equations with temperature  - dependent thermal and viscosity diffusions in general Sobolev spaces. Z. Angew. Math. Phys.: 2022, 73(1), Paper No. 16, 25 pp. 

  Correction, Z. Angew. Math. Phys.: 2024, 75(4), Paper No. 129, 4 pp. 

• X. Liao and Y. Liu, On the global regularity of three dimensional density patch for inhomogeneous incompressible viscous flow. Sci. China Math.: 2019, 62(9), 1749–1764. 

• X. Liao and P. Zhang, Global regularities of 2-D density patches for viscous inhomogeneous incompressible flow with general density: high regularity case. Ana. Theory Appl.: 2019, 35(2), 163-191. 

• X. Liao and P. Zhang, Global regularity of 2D density patches for viscous inhomogeneous incompressible flow with general density: low regularity case. Comm. Pure Appl. Math.: 2019, 72(4), 835-884.  

• X. Liao and P. Zhang, On the global regularity of the two-dimensional density patch for inhomogeneous incompressible viscous flow. Arch. Ration. Mech. Anal.: 2016, 220(3), 937-981. 

• X. Liao, On the strong solutions of the nonhomogeneous incompressible Navier-Stokes equations in a thin domain. Differential Integral Equations: 2016, 29, 167-182.  

• E. Feireisl, X. Liao and J. Málek, Global weak solutions to a class of non-Newtonian compressible fluids. Math. Methods Appl. Sci.: 2015, 38(16), 3482-3494. 

• F. Fanelli and X. Liao, Analysis of an inviscid zero-Mach number system in endpoint Besov spaces with finite-energy initial data. J. Differential Equations: 2015, 259(10), 5074-5114.

• F. Fanelli and X. Liao, The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces. Asymptot. Anal.: 2015, 93(1-2), 115-140. 

• X. Liao, A global existence result for a zero Mach number system. J. Math. Fluid Mech.: 2014, 16(1), 77-103.  
  

• R. Danchin and X. Liao, On the wellposedness of the full low-Mach number limit system in general Besov spaces. Commun. Contemp. Math.: 2012, 14(3), 1250022, 47 pages. 

Part II. Nonlinear Schrödinger equation related:

• X. Liao and R. Wegner, Global well-posedness of the NLS hierarchy with nonzero boundary condition.  arXiv:2508.14572 

• X. Cheng, C.-Y. Guo, Z. Guo, X. Liao and J. Shen, Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials.  Anal. PDE: 2024, 17(10), 3371–3446.

• X. Liao and M. Plum, Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schrödinger equation. SIAM J. Math. Anal.: 2023, 55(6), 7399-7425. 

• H. Koch and X. Liao, Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case. Adv. Math.: 2023, 420, Paper No. 108996, 61 pp.

• H. Koch and X. Liao, Conserved energies for the one-dimensional Gross-Pitaevskii equation. Adv. Math.: 2021, 377, Paper No. 107467, 83 pp.

• [1] 2010.01 -- 2013.04

  巴黎东大学       数学       博士

• [2] 2008.09 -- 2011.06

  南京大学       应用数学       硕士

• [3] 2004.09 -- 2008.06

  南京大学       数学与应用数学       学士

• [1] 2018.11 -- 2025.07

  卡尔斯鲁厄理工学院      青年教授      Organizing lectures and seminars, leading research group "Analysis and PDEs", supervising undergraduate, graduate, and doctoral candidates,  and managing administrative tasks.

• [2] 2016.01 -- 2018.09

  波恩大学      博士后      Working with Prof. Herbert Koch on the one-dimensional Gross-Pitaevskii equation

• [3] 2014.01 -- 2015.12

  中国科学院      博士后      Working with Prof. Ping Zhang on the density-patch problems in fluid mechanics

• [4] 2013.05 -- 2013.12

  布拉格查理大学      博士后      Working with Prof. Eduard Feireisl and Prof. Josef Malek on the non-Newtonian fluid models