Department of Mathematics
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Institutes
Institute for Analysis
Workgroup Nonlinear Partial Differential Equations
JProf. Dr. Xian Liao

Publications

PDEs in fluid mechanics and Free interface problems
I am in particular interested in the motion of non homogeneous fluid flows, where the density/temperature functions may have big variations and the diffusion coefficients may depend on these density/temperature variables.

X. Liao and R. Zimmermann: Global-in-time well-posedness for the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient. Submitted, 2024.
https://arxiv.org/abs/2409.06517

X. Liao and M. Zodji: Global-in-time well-posedness of the compressible Navier-Stokes equations with striated density. Submitted, 2024.
http://arxiv.org/abs/2405.11900

Zihui He and X. Liao: Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient. Commun. Contemp. Math. 26 (2024), no. 7, Paper No. 2350039.
https://doi.org/10.1142/S0219199723500396
https://arxiv.org/abs/2005.13277

X. Liao and C. Zillinger: On variable viscosity and enhanced dissipation. Nonlinearity 36 (2023), no. 11, 6071–6103.
https://iopscience.iop.org/article/10.1088/1361-6544/acfec0/meta
https://arxiv.org/abs/2110.10976

Zihui 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. 73 (2022), no. 1, Paper No. 16, 25 pp.
https://doi.org/10.1007/s00033-021-01650-3
https://arxiv.org/abs/2107.04489
->!!Correction!! https://doi.org/10.1007/s00033-024-02229-4 Z. Angew. Math. Phys. 75 (2024), no.4, Paper No. 129.

X. Liao and Yanlin Liu: On the global regularity of three dimensional density patch for inhomogeneous incompressible viscous flow. Sci. China Math. 62 (2019), no. 9, 1749–1764.
https://doi.org/10.1007/s11425-017-9196-7
arXiv:1606.05395.

X. Liao and Ping 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.
http://doi.org/10.4208/ata.OA-0004
arxiv_pdf

X. Liao and Ping Zhang: Global regularities of 2-D density patches for viscous inhomogeneous incompressible flow with general density: low regularity case. Comm. Pure Appl. Math.: 2019, 72(4), 835-884.
https://doi.org/10.1002/cpa.21782

X. Liao and Ping Zhang: On the global regularity of 2D density patch for inhomogeneous incompressible viscous flow. Arch. Ration. Mech. Anal.: 2016, 220(3), 937-981.
https://doi.org/10.1007/s00205-015-0945-z

X. Liao: On the strong solutions of the nonhomogeneous incompressible Navier-Stokes equations in a thin domain. Differential Integral Equations: 2016, 29, 167-182.
https://projecteuclid.org/euclid.die/1448323258

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.
https://doi.org/10.1002/mma.3432

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.
https://doi.org/10.1016/j.jde.2015.06.038

F. Fanelli and X. Liao: The well-posedness issue for an inviscid zero-Mach number system in general Besov spaces. Asymptot. Anal.: 2015, 93, no.1-2, 115-140.
https://content.iospress.com/articles/asymptotic-analysis/asy1290

X. Liao: A global existence result for a zero Mach number system. J. Math. Fluid Mech.: 2014, 16(1), 77-103.
https://doi.org/10.1007/s00021-013-0152-3

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.
https://doi.org/10.1142/S0219199712500228


Dispersive equations and Complete integrable systems
I am in particular interested in the nonlinear Schrödinger equations.

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.
https://doi.org/10.48550/arXiv.2207.05186
https://epubs.siam.org/doi/full/10.1137/23M1550232

Xing Cheng, Chang-Yu Guo, Zihua Guo, Xian Liao, Jia Shen: Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials.
https://arxiv.org/abs/2105.02515

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.
https://doi.org/10.1016/j.aim.2023.108996
https://arxiv.org/abs/2204.06293

H. Koch and X. Liao: Conserved energies for the one-dimensional Gross-Pitaevskii equation. Adv. Math.: 2021, 377, Paper No. 107467, 83.
https://doi.org/10.1016/j.aim.2020.107467

Other PDEs
Li-Chang Hung and X. Liao: Nonlinear estimates for traveling wave solutions of reaction diffusion equations. Jpn. J. Ind. Appl. Math. 37 (2020), no. 3, 819–830.
https://link.springer.com/article/10.1007%2Fs13160-020-00420-4