Webrelaunch 2020

Junior Research Group "Analysis of PDEs"

Welcome

There are plenty of interesting PDE models and the analysis methods adapted to them vary a lot. The group focuses on the study of the mathematical theories of various prototypical PDE models (such as Navier-Stokes/Euler equations, nonlinear Schrödinger equations), by use of the powerful analysis toolboxes (such as harmonic analysis, Fourier analysis, functional analysis).

In particular we are interested in those PDE models involving variable physical coefficients (which bring strong nonlinearities), discontinuous data (which bring singularities), nonzero boundary conditions (which bring more functional structures than the spatially-homogeneous case), singular limits (which deal with considerably different scales of parameters), etc., which should find their sources and applications in natural sciences.

(See also CRC 1173 Wave Phenomena Project A12: Dynamics of the Gross–Pitaevskii equation.)

Staff in the Junior Research Group Analysis of PDEs
Name Tel. E-Mail
+49 721 608 42616 xian.liao@kit.edu
+49 721 608 47672 robert.wegner@kit.edu
+49 721 608 46191 rebekka.zimmermann@kit.edu

Current Offering of Courses
Semester Titel Typ
Winter Semester 2023/24 Lecture
Lecture
Seminar
Summer Semester 2023 Lecture
Seminar
Seminar
Winter Semester 2022/23 Lecture
Seminar
Proseminar
Summer Semester 2022 Lecture
Seminar
Winter Semester 2021/22 Lecture
Lecture
Seminar
Seminar
Summer Semester 2021 Lecture
Seminar
Seminar
Winter Semester 2020/21 Lecture
Seminar
Seminar
Summer Semester 2020 Seminar
Seminar
Summer Semester 2019 Lecture
Winter Semester 2018/19 Lecture

Publications

(since 2019)

Xing Cheng, Chang-Yu Guo, Zihua Guo, X. Liao and Jia Shen: Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials. 2021.
https://arxiv.org/abs/2105.02515.

P. Gérard, S. Grellier and Zihui He: Turbulent cascades for a family of damped Szegö equations. Nonlinearity, 35 (9), 2022.
https://iopscience.iop.org/article/10.1088/1361-6544/ac7e13
https://arxiv.org/abs/2111.05247

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.: 2022, 73, no. 1, Paper No. 16, 25 pp.
https://doi.org/10.1007/s00033-021-01650-3
https://arxiv.org/abs/2107.04489

Zihui He and X. Liao: Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient. Commun. Contemp. Math.: 2023.
https://www.worldscientific.com/doi/abs/10.1142/S0219199723500396
https://arxiv.org/abs/2005.13277

H. Koch and X. Liao: Conserved energies for the one-dimensional Gross-Pitaevskii equation. Adv. Math.: 2021, 377, Paper No. 107467, 83.
https://www.sciencedirect.com/science/article/abs/pii/S0001870820304953

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

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

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://link.springer.com/article/10.1007%2Fs11425-017-9196-7
arXiv:1606.05395.

X. Liao and M. Plum: Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schrödinger equation. SIAM J. Math. Ana., 55 (6), 2023.
https://doi.org/10.48550/arXiv.2207.05186
https://epubs.siam.org/doi/full/10.1137/23M1550232

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://global-sci.org/intro/article_detail/ata/13112.html
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://onlinelibrary.wiley.com/doi/abs/10.1002/cpa.21782

X. Liao and C. Zillinger: On variable viscosity and enhanced dissipation. Nonlinearity, 2023.
https://iopscience.iop.org/article/10.1088/1361-6544/acfec0/meta
https://arxiv.org/abs/2110.10976

R. Wegner: Global-in-time well-posedness of the one-dimensional hydrodynamic Gross–Pitaevskii equations without vacuum. Z. Angew. Math. Phys. 74, 194 (2023).
https://link.springer.com/article/10.1007/s00033-023-02089-4

Ruoci Sun: Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds. Comm. Math. Phys.: 2021, 383 (2), 1051–1092.
https://doi.org/10.1007/s00220-021-03996-1