Webrelaunch 2020

Nachwuchsgruppe Analysis of PDEs

Willkommen

Es gibt viele interessante PDE-Modelle und die an sie angepassten Analysemethoden variieren stark. Meine Nachwuchsgruppe focussiert sich auf das Studium der mathematischen Theorien verschiedener prototypischer PDE-Modelle wie zum Beispiel Navier-Stokes/Euler-Gleichungen und nichtlineare Schrödinger-Gleichungen. Dabei finden leistungsfähige Analysemethoden wie die harmonische Analysis , Fourier-Analysis und Funktionalanalysis Verwendung.

Insbesondere interessieren wir uns für PDE-Modelle mit variablen physikalischen Koeffizienten (für starke Nichtlinearitäten), mit diskontinuierlichen Daten (für Singularitäten), für Modelle mit Randbedingungen ungleich Null (stärkere funktionale Strukturen als im räumlich-homogenen Fall) sowie mit singulären Grenzen, die sich mit erheblich unterschiedlichen Parameterskalen befassen.

Das sind nur einige Beispiele. Sie alle finden ihre Quellen und Anwendungen in den Naturwissenschaften.

(Siehe auch SFB 1173 Wave Phenomena Project A12: Dynamics of the Gross–Pitaevskii equation and related dispersive equations.)

Personen der Nachwuchsgruppe Analysis of PDEs
Name Tel. E-Mail
0721 608 42616 xian.liao@kit.edu
0721 608 47672 robert.wegner@kit.edu
0721 608 46191 rebekka.zimmermann@kit.edu

Current Offering of Courses
Semester Titel Typ
Sommersemester 2024 Lecture
Seminar
Wintersemester 2023/24 Lecture
Lecture
Seminar
Sommersemester 2023 Lecture
Seminar
Seminar
Wintersemester 2022/23 Lecture
Seminar
Proseminar
Sommersemester 2022 Lecture
Seminar
Wintersemester 2021/22 Lecture
Seminar
Seminar
Sommersemester 2021 Lecture
Seminar
Seminar
Wintersemester 2020/21 Lecture
Seminar
Sommersemester 2020 Seminar
Seminar
Sommersemester 2019 Lecture
Wintersemester 2018/19 Lecture

Publikationen

(seit 2019)

Xing Cheng, Chang-Yu Guo, Zihua Guo, Xian Liao, Jia Shen: Scattering of the three-dimensional cubic nonlinear Schrödinger equation with partial harmonic potentials. Anal. PDE: 2024, 17 (10), 3371–3446.
https://msp.org/apde/2024/17-10/p01.xhtml
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

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

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

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