Junior Research Group "Analysis of PDEs"
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.)
|JProf. Dr. Xian Liao||+49 721 608 email@example.com|
|M. Sc. Robert Wegner||+49 721 608 firstname.lastname@example.org|
|M. Sc. Rebekka Zimmermann||+49 721 608 email@example.com|
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.
P. Gérard, S. Grellier and Zihui He: Turbulent cascades for a family of damped Szegö equations. 2021.
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.
Zihui He and X. Liao: Solvability of the two-dimensional stationary incompressible inhomogeneous Navier-Stokes equations with variable viscosity coefficient. 2020.
H. Koch and X. Liao: Conserved energies for the one-dimensional Gross-Pitaevskii equation. Adv. Math.: 2021, 377, Paper No. 107467, 83.
H. Koch and X. Liao: Conserved energies for the one dimensional Gross-Pitaevskii equation: low regularity case. 2022
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.
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.
X. Liao and M. Plum: Eigenvalue analysis of the Lax operator for the one-dimensional cubic nonlinear defocusing Schrödinger equation. 2022
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.
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.
X. Liao and C. Zillinger: On variable viscosity and enhanced dissipation. 2021.
Ruoci Sun: Complete integrability of the Benjamin-Ono equation on the multi-soliton manifolds. Comm. Math. Phys.: 2021, 383 (2), 1051–1092.