Recent Projects
1. Obstacle problem: moving contact lines, lubrication
The dynamics and equilibrium of a droplet on a substrate are important problems with many practical applications such as coating, painting in industries and the adhesion of vesicles in biotechnology. Particularly, the contact line dynamics of a droplet placed on an inclined rough surface are challenging fluid mechanics problems dated back to Young in 1805. The capillary effect, which contributes the leading behaviors of the geometric motion of a small droplet, is characterized by the surface tensions on interfaces separating two different physical phases.
The geometric dynamics of the droplets are described by the mean curvature flow of the capillary surface, coupled with the moving contact lines (where three phases meet), which contributes the leading driven force for the droplet dynamics. The dynamic contact angles tend to go to the equilibrium contact angle (Young’s angle) following the contact line speed mechanism proposed by de Gennes. We focus on the 2nd order unconditionally stable numerical schemes for simulating droplets dynamics on a inclined rough surface with topological changes, such that merging and splitting.
After lubrication approximation, we also investigated the resulting fluid thin film system. A spatial variation in a thin lipid layer leads to locally elevated evaporation rates of the tear film, which in turn affects the local salt concentration in the liquid film. After considering all the contributions from evaporation, capillarity and osmolarity, a general model can capture the key features of tearfilm dynamics and rupture with powerlaw mobility functions.



Related publications:
 Gao, Y., and J.G. Liu. Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface. Interfaces and Free Boundaries. 23(1): 103–158. (Apr 19, 2021) Full text:
 Gao, Y., and J.G. Liu. Projection method for droplet dynamics on groovetextured surface with merging and splitting, SIAM Journal on Scientific Computing. 44(2): B310–B338. (Mar 21, 2022). Full text:
 Gao, Y., and J.G. Liu. Surfactantdependent contact line dynamics and droplet adhesion on textured substrates: derivations and computations, Physica D: Nonlinear Phenomena. 428: 133067. (Oct 12, 2021). Full text:
 Gao, Y., H. Ji, J.G. Liu, and T. P. Witelski. Global existence of solutions to a tear film model with locally elevated evaporation rates. Physica D: Nonlinear Phenomena. 350: 13–25. (Jul 1, 2017). Full text:
2. Motion of materials defects: dislocations and grain boundaries
Dislocations are line defects in crystalline materials and they play essential roles in understanding materials properties like plastic deformation and in the development of novel materials with robust performance.
The detailed structure in a dislocation core can be described by the PeierlsNabarro (PN) model, which is a multiscale continuum model that incorporates the atomistic effect by introducing a nonlinear potential describing the nonlinear atomistic interaction across the slip plane of the dislocation. We focus on existence, De Giorgitype 1D symmetry, uniqueness and asymptotic stability of the original 3D vectorial dislocation model.
Related publications:
 Gao, Y., J.G. Liu, T. Luo, and Y. Xiang. Revisit of the PeierlsNabarro model for edge dislocations in Hilbert space. Discrete and Continuous Dynamical Systems Series B. 26(6): 31773207. (Jul 2020)
 Gao, Y., and J.G. Liu. Long time behavior of dynamic solution to Peierls–Nabarro dislocation model. Methods and Applications of Analysis. 27(2): 161–198. (Jul 27, 2020) Full text:
 Dong, H., and Y. Gao. Existence and uniqueness of bounded stable solutions to the Peierls–Nabarro model for curved dislocations. Calculus of Variations and Partial Differential Equations. 60(2): 62. (Apr 3, 2021). Full text:
 Gao, Y., J.G. Liu, and Z. Liu. Existence and rigidity of the PeierlsNabarro model for dislocations in high dimensions, Nonlinearity. 34(11): 7778–7828. (Oct 6, 2021). Full text:
 Gao, Y. and J.M. Roquejoffre. Asymptotic stability for diffusion with dynamic boundary reaction from GinzburgLandau energy, to appear in SIAM Mathematical Analysis.
3. Langevin dynamics, rare events
Langevin dynamics with an energy landscape representing the stable or metastable states are often used to describe various physical system satisfying the fluctuationdissipation relation. More importantly, irreversible driftdiffusion processes are very common in biochemical reactions and the irreversibility in circulation balance is indeed an primary characteristic of life activities. They have a nonequilibrium stationary state (invariant measure) which does not satisfy detailed balance.
To simulate those Markov processes, designing structurepreserving numerical schemes which enjoys stochastic Qmatrix structure and symmetric decomposition are very important. Those algorithms assign the transition probability on point clouds and thus naturally give random walk approximations for a generic Markov process on a manifold. Thus combining with collected point clouds probing the manifold (induced from nonlinear dimension reduction such as diffusion map), those algorithms can be easily adapted to manifoldrelated computations such as finding the transition path in a rare event such as protein folding.
Various promising interdisciplinary applications are conducted including efficient sampling enhanced by a mixture flow, autoinbetweening image transformation, optimal controlled Monte Carlo simulation for conformational transitions and simulations of shape dynamics on manifold.






Related publications:
 Gao, Y., J. G. Liu, and N. Wu. Datadriven Efficient Solvers for Langevin Dynamics on Manifold in High Dimensions. Applied and Computational Harmonic Analysis. 62(2023):261309. (January, 2023) Full text:
 Gao, Y., G. Jin, and J.G. Liu. Inbetweening autoanimation via FokkerPlanck dynamics and thresholding. Inverse Problems and Imaging. 15(5): 843864. (Oct, 2021) Full text: Video for facial aging process:
 Gao, Y., T. Li, X. Li, and J.G. Liu. Transition path theory for Langevin dynamics on manifold: optimal control and datadriven solver. Multiscale Modeling and Simulation. 21(1):133. (January, 2023) Full text:
 Gao, Y., and J.G. Liu. Random walk approximation for irreversible driftdiffusion process on manifold: ergodicity, unconditional stability and convergence, submitted.
 Gao, Y., and J. G. Liu. Revisit of macroscopic dynamics for some nonequilibrium chemical reactions from a Hamiltonian viewpoint. Journal of Statistical Physics. 189(22). (August 22, 2022) Full text:
4. Crystal growth
Epitaxial growth is a process in which adatoms are deposited on a substrate and grow a solid thin film on the substrate. In general, PDEs modeling macroscopic dynamics in nonequilibrium dynamics are usually guided by the underlying competing mechanism at the microscopic level. Meanwhile, an effective description of the microscopic dynamics, using equilibrium Gibbs measure or nonequilibrium optimal twist measure, is suggested by macroscopic observations. We also derive the continuum limit PDE from the surface hopping process on lattice and give validations via kinetic Monte Carlo.
From a larger mesoscopic scale, Epitaxial growth is described by step flow dynamics driven by misfit elasticity between thin film and the substrate. The discrete BurtonCabreraFrank (BCF) type models have been proposed by Burton, Cabrera, Frank, Duport, Tersoff, et al.. From the macroscopic view, the governing equations for thin film growth processes are all high order degenerate parabolic equations. We focus on the analytic validation of those continuum models by studying the continuum limit from discrete model, global positive solution, strong solutions with latent singularities and longtime behavior of solutions. The facet formation for some nonlocal misfit elasticity with fat tails is still open.


Related publications:
 Gao, Y., J.G. Liu, and J. Lu. Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces. Journal of Nonlinear Science. 27(3): 873–926. (Jun 1, 2017). Full text:
 Gao, Y., J.G. Liu, and J. Lu. Weak solution of a continuum model for vicinal surface in the attachmentdetachmentlimited regime. Siam Journal on Mathematical Analysis. 49(3): 1705–1731. (Jan 1, 2017). Full text:
 Gao, Y., J.G. Liu, X. Y. Lu, and X. Xu. Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface. Calculus of Variations and Partial Differential Equations. 57(2): 55. (Apr 1, 2018). Full text:
 Gao, Y., H. Ji, J.G. Liu, and T. P. Witelski. A vicinal surface model for epitaxial growth with logarithmic free energy. Discrete and Continuous Dynamical Systems Series B. 23(10): 4433–4453. (Dec 1, 2018). Full text:
 Gao, Y., J.G. Liu, and X. Y. Lu. Gradient flow approach to an exponential thin film equation: Global existence and latent singularity. Esaim Control, Optimisation and Calculus of Variations. 25(49). (Oct 14, 2019). Full text:
 Gao, Y.. Global strong solution with BV derivatives to singular solidonsolid model with exponential nonlinearity. Journal of Differential Equations. 267(7): 4429–4447. (Sep 15, 2019). Full text:
 Gao, Y., J.G. Liu, J. Lu and J. L. Marzuola. Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects. Nonlinearity. 33(8): 38163845. (Jun 5, 2020). Full text:
 Gao, Y., X. Y. Lu, and C. Wang. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects. Advances in Calculus of Variations. 000010151520200114. (Apr 16, 2021). Full text:
 Gao, Y., A.E. Katsevich, J.G. Liu, J. Lu, and J.L. Marzuola. Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolistype transition rates. Pure and Applied Analysis. 3(4): 595612. (Feb 12, 2022). Full text:
5. Controllability, stabilization for dynamic boundary condition
My research interests also extend to noise control in building materials which derives a system of wave equation coupled with some acoustic boundary conditions. Although there has been some research on system with acoustic boundary condition, there is little result dealing with completely nonlinear oscillatory of boundary materials, especially for uniformly stabilization with only boundary damping.
Related publications:
 Gao, Y., J. Liang, and T. J. Xiao. Observability inequality and decay rate for wave equations with nonlinear boundary conditions. Electronic Journal of Differential Equations. 2017(161): 112. (Jul 4, 2017). Full text:
 Gao, Y., J. Liang, and T. J. Xiao. A new method to obtain uniform decay rates for multidimensional wave equations with nonlinear acoustic boundary conditions. Siam Journal on Control and Optimization. 56(2): 1303–1320. (Jan 1, 2018). Full text:
6. Miscellany
Related publications:
 Gao. Y., and J.G. Liu. A note on parametric Bayesian inference via gradient flows. Annals of Mathematical Sciences and Applications. 5(2): 261–282. (Oct 13, 2020) Full text:
 Gao, Y., Y. Gao, and J.G. Liu. Large Time Behavior, biHamiltonian Structure, and Kinetic Formulation for a Complex Burgers Equation. Quarterly of Applied Mathematics. 79(1): 55102. (May 21, 2020). Full text: