Welcome to Yuan Gao's HomePage
Tel: +1 919-660-2896
Address: Physics 06, Department of Mathematics, Duke University, Durham, 27708, USA.
My research in Google Scholar
My research interest is (I) PDE analysis and numerical simulations for problems in materials sciences and interface dynamics, and (II) applied stochastic analysis for the Langevin dynamics in data science and surface hopping. I mainly work on 4th order degenerated parabolic equations, coupled Ginzburg-Landau systems with dynamic boundary condition and multiscale models including phase transition and defects motion. I also work on numerical methods on contact line dynamics, Bayesian inference, manifold learning, hidden Markov model. The main tools involved are calculus of variation, convex analysis, maximal monotone operator, spectral analysis, gradient flows in metric spaces, optimal control theory, and applied stochastic calculus.
Brief Curriculum Vitae:
You can download my full CV here.
Math 1013 Calculus 1B, Fall 2017
Math 5351 Mathematical Methods in Science and Engineering I, Fall 2018
Math 353, Ordinary and Partial Differential Equations, Spring 2019
Math 557, Introduction to PDE, Spring 2019
Math 353, Ordinary and Partial Differential Equations, Spring/Fall 2020
Math 212D, Multivariable Calculus, Spring 2021
Solid thin film growth
Epitaxial growth is a process in which adatoms are deposited on a substrate and grow a solid thin film on the substrate. Epitaxial growth on crystal surface involves various structures, one of which is described by step flow dynamics driven by misfit elasticity between thin film and the substrate. The discrete Burton-Cabrera-Frank (BCF) type models have been proposed by Burton, Cabrera, Frank, Duport, Tersoff, et al.. From the macroscopic view, the governing equations for thin film grow processes are all 4th 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 long-time behavior of solutions. The detailed evolution of boundary profiles such like facets is still open.
1. Continuum limit of a mesoscopic model with elasticity of step motion on vicinal surfaces, with J.-G. Liu and J. Lu, Journal of Nonlinear Science 27 (3), 873-926, (2017)
2. Weak solution of a continuum model for vicinal surface in the attachment-detachment-limited regime,with J.-G. Liu and J. Lu, SIAM Journal on Mathematical Analysis 49 (3), 1705-1731, (2017)
3. Maximal monotone operator in non-reflexive Banach space and the application to thin film equation in epitaxial growth on vicinal surface, with J.-G. Liu, X. Y. Lu and X. Xu, Calculus of Variations and Partial Differential Equations, 57(2), (2018).
4. A vicinal surface model for epitaxial growth with logarithmic free energy, with H. Ji, J.-G. Liu and T. P. Witelski, Discrete Contin. Dyn. Syst. Ser. B. 23(10): 4433-4453, (2018).
5. Gradient flow approach to an exponential thin film equation: global existence and latent singularity, with J.-G. Liu and X. Y. Lu, ESIAM Control Optim. Calc. Var., 25:49, (2018).
Global Strong Solution With BV Derivatives to Singular Solid-on-Solid model With Exponential Nonlinearity , Yuan Gao, Journal of Differential Equations, 267(7), 4429-4447, (2019).
Analysis of a continuum theory for broken bond crystal surface models with evaporation and deposition effects , with J.-G. Liu, J. Lu, J.L. Marzuola, Nonlinearity 33, 3816-3845, (2020).
8. Regularity and monotonicity for solutions to a continuum model of epitaxial growth with nonlocal elastic effects, with X.Y. Lu and C. Wang, to appear in Adv. Calc. Var.(2021).
Tear film evolution
Fluid thin film is offten more complicated than solid thin film. 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 to capture and explore the key features of tear-film dynamics and rupture with power-law mobility functions is investigated.
9. Global existence of solutions to a tear film model with locally elevated evaporation rates, with H. Ji, J.-G. Liu and T. P. Witelski, Physica D: Nonlinear Phenomena 350, 13-25, (2017).
Exact controllability and stabilization in porous materials
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.
10. Observability inequality and decay rate for wave equations with nonlinear boundary conditions, with J. Liang and T.-J. Xiao, Electronic Journal of Differential Equations, 161, 1-12, (2017)
11. A new method to obtain uniform decay rates for multidimensional wave equations with nonlinear acoustic boundary conditions, with J. Liang and T.-J. Xiao, SIAM J. Control Optim., 56(2): 1303-1320, (2018).
Motion of defects in materials science including 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 Peierls-Nabarro (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 Giorgi-type 1D symmetry, uniqueness and asymptotic stability of the original 3D vectorial dislocation model.
12. Mathematical validation of the Peierls--Nabarro model for edge dislocations , with J.-G. Liu, T. Lao and Y. Xiang, Discrete Contin. Dyn. Syst. Ser. B.. 22, (2020).
13. Long time behavior of dynamic solution to Peierls--Nabarro dislocation model , with J.-G. Liu, Methods and Applications of Analysis, 27, 161-198 (2020).
14. Existence and uniqueness of bounded stable solutions to Peierls-Nabarro model for curved dislocation, with H. Dong, Calculus Var. Partial Differ. Equ., 60:62, (2021).
15. Existence and rigidity of the Peierls-Nabarro model for dislocations in high dimensions, with J.-G. Liu and Z. Liu, submitted.
Numerical analysis for contact line dynamics
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 a rough inclined 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.
16. Gradient flow formulation and second order numerical method for motion by mean curvature and contact line dynamics on rough surface, with J.-G. Liu, to appear
in Interfaces and Free Boundaries, (2021).
17. Projection method for droplet dynamics on groove-textured surface with merging and splitting, with J.-G. Liu, submitted.
18. Surfactant-dependent contact line dynamics and droplet
adhesion on textured substrates: derivations and computations, with J.-G. Liu, submitted.
Applied stochastic analysis
PDEs modeling macroscopic dynamics in equilibrium/non-equilibrium systems 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 non-equilibrium optimal twist measure, is suggested by macroscopic observations. We first focused on deriving the continuum limit PDE from the Markov jumping process on lattice. Then for Markov process on point clouds, which suggest an approximated intrinsic manifold (for instance by diffusion map), we simulate the Fokker-Planck equation on the manifold by constructing an approximate Voronoi tessellation. The constructed Markov process (with transition probability assigned on each adjacent data points) and the approximated energy landscape are foundations to simulate inbetweening transformations and manifold-related sampling/transition path problems in biochemical reactions.
How to resolve the underlying dynamic processes via data-driven algorithms ?
19. A note on parametric Bayesian inference via gradient flows, with J.-G. Liu, Annals. of Math. Science and Appl., 261-282 (2020) vol.5.
20. Large time behavior, bi-Hamiltonian structure and kinetic formulation for complex Burgers equation, with Y. Gao and J.-G. Liu, Quart. Appl. Math. 79,
21. Data-driven efficient solvers and predictions of conformational transitions for Langevin dynamics on manifold in high dimensions, with J.-G. Liu and N. Wu submitted.
22. Inbetweening auto-animation via Fokker-Planck dynamics and thresholding, with G. Jin and J.-G. Liu, to appear in Inverse Problems and Imaging, (2021). Video for facial aging process.
23. Analysis of a fourth order exponential PDE arising from a crystal surface jump process with Metropolis-type transition rates, with A.E. Katsevich, J.-G. Liu, J. Lu, and J.L. Marzuola, to appear in Pure and Applied Analysis, (2021).
24. Transition path theory for Langevin dynamics on manifold: optimal control and data-driven solver, with T. Li, X. Li and J.-G. Liu, submitted.
Other directions in preparation
25. Sharp interface dynamics driven by non-local energy and coarsening phenomena, with T. Luo and N. K. Yip.
26. Asymptotic stability for diffusion with dynamic boundary reaction from Ginzburg-Landau energy, with J.-M. Roquejoffre.