Welcome to Yuan Gao's HomePage
Tel: +852 9061-3428
Address: Rm 3485, Department of Mathematics, HKUST, Clear Water Bay, Kowloon, Hong Kong.
My research in Google Scholar
My research interest is the mathematical analysis of nonlinear evolution equations derived from physics problems, especially in materials science and surface science. I mainly work on 4th order degenerated parabolic equations, coupled equations with dynamic boundary condition and multiscale problems. The methods invovled are entropy method, calculus of variation, gradient flows, numerical simulation, operator theory, and control theory.
Math 1013 Calculus 1B, HKUST, Fall 2017
MATH5351 Mathematical Methods in Science and Engineering I, Fall 2018
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, to appear in ESIAM Control Optim. Calc. Var..
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.
6. 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)
Noise control using porous materials
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.
5. 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)
8. 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., 2018, 56(2): 1303-1320.
Other directions in preparation
9. Motion of defects in materials science including dislocations and grain boundaries.
10. Sharp interface dynamics driven by non-local energy and coarsening phenomena.
Brief Curriculum Vitae:
You can download my full CV here.