Y. Charles Li

Y. Charles Li
Professor
Y. Charles Li
Address: 
324 Mathematical Sciences Building
Phone Number: 
573-884-0622

Y. Charles Li's first fifteen years of research built a complete, systematic and rigorous mathematical theory of chaos in partial differential equations (with various collaborators). Then his research focus turned to turbulence and Navier-Stokes equations where his two main achievements are:  1. Resolution of Sommerfeld (turbulence) paradox (with Z. Lin), 2. A theory of rough dependence on initial data for fully developed turbulence, characterized by superfast (faster than exponential) amplification of perturbation. This theory implies short term unpredictability in fully developed turbulence. Our theory concludes that fully developed turbulence is initiated, developed and maintained by such superfast amplification of ever existing perturbations. This has recently been verified numerically in collaboration with R. Ho, A. Berera and Z. Feng. His future focus includes applications of this theory. The impact of his results on applications of PDE chaos to Navier-stokes equations, Landau-Lifschitz-Gilbert equation, and long Josephson junction equation (with Y. Lan, Z. Feng, S. Zhang et al.) is progressing.

For his work on chaos in partial differential equations, Y. Charles Li was awarded the Guggenheim Fellowship (1999) and the AMS Centennial Fellowship (1998). His Merit Prize awarded by Princeton University (1989) also supported that research.

Education: 
  • 1993 Ph.D., Princeton University
  • 1986 B.Sc., Peking University
Frequently Taught Courses: 
  • MATH 4100 Differential Equations
Research Interests: 

Chaos in Partial Differential Equations

Select Publications: 

Superfast amplification and superfast nonlinear saturation of perturbations as a mechanism of turbulence (with Richard Ho, Arjun Berera, Z. C. Feng), J. Fluid Mech., vol. 904, A27, 1-13, (2020)

Linear hydrodynamic stability, Notices of the AMS, vol.65, no.10, 1255-1259, (2018)

Infinite norm of the derivative of the solution oper ator of Euler equations, Physics of Fluids, vol.33, 035120, (2021)

The distinction of turbulence from chaos --- rough dependence on initial data, Electronic Journal of Differential Equations, Vol. 2014, No. 104, pp. 1-8, (2014)

 A Resolution of the Sommerfeld Paradox, SIAM J. Math. Anal., Vol.43, No.4, 1923-1954, (2011). (with Z. Lin)

Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions, J. Dyn. Diff. Eq., vol.15, no.4, 699-730, (2003)

Persistent Homoclinic Orbits for Nonlinear Schrödinger Equation Under Singular Perturbation, Dynamics of PDE, vol.1, no.1, 87-123, (2004)

Smale Horseshoes and Symbolic Dynamics in Perturbed Nonlinear Schrodinger Equations, Journal of Nonlinear Sciences, vol.9, 363-415, (1999).

Persistent Homoclinic Orbits for Perturbed Nonlinear Schrödinger Equation, Comm. Pure and Appl. Math.,XLIX: 1175-1255, (1996).(with D. McLaughlin, J. Shatah, and S. Wiggins)

Morse and Melnikov Functions for NLS Pde's, Comm. Math. Phys., vol.162, 175-214, (1994). (with D.McLaughlin)