Professor Weinan E

Research summary: My work draws inspiration from various disciplines of sciences and has made an impact in fluid dynamics, chemistry, material sciences, statistical physics and soft condensed matter physics. I have contributed to the resolution of some long standing scientific problems such as the Burgers turbulence problem (which was the original motivation of Burgers for proposing the well-known Burgers equation), the Cauchy-Born rule for crystalline solids (which indeed dates back to Cauchy, and provides a link between the macro- and microscopic theories of solids), and the moving contact line problem. I have also worked on building the mathematical foundation and finding effective numerical algorithms for modeling rare events, as well as other multiscale problems that occur in molecular dynamics, stochastic simulation algorithms, homogenization problems, problems with multiple time scales, complex fluids, etc. A common theme is to try bringing clarity to scientific issues through mathematics. My current interests are: Density functional theory and approximate models for the quantum many-body problem. Instabilities driven by noise. The mathematics and physics of ideal crystals Here are some examples of the work I have been involved with (click on the ``+'' sign to read more):

We have analyzed the statistical properties of solutions to the Burgers equation with random initial data and random forcing. This series of work provided answers to some of the questions that Burgers proposed back in the early 20th century, and resolved some of controversies concerning the asymptotics of the probability distribution functions for the random forced Burgers equation.

W. E and E. Vanden-Eijnden. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett., vol. 83, no. 13, pp. 2572-2575, 1999.
W. E, K. Khanin, A. Mazel and Ya. Sinai. Probability distribution functions for the random forced Burgers equation. Phys. Rev. Lett., vol. 78, no. 10, pp. 1904-1907, 1997.
M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys., vol. 172, no. 1, pp. 13-38, 1995.
M. Avellaneda, R. Ryan and W. E. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids, vol. 7, no. 12, pp. 3067-3071, 1995.

The objective here is to understand solids at the level of quantum mechanics or molecular mechanics. As a by-product, we give a rigorous derivation of the macroscopic continuum models of solids. A key ingredient in this analysis is to understand the various levels of stability conditions (quantum, classical but at the atomic level and classical but at the macro level).

We have developed a new way of studying stochastic PDEs, by viewing the stationary solutions as functionals of the stochastic forcing. This has led to a very elegant description of the stationary solutions of the stochastic Burgers equation and the stochastic passive scalar equation as well as the ergodicity of the stochastic Navier-Stokes equation.

W. E and D. Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys., vol. 108, no. 5-6, pp. 1125-1156, 2002.
W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 27-46. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
W. E and J.C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 1386-1402, 2001.
W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83-106, 2001.
W. E and J.C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 1386-1402, 2001.
W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83-106, 2001.
W. E and E. Vanden-Eijnden. Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl. Acad. Sci., vol. 97, no. 15, pp. 8200-8205, 2000.

My work on modeling rare events (joint with Weiqing Ren and Eric Vanden-Eijnden) has centered around developing the string method, which is now quite popular in cmputational chemistry and begins to get popularity in material science, as well as the transition path theory, which is a general theoretical framework for analyzing transition events in complex systems.

W. E, W. Ren, E. Vanden-Eijnden. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys., vol. 126, no. 16, 164103, 2007.
W. E and E. Vanden-Eijnden. Towards a theory of transition paths. J. Stat. Phys., vol. 123, No. 3, 503-523, 2006.
W. Ren, E. Vanden-Eijnden, P. Maragakis and W. E. Transition pathways in complex systems: Application of the finite temperature string method to the alanine dipeptide. J. Chem. Phys., vol. 123, 134109, 2005.
W. E, W. Ren and E. Vanden-Eijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109, 6688-6693, 2005.
W. E, W. Ren and E. Vanden-Eijnden. String method for the study of rare events. Phys. Rev. B, vol. 66, no. 5, 052301, 2002.

We have developed the framework of the heterogeneous multiscale method (HMM). HMM has led to very promising applications to stochastic simulation algorithms, ODEs with multiple time scales, and many other areas. It also provides a very nice framework for analyzing multiscale methods.

W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367-450, 2007.
W. E, P.B. Ming and P.-W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121-156, 2005.
W. E, D. Liu and E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 1544-1585, 2005.
W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., vol. 1, no. 3, pp. 423-436, 2003.
W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math. Sci., vol. 1, no. 1, pp. 87-132, 2003.
W. E and B. Engquist. Multiscale modeling and computation. Notices Amer. Math. Soc., vol. 50, no. 9, pp. 1062-1070, 2003.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.

We have developed the first general nonlinear model for smectic A liquid crystals and used it to study the interesting filamentary structures arising in isotropic-smectic phase transition. We have also developed models for the dynamics of membranes and polymer phase separations that are consistent with thermodynamics. In addition, we have developed models for general inhomogeneous liquid crystal polymer systems using the one-particle probability distribution function as the order parameter.

D. Hu, P. Zhang and W. E. Continuum theory of a moving membrane. Phys. Rev. E, vol. 75, no. 4, 041605, 2007.
Q. Wang, W. E, C. Liu, P.-W. Zhang. Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. Phys. Rev. E, vol. 65, no. 5, 051504, 2002.
W. E and P. Zhang. A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Methods Appl Anal., vol 13, no. 2, pp. 181-198, 2006.
D. Zhou, P. Zhang and W. E. Modified models of polymer phase separation. Phys. Rev. E, vol. 73, 061801, 2006.
C.B. Muratov and W. E. Theory of phase separation kinetics in polymer-liquid crystal systems. J. Chem. Phys., vol. 116, no. 11, pp. 4723-4734, 2002.
W. E and P. Palffy-Muhoray. Dynamics of filaments during the isotropic-smectic A phase transition. J. Nonlin. Sci., vol. 9, no. 4, pp. 417-437, 1999.
W. E. Nonlinear continuum theory of smectic-A liquid crystals. Arch. Rat. Mech. Anal., vol. 137, no. 2, pp. 159-175, 1997.
W. E and P. Palffy-Muhoray. Phase separation in incompressible systems. Phys. Rev. E, vol. 55, no. 4, pp. R3844-R3846 , 1997.
F. Otto and W. E. Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys., vol. 107, no. 23, pp. 10177-10184, 1997.

Jian-Guo Liu and I addressed long time controversies in vorticity boundary conditions and the numerical boundary layers for the projection method.

In my master degree thesis completed in 1985 under the supervision of Prof. Huang Hongci, I established some of the earliest results on a posteriori error estimates for finite element methods. I introduced the Clement interpolation technique, and proved upper and lower bounds for local error estimators.

W. E, M. Mu and H.C. Huang. A posteriori error estimates in finite element methods. Chinese Quart. J. Math., (Chinese) vol. 3, no. 1, pp. 97-107, 1988.
H.C. Huang and W. E. A posteriori error estimates for finite element methods for one-dimensional boundary value problems. Chinese Quart. J. Math., (Chinese) vol. 2, no. 1, pp. 43-47, 1987.

Under the influence of Jurgen Moser, I independently (of Fathi) developed the weak KAM theory. This was one of the first application of PDE methods to the study of dynamical systems. The most interesting aspect is to study the implication of weak solutions of the Hamilton-Jacobi equation to Hamiltonian systems. This gives an alternative (and much simplified) viewpoint for the Aubry-Mather theory.

W. E. Aubry-Mather theory and periodic solutions of the forced Burgers equation. Comm. Pure Appl. Math., vol. 52, no. 7, pp. 811-828, 1999.

Analysis and algorithms for multiscale problems

The main objective is to develop a rigorous mathematical theory for solids. This requires understanding models of solids at the electronic, atomistic and continuum level, as well as the relation between these models. Problems of interest include: (1). The crystallization problem: Why solids take the form of crystal lattice at zero temperature? (2). The Cauchy-Born rule, which serves as a connection between atomistic and continuum models of solids.

W. E and D. Li. On the crystallization of 2d hexagonal lattice. Comm. Math. Phys., submitted.
W. E and J.F. Lu. The continuum limit and QM-continuum approximation of quantum mechanical models of solids. Comm. Math. Sci., vol. 5, no. 3, pp. 679-696, 2007.
W. E and J.F. Lu. The elastic continuum limit of the tight binding model. Chinese Ann. Math. Ser. B, vol. 28, no. 6, pp. 665-676, 2007.
W. E and P.B. Ming. Cauchy-Born rule and the stability of crystalline solids: Dynamic problems. Acta Math. Appl. Sin. Engl. Ser., vol. 23, no. 4, pp. 529-550, 2007.
W. E and P. B. Ming. Cauchy-Born rule and the stability of crystalline solids: Static problems. Arch. Rat. Mech. Anal., vol. 183, no. 2, pp. 241-297, 2007.

The main objective is to understand the mathematical foundation of electronic structure analysis, to develop and analysis efficient algorithms.

W. E and J.F. Lu. The continuum limit and QM-continuum approximation of quantum mechanical models of solids. Comm. Math. Sci., vol. 5, no. 3, pp. 679-696, 2007.

S. Chen, W. E, Y. Liu and C.-W. Shu. A discontinuous Galerkin implementation of a domain decomposition method for kinetic-hydrodynamic coupling multiscale problems in gas dynamics and device simulations. J. Comput. Phys., vol. 225, no. 2, pp. 1314-1330, 2007.
W. E, B. Engquist, X. Li, W. Ren and E. Vanden-Eijnden. Heterogeneous multiscale methods: A review. Comm. Comput. Phys., vol. 2, no. 3, pp. 367-450, 2007.
W. E and J.F. Lu. Seamless multiscale modeling via dynamics on fiber bundles. Comm. Math. Sci., vol. 5, no. 3, pp. 649-663, 2007.
X. Yue and W. E. The local micro-scale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size. J. Comput. Phys., vol. 222, no. 2, pp. 556-572, 2007.
S. Chen, W. E and C.-W. Shu. The heterogeneous multiscale method based on the discontinuous galerkin method for hyperbolic and parabolic problems. Multiscale Model. Simul., vol. 3, no. 4, pp. 871-894, 2005.
W. E and B. Engquist. The heterogeneous multi-scale method for homogenization problems. Multiscale Methods in Sci. and Eng., pp. 89-110. Lect. Notes in Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005.
W. E and P.B. Ming. Analysis of the local quasicontinuum method. Frontiers and Prospects of Contemp. Appl. Math., pp. 18-32. Contemporary Appl. Math., vol. 6, Higher Education Press, Beijing, 2005.
W. E, P.B. Ming and P.-W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121-156, 2005.
W. E, D. Liu and E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 1544-1585, 2005.
W. E and B. Engquist. The heterogeneous multiscale method. Second Intl. Congress of Chinese Mathematicians. Proc. of ICCM2001, Taipei, pp. 57-74, New Studies in Advanced Mathematics, vol. 4, Intl. Press, 2004.
W. E, X. Li, E. Vanden-Eijnden. Some recent progress in multiscale modeling. Multiscale Modelling and Simulation, pp. 3-22. Lect. Notes Comput. Sci. Eng., vol. 39, Springer, Berlin, 2004.
W. E and X.-T. Li. Analysis of the heterogeneous multiscale method for gas dynamics. Methods Appl. Anal., vol. 11, no. 4, pp. 557-572, 2004.
W. E and P.B. Ming. Analysis of multiscale methods. J. Comput. Math., vol. 22, no. 2, pp. 210-219, 2004.
W. E nd X. Yue. Heterogeneous multiscale method for locally self-similar problems. Comm. Math. Sci., vol. 2, no. 1, pp. 137-144, 2004.
W. E. Analysis of the heterogeneous multiscale method for ordinary differential equations. Comm. Math. Sci., vol. 1, no. 3, pp. 423-436, 2003.
A. Abdulle and W. E. Finite difference heterogeneous multi-scale method for homogenization problems. J. Comput. Phys., vol. 191, no. 1 pp. 18-39, 2003.
L.-T. Cheng and W. E. The heterogeneous multi-scale method for interface dynamics. Recent advances in scientific computing and partial differential equations (Hong Kong, 2002), pp. 43-53, Contemp. Math., vol. 330, Amer. Math. Soc., Providence, RI, 2003.
W. E and B. Engquist. The heterogeneous multiscale methods. Comm. Math. Sci., vol. 1, no. 1, pp. 87-132, 2003.
W. E and B. Engquist. Multiscale modeling and computation. Notices Amer. Math. Soc., vol. 50, no. 9, pp. 1062-1070, 2003.
W. E, B. Engquist and Z. Huang. Heterogeneous multiscale method: A general methodology for multiscale modeling. Phys. Rev. B, vol. 67, no. 9, 092101, 2003.

T. Li, A. Abdulle and W. E. Effectiveness of implicit methods for stiff stochastic differential equations. Comm. Comput. Phys., vol. 3, no. 2, pp. 295-307, 2008.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.
W. E, D. Liu and E. Vanden-Eijnden. Analysis of multiscale methods for stochastic differential equations. Comm. Pure Appl. Math., vol. 58, No. 11, 1544-1585, 2005.
W. E and X.-T. Li. Analysis of the heterogeneous multiscale method for gas dynamics. Methods Appl. Anal., vol. 11, no. 4, pp. 557-572, 2004.

W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.

W. Guo, T. P. Schulze and W. E. Simulation of impurity diffusion in a strained nanowire using off-lattice KMC. Comm. Comput. Phys., vol. 2, no. 1, pp. 164-176, 2007.
X. Li and W. E. Variational boundary conditions for molecular dynamics simulations of crystalline solids at finite temperature: Treatment of the thermal bath. Phys. Rev. B, vol 76, no. 10, 104107, 2007.
J.Z. Yang and W. E. Generalized Cauchy-Born rules for elastic deformation of sheets, plates, and rods: Derivation of continuum models from atomistic models. Phys. Rev. B, vol. 74, no 18, 184110, 2006.
Y. Xiang, H. Wei, P.B. Ming and W. E. A generalized Peierls–Nabarro model for curved dislocations and core structures of dislocation loops in Al and Cu. Acta Materialia, in press, available online 14 January 2008.
W. E, J.-F. Lu, J.Z. Yang. Uniform accuracy of the quasicontinuum method. Phys. Rev. B, vol. 74, 214115, 2006.
X.-T. Li and W. E. Variational boundary conditions for molecular dynamics simulation of solids at low temperature. Comm. Comput. Phys., vol. 1, No. 1, 135-175, 2006.
N. Choly, G. Lu, W. E and E. Kaxiras. Multiscale simulations in simple metals: A density-functional based methodology. Phys. Rev. B, vol. 71, 094101, 2005.
X.-T. Li and W. E. Multiscale modeling of the dynamics of solids at finite temperature. J. Mech. Phys. Solids, vol. 53, 1650-1685, 2005.
W. E and X.-T. Li. Multiscale modeling of crystalline solids. Handbook of Materials Modeling, Part A, edited by S. Yip., pp. 1491-1506, Springer Netherlands, 2005.
Y. Xiang and W. E. Misfit elastic energy and a continuum model for epitaxial growth with elasticity on vicinal surfaces. Phys. Rev. B, vol. 69, no. 3, 035409, 2004.
Y. Xiang, D.J. Srolovitz, L.-T. Cheng and W. E. Level set simulations of dislocation-particle bypass mechanisms. Acta Materialia, vol. 52, no. 7, pp. 1745-1760, 2004.
Y. Xiang, L.-T. Cheng, D.J. Srolovitz and W. E. A level set method for dislocation dynamics. Acta Materialia, vol. 51, no. 18, pp. 5499-5518, 2003.
T. Schulze, P. Smereka and W. E. Coupling kinetic Monte-Carlo and continuum models with application to epitaxial growth. J. Comput. Phys., vol 189, no. 1, pp. 197-211, 2003.
Y. Xiang and W. E. Nonlinear evolution equation for the stress-driven morphological instability. J. Appl. Phys., vol. 91, no. 11, pp. 9414-9422, 2002.
W. E and Z. Huang. A dynamic atomistic-continuum method for the simulation of crystalline materials. J. Comput. Phys., vol. 182, no. 1, pp. 234-261, 2002.
W. E and Z. Huang. Matching conditions in atomistic-continuum modeling of materials. Phys. Rev. Lett., vol. 87, no. 13, 135501, 2001.
W. E and N.K. Yip. Continnum theory of epitaxial crystal growth. I. J. Stat. Phys., vol. 104, no. 1-2, pp. 221-253, 2001.
M.I. Mendelev, D.J. Srolovitz and W. E. Grain-boundary migration in the presence of diffusing impurities: simulations and analytical models. Philos. Mag. A, vol. 81, no. 9, pp. 2243-2269, 2001.
T. Schulze and W. E. A continuum model for the growth of epitaxial films. J. Crystal Growth, vol. 222, no. 1-2, pp. 414-425, 2001.
W. E and N.K. Yip. Continuum limits of step flow models. EQUADIFF 99 Proc. Intl. Conf. Differential Equations, vol. 1, 2 (Berlin, 1999), pp. 448-453, World Sci. Publishing, River Edge, NJ, 2000.
R. Caflisch, W. E, M. Gyure, B. Merriman and C. Ratsch. Kinetic model for a step edge in epitaxial growth. Phys. Rev. E, vol. 59, no. 6, pp. 6879-6887, 1999.

D. Hu, P. Zhang and W. E. Continuum theory of a moving membrane. Phys. Rev. E, vol. 75, no. 4, 041605, 2007.
W. E and P. Zhang. A molecular kinetic theory of inhomogeneous liquid crystal flow and the small Deborah number limit. Methods Appl Anal., vol 13, no. 2, pp. 181-198, 2006.
D. Zhou, P. Zhang and W. E. Modified models of polymer phase separation. Phys. Rev. E, vol. 73, 061801, 2006.
W. Ren and W. E. Heterogeneous multiscale method for the modeling of complex fluids and micro-fluidics. J. Comput. Phys., vol. 204, no. 1, pp. 1-26, 2005.
S. Succi, W. E and E. Kaxiras. Lattice boltzmann methods for multiscale fluid problems. Handbook of Materials Modeling, Part B, pp. 2475-2486, Springer Netherlands, 2005.
X. Nie, S. Chen, W. E and M. Robbins. Hybrid continuum-atomistic simulation of singular corner flow. Phys. Fluids, vol. 16, no. 10, pp. 3579-3591, 2004.
T.-J. Li, E. Vanden-Eijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. Non-Newtonian Fluid Mechanics, vol. 121, no. 2-3, pp. 117-125, 2004.
X. Nie, S. Chen, W. E and M.O. Robbins. A continuum and molecular dynamics hybrid method for micro- and nano-fluid flow. J. Fluid Mech., vol. 500, pp. 55-64, 2004.
W. E, T.-J. Li and P.-W. Zhang. Well-posedness for the dumbbell model of polymeric fluids. Comm. Math. Phys., vol. 248, no. 2, pp. 409-427, 2004.
T.-J. Li, E. Vanden-Eijnden, P.W. Zhang and W. E. Stochastic models of polymeric fluids at small Deborah number. J. Non-Newtonian Fluid Mechanics, vol. 121, 117-125, 2004.
W. E, T.-J. Li, P.-W. Zhang. Convergence of a stochastic method for the modeling of polymeric fluids. Acta Math. Appl. Sin., vol. 18, no. 4, pp. 529-536, 2002.
C.B. Muratov and W. E. Theory of phase separation kinetics in polymer-liquid crystal systems. J. Chem. Phys., vol. 116, no. 11, pp. 4723-4734, 2002.
P. Palffy-Muhoray, T. Kosa and W. E. Brownian motors in the photoalignment of liquid crystals. Appl. Phys. A, vol. 75, no. 2, pp. 293-300, 2002.
Q. Wang, W. E, C. Liu, P.-W. Zhang. Kinetic theory for flows of nonhomogeneous rodlike liquid crystalline polymers with a nonlocal intermolecular potential. Phys. Rev. E, vol. 65, no. 5, 051504, 2002.
W. E and P. Palffy-Muhoray. Dynamics of filaments during the isotropic-smectic A phase transition. J. Nonlin. Sci., vol. 9, no. 4, pp. 417-437, 1999.
W. E and P. Palffy-Muhoray. Orientational ratchets and angular momentum balance in the Janossy effect. Mol. Cryst. Liq. Cryst., vol. 320, no. 1, pp. 193-206, 1998.
W. E. Nonlinear continuum theory of smectic-A liquid crystals. Arch. Rat. Mech. Anal., vol. 137, no. 2, pp. 159-175, 1997.
W. E and P. Palffy-Muhoray. Phase separation in incompressible systems. Phys. Rev. E, vol. 55, no. 4, pp. R3844-R3846 , 1997.
F. Otto and W. E. Thermodynamically driven incompressible fluid mixtures. J. Chem. Phys., vol. 107, no. 23, pp. 10177-10184, 1997.

X. Yue and W. E. The local micro-scale problem in the multiscale modelling of strongly heterogeneous media: Effect of boundary conditions and cell size. J. Comput. Phys., vol. 222, no. 2, pp. 556-572, 2007.
W. E and B. Engquist. The heterogeneous multi-scale method for homogenization problems. Multiscale Methods in Sci. and Eng., pp. 89-110. Lect. Notes in Comput. Sci. Eng., vol. 44, Springer, Berlin, 2005.
W. E, P.B. Ming and P.-W. Zhang. Analysis of the heterogeneous multiscale method for elliptic homogenization problems. J. Amer. Math. Soc., vol. 18, no. 1, pp. 121-156, 2005.
X. Yue and W. E. Numerical methods for multiscale transport equations and application to two-phase porous media flow. J. Comput. Phys., vol. 210, no. 2, pp. 656-675, 2005.
A. Abdulle and W. E. Finite difference heterogeneous multi-scale method for homogenization problems. J. Comput. Phys., vol. 191, no. 1 pp. 18-39, 2003.

W. Ren and W. E. Boundary conditions for the moving contact line problem. Phys. Fluids, vol. 19, 022101, 2007.

B. Engquist and W. E. Large time behavior and homogenization of solutions of two-dimensional conservation laws. Comm. Pure Appl. Math., vol. 46, no. 1, pp. 1-26, 1993.
W. E and C.-W. Shu. Effective equations and the inverse cascade theory for Kolmogorov flows. Phys. Fluids A, vol. 5, no. 4, pp. 998-1010, 1993.
W. E. Propagation of oscillations in the solutions of 1-D compressible fluid equations. Comm. Partial Differential Equations, vol. 17, no. 3-4, pp. 545-552, 1992.
W. E. Homogenization of linear and nonlinear transport equations. Comm. Pure Appl. Math., vol. 45, no. 3, pp. 301-326, 1992.
W. E. Homogenization of scalar conservation laws with oscillatory forcing terms. SIAM J. Appl. Math., vol. 52, no. 4, pp. 959-972, 1992.
W. E and D. Serre. Correctors for the homogenization of conservation laws with oscillatory forcing terms. Asymptotic Anal., vol. 5, no. 4, pp. 311-316, 1992.
W. E. A class of homogenization problems in the calculus of variations. Comm. Pure Appl. Math., vol. 44, no. 7, pp. 733-759, 1991.
W. E and R.V. Kohn. The initial value problem for measure-valued solutions of a canonical 2x2 system with linearly degenerate fields. Comm. Pure Appl. Math., vol. 44, no. 8-9, pp. 981-1000, 1991.
W. E and H. Yang. Numerical study of oscillatory solutions of the gas-dynamic equations. Stud. Appl. Math., vol. 85, no. 1, pp. 29-52, 1991.
W. E and T.Y. Hou. Homogenization and convergence of the vortex method for 2-D Euler equations with oscillatory vorticity fields. Comm. Pure Appl. Math., vol. 43, no. 7, pp. 821-855, 1990.

Analysis and modeling of stochastic problems

W. E and D. Liu. Gibbsian dynamics and invariant measures for stochastic dissipative PDEs. J. Stat. Phys., vol. 108, no. 5-6, pp. 1125-1156, 2002.
W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 27-46. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
W. E and J.C. Mattingly. Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finite-dimensional approximation. Comm. Pure Appl. Math., vol. 54, no. 11, pp. 1386-1402, 2001.
W. E, J.C. Mattingly and Ya. Sinai. Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys., vol. 224, no. 1, pp. 83-106, 2001.
W. E. Stochastic hydrodynamics. Current Developments in Mathematics, 2000, pp. 109-147, Intl. Press, Somerville, MA, 2000.
W. E, K. Khanin, A. Mazel and Ya. Sinai. Invariant measures for Burgers equation with stochastic forcing. Ann. of Math., vol. 151, no. 3, pp. 877-960, 2000.
W. E and Ya. Sinai. Recent results on mathematical and statistical hydrodynamics. Russ. Math. Survey, vol. 55, no. 4, 635-666, 2000.
W. E, Yu. Rykov and Ya. Sinai. Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics. Comm. Math. Phys., vol. 177, no. 2, pp. 349-380, 1996.

W. E, W. Ren, E. Vanden-Eijnden. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys., vol. 126, no. 16, 164103, 2007.
T. Qian, W. Ren, J. Shi, W. E and P. Sheng. Numerical study of metastability due to tunneling: The quantum string method. Phys. A, vol. 379, no. 2, pp. 491-502, 2007.
W. E and E. Vanden-Eijnden. Towards a theory of transition paths. J. Stat. Phys., vol. 123, No. 3, 503-523, 2006.
W. E, W. Ren and E. Vanden-Eijnden. Transition pathways in complex systems: Reaction coordinates, iso-committor surfaces and transition tubes. Chem. Phys. Lett., vol. 413, no. 1-3, 242-247, 2005.
W. Ren, E. Vanden-Eijnden, P. Maragakis and W. E. Transition pathways in complex systems: Application of the finite temperature string method to the alanine dipeptide. J. Chem. Phys., vol. 123, 134109, 2005.
W. E, W. Ren and E. Vanden-Eijnden. Finite temperature string method for the study of rare events. J. Phys. Chem. B, 109, 6688-6693, 2005.
W. E, W. Ren, E. Vanden-Eijnden. Minimum action method for the study of rare events. Comm. Pure Appl. Math., vol. 57, no. 5, pp. 637-656, 2004.
W. E and E. Vanden-Eijnden. Metastability, conformation dynamics, and transition pathways in complex systems. Multiscale Modelling and Simulation, pp. 35-68, Lect. Notes Comput. Sci. Eng., vol. 39, Springer, Berlin, 2004.
W. E, W. Ren and E. Vanden-Eijnden. Energy landscape and thermally activated switching of submicron-sized ferromagnetic elements. J. Appl. Phys., vol. 93, no. 4, pp. 2275-2282, 2003.
W. E, W. Ren and E. Vanden-Eijnden. String method for the study of rare events. Phys. Rev. B, vol. 66, no. 5, 052301, 2002.
W. E, W. Ren and E. Vanden-Eijnden. Energy landscapes and rare events. ICM Report, vol. 1, pp. 621-630, Higher Ed. Press, Beijing, 2002.

W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales. J. Comput. Phys., vol. 221, no. 1, pp. 158-180, 2007.
W. E, D. Liu and E. Vanden-Eijnden. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys., vo. 123, 194107, 2005.

W. E and E. Vanden-Eijnden. A note on generalized flows. Phys. D, vol. 183, no. 3-4, pp. 159-174, 2003.
W. E. Stochastic PDES in turbulence theory. Proc. 1st Intl. Congress Chinese Math. (Beijing, 1998), pp. 27-46. AMS/IP Stud. Adv. Math, vol. 20, Amer. Math. Soc., Providence, RI, 2001.
W. E and E. Vanden-Eijnden. Turbulent Prandtl number effect on passive scalar advection. Phys. D, vol. 152-153, pp. 636-645, 2001.
W. E and E. Vanden-Eijnden. Statistical theory for the stochastic Burgers equation in the inviscid limit. Comm. Pure Appl. Math., vol. 53, no. 7, pp. 852-901, 2000.
W. E and E. Vanden-Eijnden. Another note on forced Burgers turbulence. Phys. Fluids, vol. 12, no. 1, pp. 149-154, 2000.
W. E and E. Vanden-Eijnden. Generalized flows, intrinsic stochasticity and turbulent transport. Proc. Natl. Acad. Sci., vol. 97, no. 15, pp. 8200-8205, 2000.
W. E and E. Vanden-Eijnden. On the statistical solution of the Riemann equation and its implications for Burgers turbulence. Phys. Fluids, vol. 11, no. 8, pp. 2149-2153, 1999.
W. E and E. Vanden-Eijnden. Asymptotic theory for the probability density functions in Burgers turbulence. Phys. Rev. Lett., vol. 83, no. 13, pp. 2572-2575, 1999.
W. E, K. Khanin, A. Mazel and Ya. Sinai. Probability distribution functions for the random forced Burgers equation. Phys. Rev. Lett., vol. 78, no. 10, pp. 1904-1907, 1997.
M. Avellaneda and W. E. Statistical properties of shocks in Burgers turbulence. Comm. Math. Phys., vol. 172, no. 1, pp. 13-38, 1995.
M. Avellaneda, R. Ryan and W. E. PDFs for velocity and velocity gradients in Burgers' turbulence. Phys. Fluids, vol. 7, no. 12, pp. 3067-3071, 1995.

C.B. Muratov, E. Vanden-Eijnden, W. E. Noise can play an organizing role for the recurrent dynamics in excitable media. Proc. Natl. Acad. Sci., vol. 104, no. 3, pp. 702-707, 2007.
C.B. Muratov, E. Vanden-Eijnden and W. E. Self-induced stochastic resonance in excitable systems. Phys. D, vol. 210, no. 3-4, pp. 227-240, 2005.
P. Palffy-Muhoray, T. Kosa, W. E. Brownian ratchets and the photoalignment of liquid crystals. Braz. J. Phys., vol.32 no.2b, pp. 552-563, Sao Paulo, 2002.
P. Palffy-Muhoray, T. Kosa and W. E. Dynamics of a Light Driven Molecular Motor. Mol. Cryst. Liq. Cryst., vol. 375, no. 1, pp. 577-592, 2002.
T. Kosa, W. E and P. Palffy-Muhoray. Brownian motors in the photoalignment of liquid crystals. Intl J. Eng. Sci., vol. 38, no. 9-10, pp. 1077-1084, 2000.
W. E and P. Palffy-Muhoray. Domain size in the presence of random fields. Phys. Rev. E, vol. 57, no. 1, pp. 135-137, 1998.

Weinan E

Research:

Analysis and algorithms for multiscale problems

Analysis and modeling of stochastic problems

Other topics

Selected Review Papers

Weinan E
Professor, Department of Mathematics and Program in Applied and Computational Mathematics Princeton University Princeton, NJ 08544-1000 U.S.A. Phone: (609)258-3683 ~ Fax: (609)258-1735 weinan@math.princeton.edu	Research Complete publication list Curriculum Vitae Picture Teaching and course development Book