Bixiang Wang

Grants

• NSF Division of Mathematical Sciences-Applied Mathematics Program DMS-0703521, $158,040 (08/2007-07/2010). [NSF Award Abstract]

  The goal of this research project is to investigate the dynamical behavior of ion flow through membrane channels in living cells. The membrane channels are three-dimensional tubular-like structures whose radii are much smaller than their lengths. The ion flow is modeled by the Poisson-Nernst-Planck (PNP) system, which consists of at least two nonlinear parabolic equations for ion concentrations, coupled with an elliptic equation for the electric potential. The PNP system is singularly perturbed by the presence of small physical parameters multiplying the highest-order derivatives. This project focuses on the effects of the singular parameters on the global dynamics of the PNP model. It combines approaches from the theory of singular perturbations, dynamical systems, and partial differential equations. The PI will (i) identify the one-dimensional limiting system of the original three-dimensional PNP system when the radius of the membrane channel approaches zero; (ii) justify the limiting system by examining the relationship between the dynamics of the limiting system and the perturbed system; (iii) study the existence and stability of the steady state solutions of the systems; and (iv) explore the dynamics of the perturbed three-dimensional system based on the dynamical information of the one-dimensional limiting system.



• Research Fellowship of Spanish Ministry of Education (1997-1999).


• Research Fellowship of DAAD (German Academic Exchange Service, 1998).


• Postdoctoral Foundation of China (1994-1996).

        My research takes on several directions: (1) partial differential equations; (2) deterministic and stochastic infinite-dimensional dynamical systems; (3) singular perturbation problems; and (4) mathematical biology.

Publications

1. Bixiang Wang, Random attractors for the stochastic FitzHugh-Nagumo system on unbounded domains, accepted by Nonlinear Analysis, 2009.


2. Bixiang Wang, Random attractors for the stochastic Benjamin-Bona-Mahony equation on unbounded domains, accepted by Journal of Differential Equations, 2008.


3. Bixiang Wang, Pullback attractors for the non-autonomous FitzHugh-Nagumo system on unbounded domains, accepted by Nonlinear Analysis, 2008.


4. Bixiang Wang, Pullback attractors for non-autonomous Reaction-Diffusion equations on unbounded domains, accepted by Frontiers of Mathematics in China, 2008.


5. Peter W. Bates, Kening Lu and Bixiang Wang, Random attractors for stochastic Reaction-Diffusion equations on unbounded domains, Journal of Differential Equations, 246 (2009), 845-869.


6. Guglielmo Fucci, Bixiang Wang and Preeti Singh, Asymptotic behavior of the Newton-Boussinesq equation in a two-dimensional channel, Nonlinear Analysis, 70 (2009), 2000-2013.


7. Bixiang Wang and Siyu Lin, Existence of global attractors for the three-dimensional Brinkman-Forchheimer equation, Mathematical Methods in the Applied Sciences, 31 (2008), 1479-1495.


8. Bixiang Wang, Uniform attractors of non-autonomous discrete Reaction-Diffusion systems in weighted spaces, Internat. J. Bifur. Chaos, 18 (2008), 695-716.


9. Timothy Trujillo, Bixiang Wang, Continuity of strong solutions of the Reaction-Diffusion equation in initial data, Nonlinear Analysis, 69 (2008), 2525-2532.


10. Bixiang Wang, Daniel W. Fussner, Chenggeng Bi, Existence of global attractors for the Benjamin-Bona-Mahony equation in unbounded domains, J. Phys. A, 40 (2007), 10491-10504.


11. Bixiang Wang, Dynamical behavior of the almost-periodic discrete FitzHugh-Nagumo systems, Internat. J. Bifur. Chaos, 17 (2007), 1673-1685.


12. Bixiang Wang, Asymptotic behavior of non-autonomous lattice systems, J. Math. Anal. Appl., 331 (2007), 121-136.


13. Weishi Liu, Bixiang Wang, Asymptotic behavior of the FitzHugh-Nagumo system, Internat. J. of Evolution Equations, 2 (2007), 129-163.


14. Bixiang Wang, Dynamics of systems on infinite lattices, J. Differential Equations, 221 (2006), 224-245.


15. Milena Stanislavova, Atanas Stefanov, Bixiang Wang, Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on R^3 , J. Differential Equations, 219 (2005), 451-483.


16. Erik Van Vleck, Bixiang Wang, Attractors for lattice FitzHugh-Nagumo systems, Physica D, 212 (2005), 317-336.


17. Kening Lu, Bixiang Wang, Upper semicontinuity of attractors for the Klein-Gordon-Schrodinger equation on unbounded domains, Internat. J. Bifur. Chaos, 15 (2005), 157-168.


18. Christopher Kumar Anand, Tamas Terlaky, Bixiang Wang, Rapid and embeddable design method for magnetic resonance image reconstruction resampling kernels, Optimization and Engineering, 5 (2004), 485-502.


19. Bixiang Wang, Uniqueness of solutions for the Ginzburg-Landau model of superconductivity in three spatial dimensions, J. Math. Anal. Appl. 266 (2002), 1-20.


20. Peter W. Bates, Kening Lu, Bixiang Wang, Attractors for lattice dynamical systems, International J. of Bifurcation and Chaos, 11 (2001), 143-153.


21. Kening Lu, Bixiang Wang, Attractors for Klein-Gordon-Schrodinger equation in unbounded domains, J. Differential Equations, 170 (2001), 281-316.


22. Wanli Yang, Bixiang Wang, On the question of global existence for the two-component reaction diffusion systems with the mixed boundary conditions, Nonl. Anal. TMA, 39 (2000), 755-766.


23. Bixiang Wang, Attractors for reaction diffusion equations in unbounded domains, Physica D, 128 (1999), 41-52.


24. Bixiang Wang, Existence of time periodic solutions for the Ginzburg-Landau equations of superconductivity, J. Math. Anal. Appl., 232 (1999), 394-412.


25. Bixiang Wang, Ning Su, Weak solutions of Ginzburg-Landau equations of superconductivity, Appl. Math. Lett., 12 (1999), No. 3, 115-118.


26. Bixiang Wang, Horst Lange, Attractors for the Klein-Gordon-Schrodinger equation, J. Math. Phys., 40 (1999), 2445-2457.


27. Bixiang Wang, Ning Su, Existence of solutions for Ginzburg-Landau equations of superconductivity in three spatial dimensions, Proceedings of Royal Society of Edinburgh, 129A (1999), 627-639.


28. Anibal Rodriguez-Bernal, Bixiang Wang, Robert Willie, Asymptotic behaviour of time-dependent Ginzburg-Landau equations of superconductivity, Math. Methods in Appl. Sciences, 22 (1999), 1647-1669.


29. Anibal Rodriguez-Bernal, Bixiang Wang, Attractors for partly dissipative reaction diffusion equations, J. Math. Anal. Appl., 252 (2000), 790-803.


30. Horst Lange, Bixiang Wang, Regularity of attractors for the Klein-Gordon-Schrodinger equation, Math. Methods in Appl. Sciences, 22 (1999), 1535-1554.


31. Anibal Rodriguez-Bernal, Bixiang Wang, Reduction of dimensions of approximate inertial manifolds by symmetry, Bull. Australian Math. Soc., 60 (1999), 319-330.


32. Anibal Rodriguez-Bernal, Bixiang Wang, Cauchy problem for the time dependent Ginzburg-Landau model of superconductivity, Proceedings of Royal Society of Edinburgh 130A (2000), 1383-1404.


33. Boling Guo, Bixiang Wang, Finite dimensional behaviour for the derivative Ginzburg-Landau equation in two spatial dimensions, Physica D, 89(1995), 83-99.


34. Bixiang Wang, Boling Guo, Attractors for the Davey-Stewartson system on R^2, J. Math. Physics, 38(1997), 2524-2534.


35. Bixiang Wang, Wanli Yang, Finite dimensional behaviour for the Benjamin-Bona-Mahony equation, J. Phys. A, 30(1997), 4877-4885.


36. Bixiang Wang, Strong attractors for the Benjamin-Bona-Mahony equation, Appl. Math. Lett., 10(1997), 23-28.


37. Bixiang Wang, Attractors and approximate inertial manifolds for the generalized Benjamin-Bona-Mahony equation, Math. Methods in Appl. Sciences, 20(1997), 189-203.


38. Hans G. Kaper, Bixiang Wang, Shouhong Wang, Determining nodes for the time dependent Ginzburg-Landau equations of superconductivity, Discrete and Continuous Dynamical Systems, 4(1998), 205-224.


39. Bixiang Wang, Regularity of attractors for the Benjamin-Bona-Mahoney equation, J. Phys. A, 31 (1998), 7635-7645.


40. Bixiang Wang, Shouhong Wang, Gevrey class regularity for the time dependent Ginzburg-Landau equations of superconductivity, Discrete and Continuous Dynamical Systems , 4 (1998), 507-522.


41. Boling Guo, Bixiang Wang, Gevrey class regularity and approximate inertial manifolds for the Newton-Boussinesq equation, Chinese Annals of Math., 19B (1998), 179-188.


42. Boling Guo, Bixiang Wang, Gevrey regularity and approximate inertial manifolds for the derivative Ginzburg-Landau equation in two spatial dimensions, Discrete and Continuous Dynamical Systems, 2(1996), 455-466.


43. Xianling Fan, Bixiang Wang, Remarks on periodic solutions of prescribed energy for singular Hamiltonian systems, Houston J. Math., 17(1991), 385-393.


44. Boling Guo, Bixiang Wang, Global existence theory for the two dimensional derivative Ginzburg-Landau equation, Chinese Sci. Bull. 43 (1998), 393--395.


45. Boling Guo, Bixiang Wang, Exponential attractors for the generalized Ginzburg-Landau equation, Acta Math. Sin., 16 (2000), 515--526.


46. Boling Guo, Bixiang Wang, Weak solutions to the two dimensional derivative Ginzburg-Landau equation, Acta Math. Appl. Sinica, 15 (1999),1-8.


47. Boling Guo, Bixiang Wang, Attractors for the long-short wave equations, J. Partial Diff. Equations, 11 (1998), 361--383.


48. Boling Guo, Bixiang Wang, Upper semicontinuity of attractors for the reaction diffusion equation, Acta Math. Scientia, 18 (1998), 139-145.


49. Boling Guo, Bixiang Wang, Global existence for the Landau-Lifshitz equation, Acta Math. Scientia, 17 (1997), 429-436.


50. Bixiang Wang, Ke Shi, On nonlinear Galerkin approximation, J. Computational Math., 17(1997), 23-35.


51. Boling Guo, Bixiang Wang, Approximate inertial manifolds for the Newton-Boussinesq equation, J. Partial Diff. Equ., 9(1996), 237-250.


52. Boling Guo, Bixiang Wang, The global solution and its long time behaviour for a class of generalized LS type equations, Progress in Natural Science, 6(1996), 533-546.


53. Boling Guo, Bixiang Wang, Approximation to the global attractor of nonlinear Schrodinger equation, Appl. Math. , J. Chinese Univ., 11(1996), 125-136.


54. Xianling Fan, Bixiang Wang, Error analysis of nonlinear Galerkin methods for Kuramoto-Sivashinsky equations, Numer. Math. J. Chinese Univ. 5 (1996), 49-61.


55. Xianling Fan, Bixiang Wang, Conservative periodic solutions of prescribed average energy for singular Hamiltonian inclusions, J. Math. Study, 28(1995), 1-10.


56. Boling Guo, Bixiang Wang, The global solution and its long time behaviour for a class of generalized LS type equations, Advances in Math.(China), 24(1995), 179-181.


57. Boling Guo, Bixiang Wang, Approximation to the global attractor of for the Landau-Lifshitz equation of the Ferromagnetic spin chain, Beijing Math., 1(1995), 164-174.

Preprints

1. Weishi Liu, Bixiang Wang, Poisson-Nernst-Planck systems for narrow tubular-like membrane channels, Submitted.

Publications Listed in Mathematical Reviews