Teaching The only good is knowledge and the only evil is ignorance. - Socrates

Previous semesters

Research What we know is a drop, what we don't know is an ocean. - Isaac Newton

Research interest

  • Partial differential equations
  • Fluid dynamics
  • Porous medium equations
  • Dynamical systems

Publications

(All papers are in peer reviewed journals. List in PDF.)
  1. Long-time asymptotic expansions for Navier-Stokes equations with power-decaying forces, (with Dat Cao,) 35 pp, submitted. [arXiv Preprint]
  2. Global estimates for generalized Forchheimer flows of slightly compressible fluids, (with Thinh Kieu,) Journal d'Analyse Mathematique (2018), 54 pp, in press. [arXiv Preprint]
  3. Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media, (with Emine Celik, Thinh Kieu,) Nonlinearity, Vol. 31, No. 8 (2018) 3617-3650. (doi:10.1088/1361-6544/aabf05)
  4. Navier and Stokes meet Poincaré and Dulac, (with Ciprian Foias, Jean-Claude Saut,) J. Appl. Anal. Comput., Volume 8, Number 3, June 2018, 727-763. (doi:10.11948/2018.727) (survey)
  5. Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces, (with Vicent Martinez,) J. Math. Anal. Appl., Volume 462, Issue 1, (1 June 2018) 84-113. (doi:10.1016/j.jmaa.2018.01.065)
  6. Generalized Forchheimer flows of isentropic gases, (with Emine Celik, Thinh Kieu,) J. Math. Fluid Mech., March 2018, Volume 20, Issue 1, 83-115. (doi:10.1007/s00021-016-0313-2)
  7. Continuity of pullback and uniform attractors, (with Eric Olson, James Robinson,) J. Differential Equations, Volume 264, Issue 6, (15 March 2018) 4067-4093. (doi:10.1016/j.jde.2017.12.002)
  8. Interior estimates for generalized Forchheimer flows of slightly compressible fluids, (with Thinh Kieu,) Advanced Nonlinear Studies, 17(4), (2017) 739-767. (doi:10.1515/ans-2016-6027)
  9. Asymptotic expansion in Gevrey spaces for solutions of Navier-Stokes equations, (with Vincent Martinez,) Asymptotic Analysis (2017), vol. 104, no. 3-4, 167-190. (doi:10.3233/ASY-171429)
  10. Fluid flows of mixed regimes in porous media, (with Emine Celik, Akif Ibragimov, Thinh Kieu,) Journal of Mathematical Physics, Volume 58 (2017), No. 2, 023102, 30 pp. (doi:10.1063/1.4976195)
  11. Maximum estimates for generalized Forchheimer flows in heterogeneous porous media. (with Emine Celik,) J. Differential Equations, Volume 262, Issue 3 (5 February 2017), 2158-2195. (doi:10.1016/j.jde.2016.10.043)
  12. Local gradient estimates for degenerate elliptic equations, (with Truyen Nguyen, Tuoc Phan,) Advanced Nonlinear Studies, Volume 16, Issue 3 (Aug 2016), 479-489. (doi:10.1515/ans-2015-5038)
  13. Generalized Forchheimer flows in heterogeneous porous media, (with Emine Celik,) Nonlinearity, Vol. 29, No. 3 (March 2016), 1124-1155. (doi:10.1088/0951-7715/29/3/1124)
  14. Stability of solutions to generalized Forchheimer equations of any degree, (with Akif Ibragimov, Thinh Kieu, Zeev Sobol,) Journal of Mathematical Sciences, Volume 210, Number 4 (2015), 476-544. (Also in "Problemy Matematicheskogo Analiza" 81, August 2015, 121-178.) (doi:10.1007/s10958-015-2576-1)
  15. On the continuity of global attractors, (with Eric Olson, James Robinson,) Proc. Amer. Math. Soc., Volume 143, Number 10 (2015), 4389-4395. (doi:10.1090/proc/12598)
  16. Gradient estimates and global existence of smooth solutions to a cross-diffusion system, (with Truyen Nguyen, Tuoc Phan,) SIAM Journal on Mathematical Analysis, Volume 47, Issue 3 (2015), 2122-2177. (doi:10.1137/140981447)
  17. A family of steady two-phase generalized Forchheimer flows and their linear stability analysis, (with Akif Ibragimov, Thinh Kieu,) J. Math. Phys. 55, Issue 12 (2014), 123101:32pp. (doi:10.1063/1.4903002)
  18. Properties of generalized Forchheimer flows in porous media, (with Thinh Kieu, Tuoc Phan,) Problems of Mathematical Analysis, Vol. 76 (August 2014), 133--194, and in Journal of Mathematical Sciences, Vol. 202 No. 2, October 2014, 259-332. (doi:10.1007/s10958-014-2045-2)
  19. Incompressible fluids in thin domains with Navier friction boundary conditions (II), J. of Math. Fluid Mech., Volume 15, Issue 2, June 2013, 361-395. (doi:10.1007/s00021-012-0123-0)
  20. One-dimensional two-phase generalized Forchheimer flows of incompressible fluids, (with Akif Ibragimov, Thinh Kieu,) J. Math. Anal. Appl., Volume 401, Issue 2, 15 May 2013, 921-938. (doi:10.1016/j.jmaa.2012.12.055)
  21. Qualitative study of generalized Forchheimer flows with the flux boundary condition, (with Akif Ibragimov,) Adv. Diff. Eq., Volume 17, Numbers 5-6, May/June 2012, 511-556.
  22. Asymptotic integration of Navier-Stokes equations with potential forces. II. An explicit Poincaré-Dulac normal form, (with Ciprian Foias, Jean-Claude Saut,) J. Funct. Anal., Vol. 260, Issue 10, (2011) 3007-3035. (doi:10.1016/j.jfa.2011.02.005)
  23. Structural stability of generalized Forchheimer equations for compressible fluids in porous media, (with Akif Ibragimov,) Nonlinearity, Volume 24, Number 1 / January 2011, 1-41. (doi:10.1088/0951-7715/24/1/001)
  24. Navier-Stokes equations with Navier boundary conditions for an oceanic model, (with George Sell,) J. Dyn. Diff. Eqn., Volume 22, Number 3 / September 2010, 563-616. (doi:10.1007/s10884-010-9189-7)
  25. Incompressible fluids in thin domains with Navier friction boundary conditions (I), J. of Math. Fluid Mech., Volume 12, Number 3 / August 2010, 435-472. (doi:10.1007/s00021-009-0297-2)
  26. Analysis of generalized Forchheimer equations of compressible fluids in porous media, (with Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov,) J. Math. Phys. 50, Issue 10, 103102:44pp (2009). (doi:10.1063/1.3204977)
  27. The normal form of the Navier-Stokes equations in suitable normed spaces, (with Ciprian Foias, Eric Olson, Mohammed Ziane,) Annales de l'Institut Henri Poincare - Analyse non lineaire, Volume 26, Issue 5, September-October 2009, 1635-1673. (doi:10.1016/j.anihpc.2008.09.003)
  28. On the helicity in 3D Navier-Stokes equations II: The statistical case, (with Ciprian Foias, Basil Nicolaenko,) Comm. Math. Physics, Volume 290, Issue 2 (2009) 679-717. (doi:10.1007/s00220-009-0827-z)
  29. A basic inequality for the Stokes operator related to the Navier boundary condition, J. of Diff. Eqn., Volume 245, Issue 9, (1 November 2008) 2585-2594, (doi:10.1016/j.jde.2008.01.024)
  30. On the helicity in 3D Navier-Stokes equations I: The non-statistical case, (with Ciprian Foias, Basil Nicolaenko,) Proc. London Math. Soc., Volume 94 Number 1 (January 2007) 53-90. (doi:10.1112/plms/pdl003)
  31. On the solutions to the normal form of the Navier-Stokes equations, (with Ciprian Foias, Eric Olson, Mohammed Ziane,) Indiana Univ. Math. J., Vol. 55, No 2 (2006) 631-686. (doi:10.1512/iumj.2006.55.2830)

Work in progress

  • Asymptotic expansions for Navier-Stokes equations with decaying forces. Part II, (with Dat Cao,) in preparation.
  • On NSE of rorating fluids, (with Ciprian Foias, Edriss Titi,) in preparation.
  • Asymptotic expansions for solutions of ODEs, (with Dat Cao, Thinh Kieu,) in preparation.
  • Some continuous dependence results for Forchheimer flows, (with Thinh Kieu,) in preparation.
  • Generalized Forchheimer flows in geophysical fluid dynamics, (with Emine Celik, Thinh Kieu,) in preparation.
  • Fluid flows in anisotropic porous media, (with Thinh Kieu,) in preparation.
  • Long-time dynamics of fluid flows in geophysical fluid models, (with Animikh Biswas,) in preparation.

  • On the normal form and determining modes for NSE, (with Ciprian Foias, Cecilia Mondaini, Bingsheng Zhang,) in preparation.
  • Navier-Stokes equations with Navier boundary conditions for thin spherical shells, (with George Sell,) in preparation, 20pp.
  • Navier-Stokes equations with Navier boundary conditions in thin domains: The reduced problem, (with George Sell,) in preparation.

Conferences

Presentations

Grants

Profile The unexamined life is not worth living. - Socrates

Education

Employment

  • 9.2014 - present, Associate professor, Texas Tech University, Lubbock, Texas
  • 9.2008 - 8.2014, Assistant professor, Texas Tech University, Lubbock, Texas
  • 9.2005 - 5.2008, Dunham Jackson assistant professor, University of Minnesota, Minneapolis, Minnesota
  • 6.2004 - 12.2004, Teaching assistant, Texas A&M University, College Station, Texas
  • 9.2000 - 8.2002, Associate instructor, Indiana University, Bloomington, Indiana
  • 9.1998 - 8.2000, Teaching assistant and research assistant, Arizona State University, Tempe, Arizona
  • 9.1997 - 8.1998, Instructor, National University, Hochiminh City, Vietnam

CV

    Links All science requires mathematics. - Roger Bacon