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

Fall 2017

  • No teaching.

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

Publication

(All papers are in peer reviewed journals)
  1. Global estimates for generalized Forchheimer flows of slightly compressible fluids, (with Thinh Kieu,) Journal d'Analyse Mathematique (2015), 41 pp, accepted. [arXiv Preprint]
  2. Generalized Forchheimer flows of isentropic gases, (with Emine Celik, Thinh Kieu,) Journal of Mathematical Fluid Mechanics (2017), 33 pp, in press. (doi:10.1007/s00021-016-0313-2) [arXiv Preprint]
  3. 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)
  4. 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)
  5. 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)
  6. 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)
  7. 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)
  8. 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)
  9. 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)
  10. 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)
  11. 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)
  12. 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)
  13. 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)
  14. 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)
  15. One-dimensional two-phase generalized Forchheimer flows of incompressible fluids, (with Akif Ibragimov, Thinh Kieu,) J. Math. Anal. Appln., Volume 401, Issue 2, 15 May 2013, 921-938. (doi:10.1016/j.jmaa.2012.12.055)
  16. 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.
  17. 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)
  18. 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)
  19. 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)
  20. 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)
  21. 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)
  22. 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)
  23. 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)
  24. 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)
  25. 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)
  26. 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)

Submitted papers

  1. Doubly nonlinear parabolic equations for a general class of Forchheimer gas flows in porous media, (with Emine Celik, Thinh Kieu,) 31 pp, submitted. [arXiv Preprint]
  2. Continuity of pullback and uniform attractors, (with Eric Olson, James Robinson,) 25 pp, revised and resubmitted (2017). [arXiv Preprint]
  3. Asymptotic expansion for solutions of the Navier-Stokes equations with non-potential body forces, (with Vicent Martinez,) 30 pp, submitted. [arXiv Preprint]
  4. Navier and Stokes meet Poincaré and Dulac, (with Ciprian Foias, Jean-Claude Saut,) 33 pp, submitted. (survey paper, dedicated to Claude-Michel Brauner) [arXiv Preprint]

Work in progress

  • 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.
  • On Foias-Saut expansions for Navier-Stokes equations, (with Dat Cao,) in preparation.
  • Generalized Forchheimer flows in geophysical fluid dynamics, (with Emine Celik, Thinh Kieu,) in preparation.
  • Some continuous dependence results for Forchheimer flows, (with Thinh Kieu,) in preparation.
  • Fluid flows in anisotropic porous media, (with Thinh Kieu,) 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