\[
\int_{\Omega_T}\Big(|u_t|^p+|DD_{x'}u|^p+ \big|D^2_d u+\frac \alpha {x_d} D_du\big|^p+ |\sqrt{\lambda} Du|^p + |\lambda u|^p\Big)x_d^{\gamma}\,dx dt
\leq N \int_{\Omega_T}|f|^px_d^{\gamma}\,dx dt.
\]
Papers (preprint and peer-reviewed publication)
Harnack inequality for singular or degenerate parabolic equations in non-divergence form (with S. Cho, and J. Fang), (2024), [arXiv:2409.09437 ].
On linear elliptic equations with drift terms in critical weak spaces (with H. Kim and T.-P. Tsai ), submitted (2023), [arXiv:2312.11215 ].
Sobolev estimates for singular-degenerate quasilinear equations beyond the \(A_2\) class (with H. Dong and Y. Sire ), The Journal of Geometric Analysis , Volume 34, article number 286, (2024) [Journal article ], [ arXiv:2305.07634 ].
Nondivergence form degenerate linear parabolic equations on the upper half space (H. Dong and H. V. Tran ), Journal of Functional Analysis (2024), [Journal article ], [arXiv:2103.08033 ].
On stationary Navier-Stokes equations in the upper-half plane (with Adrian D. Calderon and Van Le ), Acta Applicandae Mathematicae (2024), [Journal article ], [arXiv:2306.00319 ].
Weighted mixed-norm \(L_p\)-estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem (with H. Dong ), Journal of Functional Analysis (2023), [Journal article ], [arXiv:2103.08033 ].
Degenerate linear parabolic equations in divergence form on the upper half space (with H. Dong and H. V. Tran ), Transactions of the American Mathematical Society (2023), [Journal article ], [arXiv:2107.08033 ].
On trace theorems for weighted mixed norm Sobolev spaces and applications , Potentials and Partial Differential Equations: The Legacy of David R. Adams, Vol 8, Advances in Analysis and Geometry , [arXiv:2205.04941 ].
A boundary layer problem in non-flat domains with measurable viscous coefficients (with Dario V. Castillo ), Studies in Applied Mathematics , (2022), [Journal article ].
On parabolic and elliptic equations with singular or degenerate coefficients (with H. Dong ), Indiana University Mathematics Journal , 73 (2023), no. 4, 1461-1502, [arXiv ].
Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients (with H. Dong and D. Kim ), Communications in Partial Differential Equations , https://doi.org/10.1080/03605302.2022.2084627 , arXiv:1910.00380, [arXiv ].
On higher integrability estimates for elliptic equations with singular coefficients (with J. Foldes ), Funkcialaj Ekvacioj , accepted, [arXiv ].
On Masuda uniqueness theorem for Leray-Hopf weak solutions in mixed-norm spaces (with T. Robertson ), European Journal of Mechanics / B Fluids , 90 (2021), 18-28 [Journal article ], [preprint ].
Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem (with H. Dong ), Transactions of the American Mathematical Society , https://doi.org/10.1090/tran/8397 , [Journal article ], [arXiv ].
Mixed norm \(L_p\)-estimates for non-stationary Stokes systems with singular VMO coefficients and applications (with H. Dong ), Journal of Differential Equations , Volume 276, 5 (2021), 342-367, [Journal article ], [arXiv ].
Regularity for parabolic equations with singular or degenerate coefficients (with H. Dong ), Calculus of Variations and Partial Differential Equations , 60, 44 (2021) Journal Article , [arXiv ].
Weighted mixed-norm \(L_p\)-estimates for elliptic and parabolic equations in non-divergence form with singular degenerate coefficients (with H. Dong ), Revista Matemática Iberoamericana , DOI: 10.4171/rmi/1233 , arXiv:1811.06393 [arXiv ].
Existence uniqueness and regularity theory for elliptic equations with complex valued potentials (with G. Todorova and B. Yordanov ), Discrete and Continuous Dynamical Systems-Series A , 2021, 41(3): 1071-1099 [Journal article ] [preprint ].
Liouville type theorems for 3D stationary Navier-Stokes equations in weighted mixed-norm Lebesgue spaces , Dynamics of Partial Differential Equations , Vol 17, no. 3 (2020), 229-243 [Journal article ] [arXiv ].
On well-posedness of 2D dissipative quasi-geostrophic equation in critical mixed norm Lebesgue spaces (with Y. Sire ), Analysis in Theory and Applications , 36 (2) (2020), 111-127 [Journal article ] [preprint ].
Well-posedness for the Navier-Stokes equations in critical mixed-norm Lebesgue spaces , Journal of Evolution Equations, 20(2020), 553-576 [ Author shared link ] [Journal article ] [arXiv ].
Weighted Calderon-Zygmund estimates for weak solutions of quasi-linear degenerate elliptic equations , Potential Analysis , 52 (2020), 393-425, DOI: 10.1007/s11118-018-9737-z , [Author shared link ][Journal article ], [arXiv ].
Free energy in mean field of Brownian particle (with X. Chen ), Discrete and Continuous Dynamical Systems-Series A , 39 (2019), no. 2, 747-769, [Journal article ], [preprint ].
Weighted \(W^{1,p}\)-estimates for weak solutions of degenerate elliptic equations with coefficients degenerate in one variable (with Tadele Mengesha ), Nonlinear Analysis , Vol. 179 (2019), 184-236 [Journal article ][arXiv ].
Interior gradient estimates for weak solutions of quasi-linear p-Laplacian type equations , Pacific journal of Mathematics , 297-1 (2018), 195-224 [Journal article ], [preprint ].
Gradient estimates for weak solutions of linear elliptic systems with singular-degenerate coefficients (with Dat Cao and Tadele Mengesha ), AMS Contemporary Mathematics , Nonlinear Dispersive Waves and Fluids , Volume 725, 2019, 13-33, [preprint ].
Lorentz estimates for weak solutions of quasi-linear parabolic equations with singular divergence-free drifts , Canadian Journal of Mathematics , Vol 71, Issue 4 (2019), 937-982, [Journal article ], [preprint ].
Weighted \(W^{1,p}\)-estimates for weak solutions of degenerate and singular elliptic equations (with Dat Cao and Tadele Mengesha ), Indiana University Mathematics Journal , Vol. 67, No. 6 (2018), 2225-2277, [arXiv ] [Journal article ].
Regularity estimates for BMO-weak solutions of quasi-linear equations with inhomogeneous boundary conditions , Nonlinear Differential Equations and Applications NoDEA , 49 pp., DOI: 10.1007/s00030-018-0501-2, [Journal article ], [preprint ].
Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations , Journal of Differential Equations 263 (2017) 8329-8361, [Journal article ], [arXiv ].
Local \(W^{1,p}\)-regularity estimates for weak solutions of parabolic equations with singular divergence-free drifts , Electronic Journal of Differential Equations , Vol. 2017 (2017), No. 75, pp. 1-22 [Journal article ], [arXiv ].
Advection control in parabolic PDE systems for competitive populations (with K. De Silva and S. Lenhart ), 23 pp., Discrete and Continuous Dynamical Systems, Series B , Vol. 22, Issue 3 (2017), 1049-1072, [Journal article ].
Asymptotic stability of solitary waves in generalized Gross-Neveu model (with Andrew Comech and Atanas Stefanov ), Annales de l'Institut Henri Poincare (C) Non Linear Analysis , Vol. 34, Issue 1 (2017), 157-196, [Journal article ], [arXiv ].
On small energy stabilization in the NLKG with a trapping potential (with Scipio Cuccagna , and Masaya Maeda ), Nonlinear Analysis , Vol. 146 (2016), 32-58, [Journal article ], [arXiv ].
Interior gradient estimates for quasi-linear elliptic equations (with Truyen Nguyen ), Calculus of Variations and Partial Differential Equations , Vol. 55, no. 3 (2016) [Journal article ] [arXiv ].
Local gradient estimates for degenerate elliptic equations (with Luan T. Hoang and Truyen V. Nguyen ), Advanced Nonlinear Studies . 16 (2016), no. 3, 479-489, [Journal article ], [arXiv ].
Gradient estimates and global existence of smooth solutions for a system of cross-diffusion equations (with Luan T. Hoang and Truyen V. Nguyen ), SIAM Journal on Mathematical Analysis , Vol. 47, Issue 3 (2015), 2122-2177 [Journal article ], [arXiv ].
Properties of generalized Forchheimer flows in porous media (with Thinh Kieu and Luan Hoang ), Problemy Matematicheskogo Analiza, volume 76 (2014), 133-194 (in Russian), and (English translation) Journal of Mathematical Sciences , vol. 202 (2014) No. 2, 259-332. [Journal article ], [arXiv ].
Stationary Navier-Stokes equations with critically singular external forces: Existence and stability results (with Nguyen Cong Phuc ), Advances in Mathematics 241 (2013), 137?161 [Journal article ], [arXiv ].
Optimal control of advection direction on reaction-diffusion population models (with H. Finotti and S. Lenhart ), Evolution Equations and Control Theory 1 (2012), no. 1, 81-107 [Journal article ], [Preprint ].
Normal form for symmetry breaking bifurcation in the Gross-Pitaevskii equation (with D.E. Pelinovsky ), Journal of Differential Equations 253 (2012), no. 10, 2796-2824, [Journal article ], [arXiv ].
Small Solutions of nonlinear Schrödinger equations near first excited states (with Kenji Nakanishi and Tai-Peng Tsai ), Journal of Functional Analysis 263 (2012), no. 3, 703-781 [Journal article ], [Preprint ].
A Remark on global existence of solutions of shadow systems , Zeitschrift für angewandte Mathematik und Physik 63 (2012), no. 2, 395-400, [Journal article ], [Preprint ].
Stable directions for degenerate excited states of nonlinear Schödinger equations (with Stephen Gustafson ) SIAM Journal on Mathematical Analysis , Vol. 43, Issue 4 (2011), pp. 1716-1758, [Journal article ], [arXiv ].
On global existence of solutions to a cross-diffusion system , Journal of Mathematical Analysis and Applications Vol 343, Issue 2, 15 July 2008, 826-834 [Journal article ], [IMA preprint ].
On global existence of solutions to Shigesada-Kawasaki-Teramoto cross-diffusion systems in domains of arbitrary dimensions , Proceedings of the American Mathematical Society 135 (2007), 3933-3941, [Journal article ], [Preprint ].
Lagrange multipliers for functions derivable along directions in linear subspace (with Le Hai An, Pham Xuan Du and Duong Minh Duc ), Proceedings of the American Mathematical Society , 133 (2005), no. 2, 595-604 [Journal article ].
Topological degree for a class of operators and applications (with Duong Minh Duc and Nguyen Hoang Loc ) Nonlinear Analysis Vol 57 (2004), Issue 4 [Journal article ].
Generalized zeros of operators and applications (with Duong Minh Duc and Nguyen Hoang Loc ) Vietnam Journal of Mathematics 32 (2004), Special Issue, pp. 87-96 [Journal article ].
Others/unpublished notes
Weighted estimates with $A_p$ and non $A_p$ weights: Developments and related questions , lectures notes for Workshop on PDE and related topics , VIASM , 07/2022.
On a class of divergence form linear parabolic equations with degenerate coefficients (with H. V. Tran ), (2021), [arXiv:2106.07637 ] (unpublished notes).
On a class of divergence form linear parabolic equations with degenerate coefficients (with H. V. Tran ), (2021), [arXiv:2106.07637 ] (un-published notes).
Ljusternik theorem for non-differentiable functionals , B.S. Thesis, University of Science , Ho Chi Minh City, Vietnam (2000).
On Global Existence of Solutions to a Cross-Diffusion System , Ph.D. Thesis, University of Minnesota , July 2007.
Updated on 03/15 of 2022 .
The theorem on the top of this page appears in [this paper] , which is a joint work with Hongjie Dong . Below is another formula which appears in [this paper]
$$\displaystyle{\int_{\Omega_T}|x_d^\alpha \nabla u(x,t)|^px_d^{-\alpha}dxdt +\lambda^{p/2}\int_{\Omega_T} |x_d^\alpha u(t,x)|^p x_d^{-\alpha}dxdt \leq N \int_{\Omega_T}|x_d^\alpha F(t,x)|^p x_d^{-\alpha} dxdt}.$$
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