# TUOC PHAN'S PUBLICATION

Tuoc Phan
Associate Professor
Department of Mathematics
University of Tennessee - Knoxville

Office: 205 Ayres Hall
Email : phan (at) math dot utk dot edu
Phone: 865-974-4329

$$\displaystyle{\int_{\Omega_T}|x_d^\alpha \nabla u(x,t)|^px_d^{-\alpha}dxdt +\lambda^{p/2}\int_{\Omega_T} |x_d^\alpha u(t,x)|^p x_d^{-\alpha}dxdt \leq N \int_{\Omega_T}|x_d^\alpha F(t,x)|^p x_d^{-\alpha} dxdt}.$$

## Preprints

1. Degenerate linear parabolic equations in divergence form on the upper half space (with H. Dong and H. V. Tran), (2021), [arXiv:2107.08033], submitted.
2. Weighted mixed-norm $L_p$ estimates for equations in non-divergence form with singular coefficients: the Dirichlet problem (with H. Dong), submitted (2021), [arXiv:2103.08033].
3. On parabolic and elliptic equations with singular or degenerate coefficients (with H. Dong), submitted, [arXiv].
4. Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients (with H. Dong and D. Kim), arXiv:1910.00380, submitted, [arXiv].

## Published Papers (peer reviewed)

1. On higher integrability estimates for elliptic equations with singular coefficients (with J. Foldes), Funkcialaj Ekvacioj, accepted, [arXiv].
2. 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].
3. 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].
4. 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].
5. 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].
6. 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].
7. 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].
8. 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].
9. 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].
10. 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].
11. 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].
12. 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].
13. 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].
14. 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].
15. 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].
16. 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].
17. 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].
18. 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].
19. Regularity gradient estimates for weak solutions of singular quasi-linear parabolic equations, Journal of Differential Equations 263 (2017) 8329-8361, [Journal article], [arXiv].
20. 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].
21. 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].
22. 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].
23. 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].
24. 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].
25. 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].
26. 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].
27. 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].
28. 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].
29. 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].
30. 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].
31. 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].
32. 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].
33. 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].
34. 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].
35. 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].
36. 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].
37. 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].
38. 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

1. On a class of divergence form linear parabolic equations with degenerate coefficients (with H. V. Tran), (2021), [arXiv:2106.07637] (un-published notes).
2. Ljusternik theorem for non-differentiable functionals, B.S. Thesis, University of Science, Ho Chi Minh City, Vietnam (2000).
3. On Global Existence of Solutions to a Cross-Diffusion System, Ph.D. Thesis, University of Minnesota, July 2007.