# TUOC PHAN

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

$\displaystyle{\left[\int_{-\infty}^T\left(\int_{\mathbb{R}^d_+}\big(|u|^p + |Du|^p\big)x_d^\gamma dx\right)^{q/p} dt \right]^{1/q} \leq N\left[\int_{-\infty}^T \left(\int_{\mathbb{R}^d_+} \big(|F|^p +|f|^p\big) x_d^\gamma dx\right)^{q/p} dt \right]^{1/q}}.$

Welcome to Phan's homepage! I am an associate professor at the Department of Mathematics, University of Tennessee-Knoxville. I received my B. S. degree in mathematics in 2000 from the University of Science (Ho chi Minh City-Vietnam), and my Ph.D. in 2007 from the University of Minnesota (Twin Cities-Minnesota-US). My research is in partial differential equations. I study existence, uniqueness, and regularity estimates of solutions. I also work on nonlinear dynamics of solutions, and optimal control problems in mathematical biology. I love outdoor activities (walking, biking, and hiking), and enjoy hanging with my family and friends at local coffee shops. I like gardening and regularly practice Taekwondo. The above math formula is an a-priori estimate of the unknown solution $u$ of a class singular-degenerate coefficient PDEs. This formula appears in my recent work, which is a joint work with Hongjie Dong.

## Preprints

• On a class of divergence form linear parabolic equations with degenerate coefficients (with Hung Vinh Tran), (2021), [arXiv:2106.07637].
• 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].
• On Masuda uniqueness theorem for Leray-Hopf weak solutions in mixed-norm spaces (with T. Robertson), submitted (2021).
• On parabolic and elliptic equations with singular or degenerate coefficients (with H. Dong), submitted, [arXiv].
• Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients (with H. Dong and D. Kim), arXiv:1910.00380, submitted, [arXiv].
• On higher integrability estimates for elliptic equations with singular coefficients (with J. Foldes), submitted, [arXiv].

## Recent Accepted/Published Papers (click here for the full list of Phan's publication)

1. Parabolic and elliptic equations with singular or degenerate coefficients: the Dirichlet problem (with H. Dong), Transactions of the American Mathematical Society, accepted, [arXiv].
2. 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].
3. 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].
4. 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].
5. 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].
6. 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].
7. 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].