# TUOC PHAN

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

Office: 205 Ayres Hall
Email : tphan2 (at) 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, The University of Tennessee-Knoxville. I was born and raised in a rice farm in Mekong Delta in Vietnam and I attended local schools there during my childhood. I went to college in Ho Chi Minh City (known as Sai Gon) and received the bachelor of science degree majoring in mathematics in 2000 from the University of Science. Gratefully, I was awarded a teaching assistantship by the University of Minnesota (Twin Cities-Minnesota-US) and from which I obtained my Ph.D. degree in 2007. I was a postdoc at The University of British Columbia (2007 - 2010), and The University of Tennessee - Knoxville (2010 - 2012). I became an assistant professor at The University of Tennessee - Knoxville in 08/2012, and an associate professor in 08/2018. 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. Growing up and working on the farm, I have a true love for gardening. I grow a tremendous amount of organic veggies and fruits every summer. I am fond of walking, biking, and hiking, and I enjoy hanging with friends at coffee shops, talking about mathematics. The math formula above 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 and published here by the AMS. Additional information about me can be found in this article (in Vietnamese) published by Vietnamnet, and also in an interview video (in Vietnamese) in the #toan0mau series produced by professor H. V. Tran.

## Preprints

• On trace theorems for weighted mixed norm Sobolev spaces and applications, (2022), [arXiv:2205.04941].
• A boundary layer problem in non-flat domains with measurable viscous coefficients (with Dario V. Castillo), (2022), [Preprint], submitted.
• Degenerate linear parabolic equations in divergence form on the upper half space (with H. Dong and H. V. Tran), (2021), [arXiv:2107.08033], submitted.
• On a class of divergence form linear parabolic equations with degenerate coefficients (with H. V. Tran), (2021), [arXiv:2106.07637] (unpublished notes).
• 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].

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

1. On parabolic and elliptic equations with singular or degenerate coefficients (with H. Dong), Indiana University Mathematics Journal, accepted, [arXiv].
2. Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients (with H. Dong and D. Kim), Communications in Partial Differential Equations, accepted, arXiv:1910.00380, [arXiv].
3. On higher integrability estimates for elliptic equations with singular coefficients (with J. Foldes), Funkcialaj Ekvacioj, accepted, [arXiv].
4. 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].
5. 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].
6. 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].
7. 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].
8. 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].
9. 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].