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**Research interests:**

I work at the interface between physics and the theory of partial differential equations. In recent years, I have worked on the Prandtl's boundary layer theory, the kinetic theory of plasmas, the dynamics of coherent structures, the stability theory of nonlinear waves, and the weak turbulence theory. See below for details.

**Publications:**

**0. Books:**

- Stability of Prandtl's boundary layers, with E. Grenier, 300 pp., preliminary review for Springer Books, 2017.

**I. Boundary layers in fluids:**

- On nonlinear instability of Prandtl's boundary layers: the case of Rayleigh's stable shear flows (w/ E. Grenier)

pdf file, submitted. - Sublayer of Prandtl boundary layers (w/ E. Grenier)

pdf file, submitted. - Sharp bounds on linear semigroup of Navier Stokes with boundary layer norms (w/ E. Grenier)

pdf file, submitted. - Green function for linearized Navier-Stokes around a boundary layer: near critical layers (w/ E. Grenier)

pdf file, submitted. - Green function for linearized Navier-Stokes around a boundary layer: away from critical layers (w/ E. Grenier)

pdf file, submitted. - The vanishing viscosity limit for 2D Navier-Stokes in a rough domain (w/ D. Gerard-Varet, C. Lacave, and F. Rousset)

pdf file, submitted. - Prandtl boundary layer expansions of steady Navier-Stokes over a moving plate (w/ Y. Guo)

Annals of PDEs, to appear - Remarks on the inviscid limit for compressible flows (w/ C. Bardos)

Contemporary Mathematics, AMS, 2016 - Spectral instability of characteristic boundary layer flows (w/ E. Grenier and Y. Guo)

Duke Math J., 165 (2016), no. 16, 3085-3146 - Spectral instability of symmetric shear flows in a two-dimensional channel (w/ E. Grenier and Y. Guo)

Advances in Math., 292 (2016), pp. 52--110. - Spectral stability of Prandtl boundary layers: an overview (w/ E. Grenier and Y. Guo)

Analysis (Berlin) 35 (2015), no. 4, 343-355. - Boundary layers interactions in the plane parallel incompressible flows (w/ F. Sueur)

Nonlinearity, 25 (2012) 3327--3342. - Remarks on the ill-posedness of the Prandtl equation (w/ D. Gerard-Varet)

Asymptotic Analysis, 77 (2012), no. 1-2, 71--88. - A note on the Prandtl boundary layers (w/ Y. Guo)

Comm. Pure Appl. Math., 64 (2011), no. 10, 1416--1438.

**II. Frameworks for ill-posedness caused by spacetime ellipticity (Lax-Mizohata's theorem)**

- The onset of instability in first order systems (w/ N. Lerner and B. Texier)

J. Euro Math Society, to appear - Illposedness of the hydrostatic Euler and singular Vlasov equations (w/ Han-Kwan)

Arch. Ration. Mech. Anal., 221 (2016), no. 3, 1317-1344.

**III. Kinetic theory of plasmas:**

- Long time estimates of the Vlasov-Maxwell system in the non-relativistic regime (w/ D. Han-Kwan and F. Rousset)

pdf file, submitted. - The Maxwell-Boltzmann approximation for ion kinetic modeling (w/ C. Bardos, F. Golse, and R. Sentis)

pdf file, submitted. - Instabilities in the mean field limit (w/ Han-Kwan)

Journal of Statistical Physics, 162 (2016), no. 6, 1639-1653. - Illposedness of the hydrostatic Euler and singular Vlasov equations (w/ Han-Kwan, also listed above)

Arch. Ration. Mech. Anal., 221 (2016), no. 3, 1317-1344. - Instability of Vlasov-Maxwell systems in the classical and quasineutral limits (w/ Han-Kwan)

SIAM J. Math. Anal., 48 (2016), no. 5, 3444-3466 - Global magnetic confinement for the 1.5D Vlasov-Maxwell system (w/ T. Nguyen and W. Strauss)

Kinetic and Related Models, 8 (2015), no. 1, 153-168. - Stability Analysis of a Hot Plasma in a Solid Torus (w/ W. Strauss)

Arch. Ration. Mech. Anal., 211 (2014), no. 2, 619-672. - Stability analysis of collisionless plasmas with specularly reflecting boundary (w/ W. Strauss)

SIAM J. Math. Anal., 45(2013), no. 2, 777--808.

**IV. Quantum kinetic theory:**

- On a quantum Boltzmann type equation in Zakharov's wave turbulence theory (w/ M.-B. Tran)

pdf file, submitted. - Uniform in time lower bound for solutions to a Quantum Boltzmann of bosons (w/ M.-B. Tran)

pdf file, submitted.

**V. Dynamics of coherent structures:**

- Nonlinear stability of source defects in oscillatory media (w/ M. Beck, B. Sandstede, and K. Zumbrun)
- Nonlinear stability of source defects in the complex Ginzburg-Landau equation (w/ M. Beck, B. Sandstede, and K. Zumbrun)

Nonlinearity, 27 (2014) 739-786 - Toward nonlinear stability of sources via a modified Burgers (w/ M. Beck, B. Sandstede, and K. Zumbrun)

Physica D, 241 (2012), no. 4, 382--392.

**VI. Dynamics of shock waves and noncharacteristic boundary layers:**

- Multi-dimensional stability of Lax shocks in hyperbolic-elliptic coupled systems

Journal of Differential Equations, 252 (2012), no. 1, 382--411. - On asymptotic stability of noncharacteristic viscous boundary layers

SIAM J. Math. Anal., 42 (2010), no. 3, 1156--1178 - Stability of multi-dimensional viscous shocks for symmetric systems with variable multiplicities

Duke Math. J. 150 (2009), no. 3, 577--614. - Long-time stability of multi-dimensional noncharacteristic viscous boundary layers (w/ K. Zumbrun)

Comm. Math. Phys., 299 (2010), no. 1, 1--44. - Stability of radiative shock profiles for hyperbolic-elliptic coupled systems (w/ R. Plaza and K. Zumbrun)

Physica D. 239 (2010), no. 8, 428--453. - Stability of scalar radiative shock profiles (w/ C. Lattanzio, C. Mascia, R. Plaza, and K. Zumbrun)

SIAM J. Math. Anal. 41 (2009/10), no. 6, 2165--2206. - Long-time stability of large-amplitude noncharacteristic boundary layers of general hyperbolic-parabolic conservation laws (with K. Zumbrun)

J. Math. Pures Appl. (9) 92 (2009), no. 6, 547--598. - Spectral stability of noncharacteristic isentropic Navier--Stokes boundary layers (with N. Costanzino, J. Humpherys, and K. Zumbrun)

Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537--587.

**VII. Elliptic and parabolic regularity theory:**

- Regularity and coexistence problems for strongly coupled elliptic systems (with D. Le and L. Nguyen)

Indiana Univ. Math. J., 56 (2007), no. 4, 1749--1791 - Global attractors and uniform persistence for cross diffusion parabolic systems (with D. Le)

Dyn. Sys. and Apps. 16 (2007), no. 2, 361--377. - Everywhere regularity for degenerate cross diffusion systems (with D. Le)

Comm. in PDEs, 31 (2006), no. 1-3, 307--324. - Persistence for a class of triangular cross diffusion parabolic systems (with D. Le)

Adv. Non. Stud., 5 (2005), no. 4, 493--514. - Global existence for a class of triangular parabolic systems on domains of arbitrary dimension (with D. Le)

Proc. of AMS, 133(2005), 1985-1992. - Shigesada-Kawasaki-Teramoto model on higher dimensional domains (with D. Le and L. Nguyen)

Elec. J. Diff. Eqs., 2003, No. 72.

**VIII. Others: **

- Topography influence on the Lake equations in bounded domains (with C. Lacave and B. Pausader)

J. Math. Fluid Mech., 16 (2014), no. 2, 375--406.

Back to Toan Nguyen's homepage.

Visit Toan Nguyen's blog: Snapshots in Mathematics!.