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BROWN UNIVERSITY |
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Toan Nguyen
Prager Assistant Professor
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Boundary
Layers
Shock Waves
Coherent
Structures
Kinetic Theory
Other
Works |
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Research Interests
Partial
differential equations, Fluid dynamics (e.g.,
Incompressible Navier-Stokes),
Systems of conservation laws (Compressible Navier-Stokes
and MHD; Radiative hyperbolic-elliptic systems), Kinetic
theory (Vlasov - Maxwell; Boltzmann), Stability theory
and Dynamical system, Traveling wave solutions, Viscous
shock waves and Boundary layers.
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Boundary
layers in Fluid Dynamics:
In fluid dynamics, one of the most classical and
challenging issues is to completely understand the
dynamics of fluid flows past solid bodies (e.g.,
aircrafts, ships, etc...), especially when the viscosity
or the inverse of the physical Reynolds number of the
fluid is small. The theory of boundary layers was
introduced and developed to simplify the dynamics of a
viscous fluid near the boundary by dividing it into two
regions: one near the boundary (or so-called the
boundary layer region), where viscosity is significant,
and a second one away from the boundary where the fluid
is essentially inviscid. One of the great achievements
was then the discovery of the boundary layer equations,
which significantly simplify the Navier-Stokes equations
near the boundary (for example, the pressure is no
longer an unknown quantity inside the boundary layer,
and is completely determined from the outer Euler flow
via the well-known Bernoulli's law). Since then, study
of boundary layer solutions has become a physical and
practical problem that greatly interests many
physicists, and especially, aerodynamicists. Below are
my contributions to the theory of boundary layers.
1. Prandtl boundary layers
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A
note on the Prandtl boundary layers (with Y. Guo)
Comm. Pure Appl.
Math. to appear. pdf file.
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Remarks on the ill-posedness of the Prandtl equation
(with D. Gérard–Varet)
Asymptotic Analysis, to appear. pdf file.
2. Inflow/outflow boundary layers
In aerodynamics, suction and blowing implementation is
widely used on airfoils as a very effective means to
reduce the friction drag, stabilize laminar boundary
layers, and delay the transition from laminar to
turbulent flows. This implementation creates nonzero
normal velocity on the boundary, and thus changes the
characteristic of boundary layer solutions as compared
to the classical no-slip boundary conditions. The
corresponding solutions are often called suction/blowing
or inflow/outflow boundary layers. Physically, their
stability properties are important, for instance, in
controlling the boundary layers and delaying the
phenomenon of boundary layer separations. My work in
this research area includes
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On asymptotic stability of
noncharacteristic viscous boundary layers.
SIAM J. Math. Analysis, 42 (2010), no. 3, 1156–1178 pdf file
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Long-time stability of
multi-dimensional noncharacteristic viscous boundary
layers (with K. Zumbrun).
Communications in Mathematical
Physics, 299 (2010), no. 1, 1–44. pdf file
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Long-time stability of
large-amplitude noncharacteristic boundary layers of
general hyperbolic-parabolic conservation laws (with
K. Zumbrun).
J. Maths. Pures et Appliquées, (9) 92
(2009), no. 6, 547–598. pdf file
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Spectral stability of
noncharacteristic isentropic Navier--Stokes boundary
layers (with N. Costanzino, J. Humpherys, K. Zumbrun).
Arch. Ration. Mech. Anal. 192 (2009), no. 3, 537--587. pdf file
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Stability of
shock waves:
Hyperbolic conservation
laws are systems of PDEs that include many of the most
fundamental physical principles such as conservations of
mass, momentum, and energy. In such an ideal
(hyperbolic) model, shock waves are known to exist, and
determination of their physical admissibility is the
central issue in theory of conservation laws. Relating
to the so-called entropy admissible condition,
hyperbolic-parabolic (or viscous) conservation laws are
introduced as an approximation of the hyperbolic system
with small dissipation or regularization (such as
viscosity and heat dissipation in context of gas
dynamics or magnetohydrodynamics). There are natural
traveling waves in the latter system that are associated
with inviscid shock waves. These are called viscous
shock profiles, and their stability and dynamics play a
crucial role in studies of validity of the viscous
approximations, convergence in the inviscid limit, or
convergence of numerical approximation schemes.
1. Viscous planar shocks
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Stability of multi-dimensional
viscous shocks for symmetric systems with variable
multiplicities,
Duke Math. Journal, 150
(2009), no. 3, 577–614.
pdf file
2. Radiative shock waves
There is another common
regularization in the theory of hyperbolic conservation
laws that consists of the inviscid system coupled with
an elliptic system. Such a coupled system (similar to
the standard Euler--Poisson model used in plasma
physics) is widely accepted, for example, in the theory
of radiative hydrodynamics to model an inviscid
compressible gas interacting with radiation through
energy exchanges. Traveling waves or so-called radiative
shock profiles associated with inviscid shocks are also
known to exist in this context, and their stability is
of great physical interest. My work in this research
area includes
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Multi-dimensional stability of Lax shocks in
hyperbolic-elliptic coupled systems.
J. Diff. Eqs., to appear. pdf file
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Stability of radiative shock profiles
for hyperbolic-elliptic coupled systems (with R.
Plaza and K. Zumbrun).
Physica D,
239 (2010), no. 8, 428–453. pdf
file
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Stability of scalar radiative shock
profiles (with C. Lattanzio, C. Mascia, R. Plaza,
and K. Zumbrun),
SIAM J. Math. Analysis,
41 (2009/10), no. 6, 2165–2206.
pdf file
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Coherent structures:
Patterns are ubiquitous in nature, and many interesting
patterns can be found as spatially periodic traveling
waves or wave trains of certain partial differential
equations (e.g., reaction-diffusion systems). A coherent
structure or defect is formed by two co-existing
patterns which are separated in an organized fashion by
an interface or so-called core of the defect. Defects
can be found in many biological, chemical, and physical
experiments. Mathematically, they can be described as
special solutions of the PDEs that are time-periodic in
an appropriate moving frame and spatially asymptotic at
infinities to (generally different) wave trains.
Motivated by the mentioned practical applications, I am
interested in investigating the formation of these
defects, their stability and nonlinear dynamics, and
especially, how stable they can be under small
disturbances in nature. My contribution to this research
area includes
- Toward
nonlinear stability of sources via a modified
Burgers equation (with M. Beck, B. Sandstede, and K.
Zumbrun)
Physica D, to
appear. pdf file.
Kinetic Theory:
Updating....
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Other works:
1.
Boundary-layers interactions
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Boundary layers interactions in the plane parallel
incompressible flows (with Franck Sueur)
Preprint 2011. pdf file
2. ill-posedness
via a semiclassical approach
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Instantaneous loss of hyperbolicity and ill-posedness
for quasi-linear first-order systems (with N. Lerner
and B. Texier)
Manuscript 2010.
3. Regularity theory for
coupled elliptic and parabolic systems.
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Designed
by Thanh Tran © 2011 |
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