BROWN UNIVERSITY
 
Toan Nguyen
Prager Assistant Professor
 
   
  Boundary Layers
Shock Waves
Coherent Structures Kinetic Theory
Other Works		  


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.
 

    


 
     


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

  • A note on the Prandtl boundary layers (with Y. Guo)
    Comm. Pure Appl. Math. to appear. pdf file.

  • 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

  
   

 

  
     
     
 

 

  
       
     


 
 


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 

  • 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

  • Multi-dimensional stability of Lax shocks in hyperbolic-elliptic coupled systems.
    J. Diff. Eqs., to appear. pdf file

  • 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

  • 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


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....

 

Other works:

1. Boundary-layers interactions

2. ill-posedness via a semiclassical approach

  • 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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