Aimin Huang

Aimin Huang

## Resources

=== Academic websites:
*[[http://www.ams.org/mathscinet/index.html|AMS MathSciNet]] (Note: See Mathematical review of your paper)
*[[http://apps.webofknowledge.com/|Web of Knowledge]]
**[[http://admin-apps.webofknowledge.com/JCR/JCR|Journal Citation Reports|]] (Note: Create a Session first by following the instruction)
*[[http://genealogy.math.ndsu.nodak.edu/|Mathematics Genealogy Project]] (Note: See your academic grandfather)
*[[http://www.math.ntnu.no/conservation/|Conservation Laws]] (Note: Papers about new and recent results concerning)
*[[http://arxiv.org/archive/math/|arXiv.org]] (Note: most new results will appear here)
*[[http://www.mathreference.com/|Math Reference Project]]
*[[http://www.ima.umn.edu/videos/|Institute for Mathematics and its Applications (IMA)]]
*[[http://www.claymath.org/programs/summer_school/2008/|Evolution Equation]]
=== Mathematicians' webpages:
*[[http://terrytao.wordpress.com/|Terence Tao's Math Blog]]
*[[http://www.math.uconn.edu/~kconrad/blurbs/|Algebra: Group-Ring-Field]]
*[[http://legacy.orie.cornell.edu/~aslewis/publications/01-hyperbolic.pdf|Hyperbolic Polynomial and Convex Analysis]]

## Research

My doctoral research activity starts with the analysis of the Partial Differential Equations (PDEs) arising from geophysical fluid dynamics, with focus on the [[shallow water equations]], and the [[primitive equations]] in non-smooth domains. These equations have wide applications, such as weather prediction, and oceanography (oceanic modeling). Furthermore, I extend my research to mixed initial and boundary value problems of hyperbolic PDEs in non-smooth domains.
Recently, I am working on the global attractor of 2D Boussinesq system with variable viscosity and diffusivity (see [[http://arxiv.org/abs/1403.1351|arXiv:1403.1351]]), the exact boundary controllability of the linear hyperbolic systems, and the global well-posedness of the 2D Euler-Boussinesq system in planar nonsmooth domains.
I am also interested in Numerical Simulation, Stochastic Processes, Harmonic analysis, Schr$\ddot{\text{o}}$dinger equations.
My research statement is [[http://pages.iu.edu/~aimhuang/Aimin-RS.pdf|here]], see also [[Highlights| highlights of my research|color=red]].

{{profile.jpg|align=right alt=mypicture title=mypicture id=profilepicture}}
I am Aimin Huang (黄爱民), currently a graduate student in the [[http://math.indiana.edu|math department]] of [[http://www.indiana.edu|Indiana University]], and also a member of the [[http://www.indiana.edu/~iscam/|Institute for Scientific Computing and Applied Mathematics]]. My thesis advisor is Professor [[http://pages.iu.edu/~temam|Roger Temam]].
I successfully defensed on April 2014 and I am Dr. Huang now. Here is my [[Local:Defense-IU.pdf|defense talk]]
Here is my [[Local:Aimin-CV.pdf|curriculum vitae]], [[Local:Aimin-RS.pdf|research statement]], and my research [[Highlights|highlights|color=red;font-weight=bold]].
=== Contact
Department of Mathematics, Indiana University, \\
831 East 3rd St, Rawles Hall, Bloomington, IN, 47405
|//Phone (office)//:| (812) 855-0128|
|//Email//:| aimhuang(at)indiana.edu|
|//Website//:| [[http://www.7starsea.com|http://www.7starsea.com]]|
=== Educations
*Ph.D., Department of Mathematics, Indiana University, Bloomington, Indiana, May 2014
**//Major//: Applied Mathematics, //Minor//: Scientific Computing
**//Advisor//: [[http://mypage.iu.edu/~temam|Roger Temam]]
*M.A., Department of Mathematics, Indiana University, Bloomington, Indiana, 2011
*B.S., Department of Mathematics, University of Science and Technology of China, Hefei, China, 2008
**//Major//: Mathematics
**//Advisor//: [[http://math.ustc.edu.cn/Ch/faculty/index.php?accord=faculty&f_name=%C2%E9%CF%A3%C4%CF|Xinan Ma]]

## Hyperbolic Partial Differential Equations

=== Books about Hyperbolic PDEs:
*Benzoni-Gavage, Sylvie and Serre, Denis, [[http://www.oxfordscholarship.com/oso/public/content/maths/9780199211234/toc.html|Multi-dimensional Hyperbolic Partial Differential Equations: First-order Systems and Applications]]
*Lars Garding - [[http://www.springerlink.com/content/538q5q871205hw85/fulltext.pdf|Linear Hyperbolic PDEs with Constant Coefficients]]
*Peter D. Lax, [[http://www.ams.org/bookstore-getitem/item=cln-14|Hyperbolic Partial Differential Equations]]
=== International Conference on Hyperbolic Problems: Theory, Numerics, Applications:
*14th, June 25th-29th, 2012, University of Padua, Italy, [[http://www.hyp2012.eu/]]
*13th, June 15th-19th, 2010, Beijing, China, [[http://escience.amss.ac.cn/dct/Wiki.jsp?page=Hyp2010]]

## shallow water equations

The shallow water equations (SWEs) is also called Saint Venant equations.
=== Model of SWEs
Shallow water equations can be used to model Rossby and Kelvin waves in the atmosphere, rivers, lakes and oceans as well as gravity waves in a smaller domain (e.g. surface waves in a bath). It can be also used to model the propagation of disturbances (e.g. tsunamis) in water and other incompressible ﬂuids, or to describe the horizontal structure of an atmosphere. For example, the depth of the Indian Ocean is two or
three kilometers which we usually do not think of being shallow, but the devastating tsunami in the Indian Ocean on December
26, 2004 involved waves that were dozens or hundred of kilometers long. So the
shallow water approximation provides a reasonable model in this situation (see [[http://www.mathworks.com/moler/exm/chapters/water.pdf]]).
=== Results on SWEs
The 2-D fully nonlinear inviscid SWEs in the Eulerian variables read
\begin{equation}
(1.1)\hspace{10pt}
\begin{cases}
u_t+uu_x + vu_y + g\phi_x -fv= 0, ~\\
v_t+uv_x + vv_y + g\phi_y + fu= 0, ~\\
\phi_t+u\phi_x + v\phi_y + \phi(u_x+v_y) = 0;
\end{cases}
\end{equation}
here
$u$ and $v$ are the two horizontal components of the velocity, $\phi$ is the height of the water (or atmosphere), and $g$ is the gravitational acceleration, $f$ is the Coriolis parameter due to the self rotation of earth.
The first and second equations (1.1) are derived from the equations of conservation of horizontal momentum, and the third one expresses the conservation of mass.
I have worked on studying the initial and boundary value problems associated with the inviscid SWEs. The major motivation is that these problems are related to the problems of Limited Area Models in geophysical fluid dynamics. In order to understand the fully nonlinear inviscid SWEs, in [ [[Publications|1]] ], I started with the 1d supercritical SWEs in an interval, and then continued in [ [[Publications|2]] ] to study the linearized 2d inviscid SWEs in a rectangle, and finally in [ [[Publications|3]], [[Publications|4]] ] studied two modes of the nonlinear 2d inviscid SWEs, where the supercritical mode is studied in a rectangle and the subcritical mode in a channel with periodicity.
=== Derivation of SWEs
==== 1. Newtonian formulation of SWEs
The shallow water equations are a set of [[Hyperbolic Partial Differential Equations]] that describe the flow below a pressure surface in a fluid. The basic assumption for SWEs is that the horizontal length scale is much greater than the vertical length scale, which will imply that the horizontal velocity field is constant throughout the depth of the fluid. The rigorous derivation of SWEs are based on conservation of mass and conservation of momentum.
See [[http://en.wikipedia.org/wiki/Shallow_water_equations|Wikipedia]] for more details and the following for the rigorous derivation of SWEs:
*http://users.ices.utexas.edu/~arbogast/cam397/dawson_v2.pdf
*http://physics.nmt.edu/~raymond/classes/ph332/notes/shallowgov/shallowgov.pdf
==== 2. Hamiltonian formulation of SWEs
come soon

## Publications

**Note**: Please feel free to request the articles (pdf) that have published/accepted.
#[HPT11] A. Huang, M. Petcu and R. Temam, The one-dimensional supercritical [[shallow water equations]] with topography, Annals of the University of Bucharest (Mathematical Series), **2 (LX)** (2011), 63-82. [[http://fmi.unibuc.ro/ro/anale/matematica/1_2011/huang_petcu_temam.pdf|link]]
#[HT12a] A. Huang and R. Temam, The linearized 2d inviscid [[shallow water equations]] in a rectangle: boundary conditions and well-posedness, Arch. Rational Mech. Anal. **211** (2014), no. 3, [[http://dx.doi.org/10.1007/s00205-013-0702-0|10.1007/s00205-013-0702-0]], see also [[http://arxiv.org/abs/1209.3194|arXiv:1209.3194]].
#[HPT12] A. Huang, M. Petcu and R. Temam, The nonlinear 2d supercritical inviscid [[shallow water equations]] in a rectangle, Asymptotic Analysis, **93** (2015), 187-218.
#[HT12b] A. Huang and R. Temam, The nonlinear 2d subcritical inviscid [[shallow water equations]] with periodicity in one direction, Communications on Pure and Applied Analysis, **13** (2014), no. 5, 2005-2038, [[http://dx.doi.org/10.3934/cpaa.2014.13.2005|DOI:10.3934/cpaa.2014.13.2005]].
#[HT13] A. Huang and R. Temam, The linear [[Hyperbolic Partial Differential Equations|hyperbolic initial and boundary value problems]] in a domain with corners, Discrete and Continuous Dynamical System - Series B, **19** (2014), no. 6, 1627-1665, [[http://dx.doi.org/10.3934/dcdsb.2014.19.1627|DOI:10.3934/dcdsb.2014.19.1627]], see also [[http://arxiv.org/abs/1310.5757|arXiv:1310.5757]].
#[HP13] A. Huang and D. Pham, The evolution of the semi-linear hyperbolic equation in a bounded domain, Asymptotic Analysis, **84** (2013), 123–146. [[http://dx.doi.org/10.3233/asy-131168|DOI:10.3233/ASY-131168]].
#[Hua13] A. Huang, Existence of solution for linear hyperbolic initial-boundary value problems in a rectangle, **accepted** by Applicable Analysis, [[http://dx.doi.org/10.1080/00036811.2014.957193|DOI:10.1080/00036811.2014.957193]]. [[http://www.tandfonline.com/eprint/A4cQuN7JYX52YDB9MCXI/full|Download article]].
#[CHKTZ] M. Coti Zelati, A. Huang, I. Kukavica, R. Temam, and M. Ziane, The primitive equations of the atmosphere in presence of vapor saturation, Nonlinearity, **28** (2015), no. 3, 625-668, [[http://dx.doi.org/10.1088/0951-7715/28/3/625|DOI:10.1088/0951-7715/28/3/625]], see also [[http://arxiv.org/abs/1406.3165|arXiv:1406.3165]].
#[BH14] A. Bousquet and A. Huang, Finite volume approximation of the [[shallow water equations]] in hyperbolic mode, International Journal of Numerical Analysis and Modeling, **11** (2014), no. 4, 816--840.
#[Hua14a] A. Huang, The global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity, Nonlinear Analysis Series A: Theory, Methods & Applications, **113** (2015), 401--429, see also [[http://arxiv.org/abs/1403.1351|arXiv:1403.1351]].
#[Hua14b] A. Huang, The 2D Euler-Boussinesq equations in planar polygonal domains with Yudovich’s type data, Communications in Mathematics and Statistics, **2** (2014), no. 3-4, 369--391, see also [[http://arxiv.org/abs/1405.2631|arXiv:1405.2631]].
#[HT15] A. Huang and R. Temam, The 2d nonlinear fully hyperbolic inviscid shallow water equations in a rectangle, Journal of Dynamics and Differential Equations, 2015, [[http://dx.doi.org/10.1007/s10884-015-9507-1|DOI:10.1007/s10884-015-9507-1]]

## Teaching

==== Teaching Assistant at Indiana University Bloomington:
#Assistant for Calculus III M311, Spring 2014
# Instructor for Pre-Calculus Mathematics [[http://mypage.iu.edu/~aimhuang/m25.html|M025]], Fall 2012.
# Assistant for Finite Mathematics M118, Fall 2011 and Fall 2012.
# Grader for Differential Geometry M435, Spring 2012.
# Instructor for [[http://mypage.iu.edu/~aimhuang/algebra-2011.html|Algebra in the JumpStart]] program for Ph.D. qualifying exams, Summer 2011.
# Assistant for Calculus M211-M212, Fall 2010, Spring 2011 and Fall 2013.
# Grader for Numerical Analysis M472, Spring 2010.

## Page Not Found

The page you requested does not exist! Please goto [[About me|Homepage]].

## primitive equations

The primitive equations (PEs) are a set of nonlinear differential equations governing the motion of the atmosphere and the ocean. They are derived from the conservation equations from physics, namely conservation of mass, momentum, energy and some other quantities as humidity and salinity, with the hydrostatic assumption in the vertical direction. This assumption is due to the smallness of the aspect ratio between the vertical and horizontal length scales, and is rigorously justified in \cite{AG01}. The (viscous) primitive equations of the atmosphere read
\begin{align}
&\f{\p v}{\p t} + \nabla_{v} v + \omega \f{\p v}{\p p} + f k\times v + \nabla \Phi - L_{v} v = F_{ v},~\\
&\f{\p\Phi}{\p p} + \f{RT}{p}=0.~\\
&\text{div}\, v + \f{\p\omega}{\p p}=0,~\\
&\f{\p T}{\p t} + \nabla_{v}T + \omega \f{\p T}{\p p} - \f{R\overline T}{c_p p}\omega - L_T T =F_T,~\\
&\f{\p q}{\p t} + \nabla_{ v}q + \omega \f{\p q}{\p p} - L_q q = F_q,~\\
&p=R\rho T.
\end{align}
Here$u=(v,\omega)$ is the three-dimensional velocity vector, $\rho, p, T$ are the density, pressure and temperature, and $q$ is the humidity (measuring the amount of vapor in the air); $f$ is the Coriolis force parameter, $k$ is the unit vector in the direction of the poles (from south to north); $L_{v}$, $L_T$ and $L_q$ are the Laplace operators, with suitable eddy viscosity coefficients, for example
$$L_{v}v = \mu_{v}\Delta v +\nu_{ v}\f{\p}{\p p}[ (\f{gp}{R\overline T})^2 \f{\p v}{\p p}];$$
$F_T$ corresponds to the heating of the sun, whereas $F_{v}$ and $F_q$ which vanish in reality, are added here for mathematical generality.

## Skills

=== Language skills
* Chinese: mother-tongue
* English: ﬂuent in written and spoken
=== Computer skills
*Experienced with Hadoop Cloud Framework
*Experienced with Scientific Computing Languages: C/C++, Java, Fortran, and Softwares: MatLab, Mathematica
*Experienced with Linux/Unix Operating System, MySQL Database, and Python, Shell Script, and Regular Expression
*Proficient in Web Development with XAMPP, PHP (Zend Framework), Javascript (jQuery, Prototype, Mathjax), Ajax, JSON, HTML/CSS, Wiki (Creole)

## Awards

==== Indiana University Bloomington (IUB)
# National Science Foundation Award, Sponsor: Prof. Roger Temam, Summer 2012, Summer 2013, Summer 2014
# The Glenn Schober Memorial Travel Award, 2014
# The Award for Excellence in Research, 2013
# NSF Graduate Student Fellowship, Sponsor: Prof. Roger Temam, Spring 2013
# IU College of Arts & Sci. Associate Instructor and Fee Scholarships, 2008-2014
# IU Graduate Student Travel Award, 2012
# National Science Foundation Award, Sponsor: Prof. Michael Jolly, Summer 2010
==== University of Science and Technology of China (USTC)
# Excellent Graduate, 2008
# [[http://www.tangfoundation.org/|Cyrus Tang Scholarship]] for Personal Development and Community Service, 2004-2007
# Excellent Student Scholarship, 2005
# Outstanding Freshman Scholarship, 2004

## Talks

#"Initial and boundary value problems for PDEs from geophysical fluid dynamics", at USTC, Hefei, China, May 22, 2014
#"Evolution Semi-linear Hyperbolic Equations in a Bounded Domain", in the SIAM Conference on Analysis of PDE, Dec. 10, 2013 [[Local:Semi-Linear-Hyperbolic-Du-Huang.pdf|Download the talk]]
# "The global attractor for the solutions to the 2D Boussinesq system with variable viscosity and thermal diffusivity", in the Dissipative Systems Seminar, Indiana University, Nov. 19, 2013
# "The well-posedness of linear hyperbolic systems in a domain with corners", in the PDE/Applied Math Seminar, Indiana University, Oct. 28, 2013 and in the SIAM Conference on Analysis of PDE, Dec. 9, 2013
#"Existence of the solutions for linear hyperbolic initial and boundary value problems in a rectangle", in AMS Southeastern Sectional Meeting, Louisville, KY., Oct. 5, 2013
#"Introduction to Stochastic Integration" in Professor Temam's Select Topics in Applied Math Class, Indiana University, Sep 20, 27, 2013
# **Organizing** an informal group/seminar on "[[Stochastic PDEs]]", Indiana University, Spring, 2013
#A **series** of lectures on "The nonlinear 2D subcritical inviscid shallow water equations", in Professor Temam's Evolution Equations Class, Indiana University, Mar. 19, Apr. 4, 11, 18, 2013
#A **series** of lectures on "The linear hyperbolic initial boundary value problem in a domain with corners", in Professor Temam's Evolution Equations Class, IU, Jan. 24th, 31th, Feb. 14th, 21th, Mar. 7th, 2013
# "The nonlinear 2D supercritical inviscid shallow water equations in a rectangle", in the PDE/Applied Math Seminar, IU, Nov. 26th, 2012
# "Introduction to the conservation laws for the Schr$\ddot{\text{o}}$dinger equation", in the Dissipative Systems Seminar, IU, Sep. 11th, 18th, 2012
# "The linearized 2D inviscid shallow water equations in a rectangle: boundary conditions and
well-posedness", in the 9th AIMS conference, Orlando, FL., Jul. 2nd, 2012
# A **series** of lectures on "Introduction to BMO space, and H. Koch, D. Tataru's paper: Well-posedness for the Navier-Stokes equations", in Professor Temam's Continuum Mechanics Class, IU, Mar. 22th, 29th, and Apr. 12th, 2012
# A **series** of talks on "Recent Developments in the Navier-Stokes Problem", in the Dissipative Systems Seminar, IU, May 19th, 26th, and Jun. 9th, 16th, 2010

## Stochastic PDEs

I am organizing an informal study group/seminar on "Stochastic PDEs", and we focus on the textbook [[http://dis.unal.edu.co/~gjhernandezp/sim/lectures/StochasticModels/SDE/Oksendal%20B.%20-%20Stochastic%20Differential%20Equations%20(5th%20ed.).pdf|Stochastic Differential Equations]] by Bernt Øksendal [Øks00].
Here is a list of talks on "Stochastic PDEs":
# **Problems session I** from [Øks00, Chapter 2] \\**Speaker**: Wenru Huo\\ **Date**: May 10, 2013
# **The Dynkin formula** from [Øks00, Chapter 7] \\**Speaker**: Ning Yang\\ **Date**: May 7, 2013
# **Generator of diffusion process** from [Øks00, Chapter 7] \\**Speaker**: Ning Yang\\ **Date**: Apr. 26, 2013
# **Hitting distribution** from [Øks00, Chapter 7], [[http://career.7starsea.com/Stochastic-Note-Yao.pdf|Notes]]\\ **Speaker**: Jinghua Yao \\ **Date**: Apr. 22, 2013
# **Strong markov property of diffusion process** from [Øks00, Chapter 7] \\ **Speaker**: Ping Zhong\\ **Date**: Apr. 18, 2013
# **Markov property of diffusion process** from [Øks00, Chapter 7] \\ **Speaker**: Ping Zhong \\ **Date**: Apr. 11, 2013
# **Stochastic differential equations** from [Øks00, Chapter 5] \\ **Speaker**: Xiaoyan Wang \\ **Date**: Apr. 9, 2013
# **Itō formula and martingale representation theorem** from [Øks00, Chapter 4]\\ **Speaker**: Ning Yang\\ **Date**: Apr. 2, 2013
# **Stochastic differential equations** from [Øks00, Chapter 5] \\ **Speaker**: Xiaoyan Wang\\ **Date**: Mar. 26, 2013
# **Itō integrals** from [Øks00, Chapter 3] \\ **Speaker**: Ning Yang\\ **Date**: Mar. 22, 2013
# **Itō Integrals** from [Øks00, Chapter 3] \\ **Speaker**: Ning Yang\\ **Date**: Mar. 16, 2013
# **Introduction to stochastic PDEs** from [Øks00, Chapter 2] \\ **Speaker**: Ning Yang\\ **Date**: Mar. 14, 2013
=== References
[Øks00] Bernt Øksendal, Stochastic differential equations. An introduction with applications. Fifth edition. Universitext. Springer-Verlag, Berlin, 1998. xx+324 pp. ISBN: 3-540-63720-6.

## Projects

* Numerical Simulation of Partial Differential Equations (PDE) (Burgers, Wave, SWEs), 2011, 2013
** Analyzed the Convergence Rate
** Implemented Fastest Fourier Transform Scheme (FFTW) and Runge-Kutta Method
** Implemented Crank Nicolson Scheme
** Implemented Alternating Direction Implicit (ADI) Scheme
* Program and Time Large Matrices Product in Parallel, 2012
** Utilized the Message Passing Interface (MPI)
** Analyzed the Optimal Rate between Number of Processes and Dimension of Matrices
*Developed Web Based Content Management System, 2011-2012
\\
Built Study-Abroad Platform for Applicants to Share their Information, 2007
** Designed the Back-end using PHP (Zend Framework), Python, and MySQL
** Designed the Front-end using Html, Javascript (jQuery, Prototype), CSS, Ajax
** Used Wiki (Creole) and Mathjax to display mathematical expressions

## Highlights

=== Highlights of Research
**2012-2013.** We proposed a systematic way to tackle the difficult problem - the well-posedness of the hyperbolic initial and boundary value problem in non-smooth domains.
In the articles [[Publications|HT12a, HT13]], we first studied in [[Publications|HT12a]] the 2D linearized shallow water equations with constant coefficients in a rectangle (non-smooth domain) since we want to stay close to our initial motivation of this study in Local Area Models in ocean and atmospheric science, and we then generalized in [[Publications|HT13]] to study the general hyperbolic initial and boundary value problem in a rectangle, where we also give several remarks that the rectangular domain could be replaced by any curvilinear polygonal domains (whose boundaries are made of piecewise $\mathcal C^1$-curves). We also note that in [[Publications|HT13]], we tackle with both constant and variable coefficients and we have to utilize results for Beltrami equations and quasi-conformal mapping from complex analysis.
The key ingredient in [[Publications|HT12a, HT13]] is simultaneous diagonalization by congruence, see [[Publications|HT13, Appendix A]], which allows us to reduce the full system to two elementary modes which we call the hyperbolic and elliptic modes.
Following the article [[Publications|HT12a]], we studied the 2D nonlinear shallow water equations in two special cases, see the articles [[Publications|HPT13, HT12b]], and we also implement finite-volume scheme for the 2D linearized shallow water equations, see [[Publications|BH14]] and also the [[http://career.7starsea.com/pdf/FV-Hyperbolic.pdf|hand-written notes]] for a short introduction to the finite-volume for general 2D hyperbolic system; following the article [[Publications|HT13]], we studied other boundary conditions which only lead to existence of solutions, see [[Publications|Hua13]].
**2014.** We started the project on the Boussinesq system and we currently studied the global attractor of the 2D Boussinesq system with variable viscosity and diffusivity, see [[Publications|Hua14a]], and also studied the global well-posedness of the 2D Euler-Boussinesq system with zero viscosity and positive diffusivity in planar polygonal domains with piecewise $\mathcal C^2$ curves, see [[Publications|Hua14b]].