# Chapter 0. Introduction

Please have a look at the About and Table of Contents also.

There have been very few rigorous mathematical results on Boltzmann equations arising in wave turbulence and quantum kinetic theories, despite their central roles in several aspects of sciences and engineering, due to their complexities in comparison with the classical Boltzmann equations. One of my research interests is to understand the rigorous mathematical properties of these Boltzmann equations.

1. Weak turbulence:

See also Terrence Tao’s blog or this AIM workshop .

Wave turbulence is a branch of science studying the out-of-equilibrium statistical mechanics of random nonlinear waves of all kinds and scales. Despite the fact that wave fields in nature are enormous diverse; to describe the processes of random wave interactions, there is a common mathematical concept. The development of this mathematical concept and the applications is an established integral part of nonlinear science. Wave turbulence systems are important in a vast range of physical examples (cf. Zakharov, L’vov, Falkovich, Kolmogorov spectra of turbulence I: Wave turbulence, Springer 2012; Nazarenko, Wave turbulence, Springer 2011; Newell, Rumpf, World Scientific Series on Nonlinear Science Series, 2013.):

• water surface gravity and capillary waves;
•  inertial waves due to rotation and internal waves on density stratifications, which are important in planetary atmospheres and oceans;
• Alfvén Wave Turbulence in solar wind;
• planetary Rossby waves, which are important for the weather and climate evolutions;
• waves in Bose–Einstein condensates and in Nonlinear Optics;
• waves in plasmas of fusion devices;
• phonon interactions in anharmonic crystal lattices, solid state physics;
• waves on vibrating elastic plates, and many others.

Rigorous derivations and understanding properties of weak turbulence kinetic equations is a subject with a lot of interest for the recent years.

There are two types of wave turbulence kinetic equations.

The 4-wave turbulence kinetic equation: $\partial_t f(t,p) = C_{4W}[f](t,p), f(0,k)=f_0(k),$ $C_{4W}[f](t,p)=\iiint_{\mathbb{R}^{3\times3}}|V_{p,p_1,p_2,p_3}|^2\delta(p+p_1-p_2-p_3)\delta(\omega(p)+\omega(p_1)-\omega(p_2)-\omega(p_3))[f_2f_3(f_1+f) -ff_1(f_2+f_3)dp_1dp_2dp_3.$

The most striking element of the theory of wave turbulence is the that the solutions of the 4- wave kinetic equation can describe the large time dynamic of high Sobolev norm $H^s$ of the solution $v$ of the cubic nonlinear Schr\”odinger equation on the torus $-i\partial_t v + \frac{1}{2\pi}\Delta v = |v|^2v, v(t=0)= \epsilon v_0, x\in\mathbb{T}^d.$

The 3-wave turbulence kinetic equation: $\partial_tf(t,p) =C_{3W}[f](t,p), f(0,p)=f_0(p),$ $C_{3W}[f](p) = \iint_{\mathbb{R}^{2d}} \Big[ R_{p,p_1,p_2}[f] - R_{p_1,p,p_2}[f] - R_{p_2,p,p_1}[f] \Big] dp_1dp_2$ $R_{p,p_1,p_2} [f]:= |V_{p,p_1,p_2}|^2\delta(p-p_1-p_2)\delta(\omega(p) -\omega({p_1})-\omega({p_2}))(f_1f_2-ff_1-ff_2),$

where $|V_{p,p_1,p_2}|^2$is the given transition probability, $\omega(p)$ is the dispersion relation and we have used the short-hand notation $f = f(t,p)$ and $f_j = f(t,p_j)$. Similar to the 4-wave kinetic equation, the 3-wave kinetic equation can be used to describe the long time dynamics of high Sobolev norms $H^s$ of solutions to the water wave equation (Zakharov’1965) $\Delta \Phi(t,x,y,z) \ = \ 0, \mbox{ for } z<\zeta(t,x,y), (t,x,y,z)\in\mathbb{R}_+\times\mathbb{R}^3, \zeta_t(t,x,y)-\Phi_z(t,x,y,z) \ = \ -\zeta_x(t,x,y)\Phi_x(t,x,y,z) -\zeta_y(t,x,y)\Phi_y(t,x,y,z)\Big{|}_{z=\zeta}, \Phi_t(t,x,y,z) -\alpha (\zeta_{xx}(t,x,y)+\zeta_{yy}(t,x,y)) \ = \ \frac{|\nabla\Phi(t,x,y,z)|^2}{2}\Big{|}_{z=\zeta}, \Phi(t,x,y,z)\Big{|}_{z=-\infty}\ = \ 0,$

or several other equations (Lvov, Tabak’2001, Majda, Introduction to PDEs and waves for the atmosphere and ocean, 2003).

Our recent contributions include:

i) Wave turbulence for capillary water wavesChapter 9

ii) Wave turbulence for internal waves in the ocean: Chapter 10

iii) Optimal local well-posedness theory for the 4-wave equation: Chapter 11

2. Quantum Boltzmann:

After the production of the first Bose-Einstein Condensates (BECs), that led to the 2001 Nobel Prize of Physics, there has been an explosion of physics research on the kinetic theory associated to BECs and their thermal clouds. Quantum kinetic theory is both a genuine kinetic theory and a genuine quantum theory. In which, the kinetic part arises from the decorrelation between different momentum bands. Cover of Scientific American

There are also two types of quantum Boltzmann collision operators, which are very similar to the wave turbulence collision operators.

The 4-wave quantum Boltzmann collision operators $C_{22}[f](t,p) = \iiint_{\mathbb{R}^{3\times3}}|\mathcal{M}_{p,p_1,p_2,p_3}|^2\delta(p+p_1-p_2-p_3)\delta(\omega(p)+\omega(p_1)-\omega(p_2)-\omega(p_3))[f_2f_3(f_1+f+1) -ff_1(f_2+f_3+1)dp_1dp_2dp_3.$

The 3-wave quantum Boltzmann collision operators $C_{12}[f](p) = \iint_{\mathbb{R}^{2d}} \Big[ \mathcal{R}_{p,p_1,p_2}[f] - \mathcal{R}_{p_1,p,p_2}[f] - \mathcal{R}_{p_2,p,p_1}[f] \Big] dp_1dp_2$ $\mathcal{R}_{p,p_1,p_2} [f]:= |\mathcal{M}_{p,p_1,p_2}|^2\delta(p-p_1-p_2)\delta(\omega(p) -\omega({p_1})-\omega({p_2}))(f_1f_2-ff_1-ff_2-f).$

This blog is devoted to the description of our recent research results on quantum kinetic theory, using several different points of view:

i) Kinetic equations approach:

– The Cauchy problem: Chapter 1

– Hydrodynamics limits: Chapter 8

– Convergence to equilibrium: Chapter 4

– Positivity: Chapter 2

ii) Dispersive equations approach:

– Scattering theory: Chapter 3, Chapter 7,

iii) Dynamical systems approach:

– The analog between the global attractor conjecture (GAC) in chemical reaction network and the convergence to equilibrium of quantum kinetic equations: Chapter 5

iv) Quantum field theory approach:

– The Peletmiskii-Yatchenko derivation: Chapter 6

3. Some aspects of my works on Scientific Computing-Computational Sciences and Uncertainty Quantification-Sensitivity Analysis:

i) Iterative Hybridized Discontinous Galerkind Algorithms: Extra Chapter 1 (simulations included), and Extra Chapter 2 Our simulations for solutions of a hyperbolic system using our new iHDG method

References for the pictures:

 Sriramkrishnan Muralikrishnan, Minh-Binh Tran, Tan Bui-Thanh. An Iterative HDG Framework for Partial Differential Equations. SIAM Journal on Scientific Computing.

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