This post is based on my paper We investigate the local well-posedness theory for the space-homogeneous 4-wave kinetic equation $latex \partial_t f(t,p) = \mathcal{Q}[f](t,p), \mbox{ on } \mathbb{R}_+\times\mathbb{R}^3, (1)$ $latex f(0,p) = f_0(p) \mbox{ on } \mathbb{R}^3. $ The trilinear operator $latex \mathcal{Q}$ is given by $latex \mathcal{Q}[f](p) = \iiint_{\mathbb{R}^{3\times3}}\delta(p+p_1-p_2-p_3)\delta(\omega + \omega_1 -\omega_2 - \omega_3) … Continue reading Chapter 11: Optimal local well-posedness theory for the kinetic wave equation.

# Category: Wave Turbulence Quantum Kinetics

# Chapter 10: On the wave turbulence theory for stratified flows in the ocean

This post is based on my paper . After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of … Continue reading Chapter 10: On the wave turbulence theory for stratified flows in the ocean

# Chapter 9: Wave turbulence theory for capillary water waves

This post is based on my paper Although many studies have been carried on to understand the Hasselmann-Zakharov weak turbulence equation for capillary waves since its derivation in the 60's, the question about the existence and uniqueness of solutions to the equation still remains unanswered, due to the complexity of the equation. Our work provides a … Continue reading Chapter 9: Wave turbulence theory for capillary water waves

# Chapter 8: Quantum hydrodynamic approximations to finite temperature trapped Bose gases

This post is based on my paper . At temperature $latex T$, bosons of mass $latex m$ can be regarded as quantum-mechanical wavepackets which have an extent on the order of a thermal de Broglie wavelength $latex \lambda_{dB} = \left(\frac{2 \pi \hbar^2}{m k_B T}\right)^\frac12$, where $latex k_B$ is the Boltzmann constant. The de Broglie wavelength … Continue reading Chapter 8: Quantum hydrodynamic approximations to finite temperature trapped Bose gases

# Chapter 7: Coupling kinetic and Schrodinger equations – A simplified model.

This post is based on my paper. When a bose gas is cooled below the Bose-Einstein critical temperature, the Bose-Einstein condensate is formed, consisting of a macroscopic number of particles, all in the ground state of the system. As we have already known from previous chapters (Chapters 1 &3), a finite temperature trapped Bose gas … Continue reading Chapter 7: Coupling kinetic and Schrodinger equations – A simplified model.

# Chapter 6: Derivation of quantum kinetic equations

This post is based on my work Let us recall from the previous chapters that the BEC occupies the lowest quantum state, at that point macroscopic quantum phenomena become apparent. Above the BEC are excited atoms, which occupy higher quantum states. The are two types of interactions: excited atoms - excited atoms: $latex C_{22}$ (Uehling-Ulenbeck term), … Continue reading Chapter 6: Derivation of quantum kinetic equations

# Chapter 5: Chemical reaction network and quantum kinetic theories. The key of a successful marriage.

This post is based on my paper.. In this chapter, I will try to answer the following conjecture: ''What is the key for the marriage of chemical reaction network and quantum kinetic theories to be successful?'' The behavior of reactor vessels used in chemical engineering are often based on systems of nonlinear ordinary differential equations (ODEs). … Continue reading Chapter 5: Chemical reaction network and quantum kinetic theories. The key of a successful marriage.