In weak turbulence theory, the Kolmogorov-Zakharov spectra is a class of time-independent solutions to the kinetic wave equations. In this paper, we construct a new class of time-dependent solutions to those kinetic equations. These solutions exhibit the interesting property that the energy is cascaded from small wavenumbers to large wavenumbers. We can prove that starting … Continue reading Chapter 12: On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov solutions

# Chapter 11: Optimal local well-posedness theory for the kinetic wave equation.

This post is based on my joint paper with Pierre Germain and Alexandru D. Ionescu. 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 … Continue reading Chapter 11: Optimal local well-posedness theory for the kinetic wave equation.

# Extra Chapter 2: iHDG – Iterative Hybridized Discontinuous Galerkin method (Part 2)

This post is based on my joint work with Tan Bui-Thanh and Sriramkrishnan Muralikrishnan. We continue our development of the previous work on iHDG method iHDG: An Iterative HDG Framework for Partial Differential Equations. SIAM Journal on Scientific Computing. Video 6: Our simulations for a hyperbolic system using our new iHDG We show that … Continue reading Extra Chapter 2: iHDG – Iterative Hybridized Discontinuous Galerkin method (Part 2)

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

This post is based on my joint paper with Leslie M. Smith and Irene M. Gamba. 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 … 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 joint paper with Toan Nguyen. 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 … 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 joint paper with Shi Jin. 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 … Continue reading Chapter 8: Quantum hydrodynamic approximations to finite temperature trapped Bose gases

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

This post is based on my joint paper with Avy Soffer. 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 … Continue reading Chapter 7: Coupling kinetic and Schrodinger equations – A simplified model by Soffer and Tran.