Chapter 13: A reaction network approach to the theory of acoustic wave turbulence

This post is based on my paper  Let us consider the acoustic wave turbulence kinetic equation $latex    \partial_tf \ = \ Q[f] $ where $latex   Q[f]$ is the collision term, describing pure resonant three-wave interactions. The equation is a three-wave kinetic one, in which the collision operator is of the form $latex   … Continue reading Chapter 13: A reaction network approach to the theory of acoustic wave turbulence

Chapter 12: On the energy cascade of acoustic wave turbulence: Beyond Kolmogorov-Zakharov solutions

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 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.

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

This post is based on my work 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 the new approach has several advantages, for … 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 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