Related papers: A Multiscale Camassa--Holm Equation
In this paper, multi-component generalizations to the Camassa-Holm equation, the modified Camassa-Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schr\"odinger equation, the complex…
This paper introduces the r-Camassa-Holm (r-CH) equation, which describes a geodesic flow on the manifold of diffeomorphisms acting on the real line induced by the W1,r metric. The conserved energy is for the problem is given by the full…
We consider in this paper a challenging problem of simulating fluid flows, in complex multiscale media possessing multi-continuum background. As an effort to handle this obstacle, model reduction is employed. In \cite{rh2}, homogenization…
Wave turbulence formalism for long internal waves in a stratified fluid is developed, based on a natural Hamiltonian description. A kinetic equation appropriate for the description of spectral energy transfer is derived, and its…
In this paper, we consider the quasi-gas-dynamic (QGD) model in a multiscale environment. The model equations can be regarded as a hyperbolic regularization and are derived from kinetic equations. So far, the research on QGD models has been…
In the following we study the qualitative properties of solutions to the geodesic flow induced by a higher order two-component Camassa-Holm system. In particular, criteria to ensure the existence of temporally global solutions are…
The Henon-Heiles Hamiltonian was introduced in 1964 as a mathematical model to describe the chaotic motion of stars in a galaxy. By canonically transforming the classical Hamiltonian to a Birkhoff-Gustavson normalform Delos and Swimm…
We discuss recent results obtained by the authors, regarding the analysis of the magneto-geostrophic equation: a model proposed by Moffat and Loper to study the geodynamo and turbulence in the Earth's fluid core. We conclude this review by…
Misiolek has shown that the Camassa-Holm (CH) equation is a geodesic flow on the Bott-Virasoro group. In this paper it is shown that the Camassa-Holm equation for the case $\kappa =0$ is the geodesic spray of the weak Riemannian metric on…
We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime…
In this article, we show how to embed the so-called CH2 equations into the geodesic flow of the Hdiv metric in 2D, which, itself, can be embedded in the incompressible Euler equation of a non compact Riemannian manifold. The method consists…
An internal energy function of the mass density, the volumetric entropy and their gradients at n-order generates the representation of multi-gradient fluids. Thanks to Hamilton's principle, we obtain a thermodynamical form of the equation…
Markov chain Monte Carlo methods explicitly defined on the manifold of probability distributions have recently been established. These methods are constructed from diffusions across the manifold and the solution of the equations describing…
MFC is an open-source tool for solving multi-component, multi-phase, and bubbly compressible flows. It is capable of efficiently solving a wide range of flows, including droplet atomization, shock-bubble interaction, and gas bubble…
A system of simplified equations is proposed to govern the feedback interactions of large-scale flows present in laminar-turbulent patterns of transitional wall-bounded flows, with small-scale Reynolds stresses generated by the…
Numerical treatment of the problem of two-dimensional viscous fluid flow in and around circular porous inclusions is considered. The mathematical model is described by Navier-Stokes equation in the free flow domain $\Omega_f$ and nonlinear…
The exact 1+3 covariant dynamical fluid equations for a multi-component plasma, together with Maxwell's equations are presented in such a way as to make them suitable for a gauge-invariant analysis of linear density and velocity…
In this paper, we study a multiscale method for simulating a dual-continuum unsaturated flow problem within complex heterogeneous fractured porous media. Mathematically, each of the dual continua is modeled by a multiscale Richards equation…
We describe the physical hypothesis in which an approximate model of water waves is obtained. For an irrotational unidirectional shallow water flow, we derive the Camassa-Holm equation by a variational approach in the Lagrangian formalism.
Generalised Hydrodynamics (GHD) describes the large-scale inhomogeneous dynamics of integrable (or close to integrable) systems in one dimension of space, based on a central equation for the fluid density or quasi-particle density: the GHD…